- Email: [email protected]
The effect of human errors on an integrated stochastic supply chain model with setup cost reduction and backorder price discount
Journal Pre-proof The effect of human errors on an integrated stochastic supply chain model with setup cost reduction and backorder price discount Sunil Tiwari, Nima Kazemi, Nikunja Mohan Modak, Leopoldo Eduardo Cárdenas-Barrón, Sumon Sarkar
PII: DOI: Reference:
S0925-5273(20)30044-X https://doi.org/10.1016/j.ijpe.2020.107643 PROECO 107643
To appear in:
International Journal of Production Economics
Received date : 24 April 2019 Revised date : 13 January 2020 Accepted date : 14 January 2020 Please cite this article as: S. Tiwari, N. Kazemi, N.M. Modak et al., The effect of human errors on an integrated stochastic supply chain model with setup cost reduction and backorder price discount. International Journal of Production Economics (2020), doi: https://doi.org/10.1016/j.ijpe.2020.107643. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier B.V.
*Title page including author details Click here to download Title page including author details: Title Page.pdf
Journal Pre-proof The effect of human errors on an integrated stochastic supply chain model with setup cost reduction and backorder price discount Sunil Tiwari a Department of Industrial Systems Engineering and Management, National University of Singapore, 1 Engineering Drive 2, 117576, Singapore
Nima Kazemi b
Centre for Transportation and Logistics, Massachusetts Institute of Technology, Cambridge, MA
pro
b
of
Email: [email protected]; [email protected]
02142, USA
Email: [email protected] Nikunja Mohan Modak c
Palpara Vidyamandir, Chakdaha, West Bengal, India
re-
c
Email: [email protected] Leopoldo Eduardo Cárdenas-Barrón d* Department of Industrial and Systems Engineering, School of Engineering and Sciences,
urn al P
d
Tecnológico de Monterrey, E. Garza Sada 2501 Sur, C.P. 64849, Monterrey, Nuevo León, México Email: [email protected] Sumon Sarkar e
e
Department of Mathematics, Jadavpur University, Kolkata - 700032, India Email: [email protected]
Jo
a
Journal Pre-proof *Manuscript Click here to view linked References
The effect of human errors on an integrated stochastic supply chain model with setup cost reduction and backorder price discount Sunil Tiwari1 , Nima Kazemi1 , Nikunja Mohan Modak1 , Leopoldo Eduardo C´ ardenas-Barr´ on1 , Sumon Sarkar1 a Department
pro
of
of Industrial Systems Engineering and Management, National University of Singapore, 1 Engineering Drive 2, 117576, Singapore b Centre for Transportation and Logistics, Massachusetts Institute of Technology, Cambridge, MA 02142, USA c Palpara Vidyamandir, Chakdaha, West Bengal, India d Department of Industrial and Systems Engineering, School of Engineering and Sciences, Tecnol´ ogico de Monterrey, E. Garza Sada 2501 Sur, C.P. 64849, Monterrey, Nuevo Le´ on, M´ exico e Department of Mathematics, Jadavpur University, Kolkata - 700032, India
Abstract
This study develops an integrated vendor-buyer inventory model for a supply chain where quality issues and
re-
human error affect its coordination. Each lot shipped to the buyer contains defective items, with the rate, which randomly changes from a lot to the other. The buyer inspects every shipped lot to segregate defective items. However, the inspection process at the buyer’s end goes wrong in the classification of defective and nondefective items. On the other hand, the buyer may run out of inventory, but in order to avoid lost sales, he/she
urn al P
offers a price discount on the backlogged items to his/her customers. Due to the vendor-buyer relationship, the buyer invests in reducing the setup cost of the vendor. Supply chain’s lead-time is considered variable, and two models are developed based on the probability distribution of the lead-time demand. In the first model, it is assumed that lead-time demand follows a normal distribution, while in the latter one, it does not follow any certain distribution. Two models are developed to determine the joint optimal decision variables that minimize the total cost of the supply chain. Two iterative algorithms are developed to obtain the optimal solution for both models. A set of numerical analysis and sensitivity analysis are conducted to gain insights. Keywords: Defective items, setup cost reduction, backorder price discount, inspection error, variable lead-time, distribution-free approach.
1. Introduction
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Supply chain management (SCM) encompasses different entities that make various strategic, tactical, and operational decisions at different stages, aiming at creating a competitive advantage for their business (Lambert and Enz, 2017). SCM is thus concerned with the coordination of material, information, and money along with a network of companies whose objective is to achieve better performance (Marchi et al., 2016). The growing structure of supply chains due to the involvement of various entities makes them more dynamic, yet so vulnerable ∗ Corresponding
author
Preprint submitted to Elsevier
January 13, 2020
Journal Pre-proof
to external risk and vulnerability factors (Kurniawan et al., 2017; Scholten and Fynes, 2017, Behzadi et al., 2018; Dominguez et al., 2018). This calls for effective SCM coordination and involvement of all parties in decision making to ensure that the whole supply chain (SC) works toward a joint objective (Glock, 2012a). One of the favorable research streams in SC is a coordination of inventory replenishment decisions among SC
of
members, which is the topic that has shown rapid growth over the recent years (Arshinder et al., 2011, Heydari et al., 2017). In inventory management literature, this type of model is recognized as the joint economic lot size (JELS). The so-called JELS models simulate today’s business practices (e.g., automotive, apparel, grocery)
pro
where there exists a long relationship between buyers and suppliers. The underlying assumption in the JELS models is that the buyer and supplier tend to make a collaborative decision on their inventory replenishment (Glock, 2012a). As joint inventory replenishment is a prevalent practice in SCs, there is a growing necessity to develop the decision support models that could help to gain insight into deriving the optimum policies for
re-
such cases.
Product quality is of high strategic importance to the buyer-supplier relationship and is one of the significant factors that both sides consider in their joint inventory replenishment. To ensure a high quality of buying products, the buying companies mostly carry out an inspection to ensure the high quality of incoming raw
urn al P
materials or products (Khan et al., 2017a). Even though developing new technologies could have led to the automation of quality inspection, many companies are still relying on human inspectors. When a human operator is involved in SC processes, his/her characteristic will affect the performance of the SC. Thus, ignoring human characteristics in SC operations may lead to making decisions that are sub-optimal and have financial burdens (Glock et al., 2017).
Lead-time is another concern for coordinating inventories between a buyer and a supplier. Setting up lead-time strategies may have several direct and indirect impacts on an SC, for instance, on inventory costs or customer service, among others (Deros et al., 2008). A prominent example of lead-time reduction has observed in a Japanese car manufacturing company where just-in-time production has led to a lead-time reduction, which consequently has increased efficiency and competitiveness of the company. The lead-time reduction is particularly significant when the SC encounters an uncertain demand, as high lead-time increases the risk of stock-out (De Treville et al., 2004). In addition to lead-time reduction, this research is also motivated by the
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
practice of the buyers who invest money to reduce the setup cost of the supplier company. Reducing setup costs of manufacturing companies could improve their operational performance that would positively influence the performance of the entire SC. The last issue that this paper is taking into account is the stock-out situation. Customer behavior is appeared to be different in stock-out cases, depending on the type of industry and the nature of the business (Bijvank and Vis, 2011). While some prefer to source their product from another supplier, the others choose
2
Journal Pre-proof
to wait until the order is ready for shipment (Krafft and Mantrala, 2006). However, in some certain industries, such as fashionable goods, hi-fi equipment, cosmetics, and apparel, customers are ready to wait for backorders up to a certain period (Krafft and Mantrala, 2006). In such cases, buyers try to convince their customers by offering a price discount on the backordered items.
of
Our Contribution:
The purpose of this paper is to merge four research streams, namely the JELS model, human error in inspection, variable lead-time, set up cost reduction into a single research frame. Thus, this paper presents
pro
a vendor-buyer integrated inventory model with quality issues and variable lead-time in which the buyer is willing to invest in the vendor’s manufacturing facility to improve the quality. Human errors in the inspection are considered at the buyer side to represent a real industrial setting where human error is unavoidable. By considering these four operational elements, this paper investigates how these elements affect buyer-supplier
re-
on optimal inventory replenishment. As the literature review in the next section shows, this is the first paper in the literature that addresses these operational factors concurrently in a JELS model. Besides that, this paper investigates two states for lead-time demand. In the first case, we assume that demand follows a normal distribution. In the second one, we assume that the distribution of lead-time demand is known. Considering
urn al P
the assumption and settings of the model, we try to answer the following questions: 1. What would be the optimal ordering policy for an industry if demand during replenishment lead-time is stochastic?
2. What would be the appropriate price discount strategy, if adopted by the buyer during the shortage period to secure customer demand?
3. What would be the exact investment amount to reduce setup costs? 4. What is the impact of misclassification errors on decision-making? 5. Which methodology is the best to obtain the global minimum solution and what steps a manager will adopt when lead-time demand distribution is unknown?
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
The remaining parts of the paper are organized as follows. Section 2 discusses the relevant literature and the research gap that this study tries to cover. The next section provides the notations, assumptions, and the details of the mathematical model. Section 4 presents the solution algorithms developed to optimize the mathematical model. Section 5 illustrates the numerical study, and finally, Section 6 provides the conclusions and future research suggestions.
3
Journal Pre-proof
2. Literature Review The literature review section investigates the four research streams which (i.e., the JELS model, inspection errors in quality, setup cost reduction and lead-time variation) that build the main structure of the paper at hand. We will initially discuss the four different research streams individually. At the end of this section, we
of
will discuss the research streams together and synthesize them to specify the research gap and position this study in the literature.
pro
2.1. The JELS model
The JELS models are one of the mathematical models in the area of SC that illustrate how inventory replenishment decision impacts the profit of the entire SC. The literature on the JELS models has advanced over the years since the seminal paper of Goyal (1977), who developed a model with the simple assumption
re-
that the production quantity at the supplier and the order quantity at the buyer are equal. Subsequently, many researchers have tried to relax the simplistic assumption used Goyal (1977) by working on the structured delivery between supplier and buyer (Eben-Chaime, 2004, David and Eben-Chaime, 2008, Ertogral et al., 2007). The model of Goyal (1977) was also extended to the SCs that include more than one company in each
urn al P
stage, or multiple companies in every stage (Zavanella and Zanoni, 2009, Mokhtari and Rezvan, 2017, Chen and Sarker, 2010). Recent advancement of the JELS model include permissible delay in payments (Aljazzar et al., 2016), vendor-managed inventory (De Giovanni et al., 2019, Sainathan and Groenevelt, 2019), imperfect quality (Salameh and Jaber 2000, Wee et al., 2007, Wee and Widyadana 2013), environmental issues (Tiwari et al., 2018), and sustainability (Jawad et al., 2018). 2.2. Inspection errors in quality
Human error is regarded as one of the factors influencing the system’s performance in logistics and inventory systems (Grosse et al., 2015, Kazemi et al., 2015). Even though companies tend to automate their logistics processes, many of them, particularly small-sizes, still prefer to operate manually (Grosse et al., 2015). Thanks to the factors such as work environment, task characteristics, and worker’s capability, a manual task might sometimes be performed erroneously. When it comes to the quality inspection process, it may not be error-free
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
as it is predominantly a manual task. An operator engaged in inspecting the shipped or produced batches might sort perfect quality product as a defective item and vice versa. The first type of error is known as Type I, while the latter is called Type II. (i) Type I error - rejecting a non-defective item as defective and (ii) Type II error - accepting a defective item as non-defective. These misclassification errors have an obvious effect on the inventory holding cost, warranty cost and more importantly, on the customer’s goodwill. Hence, ignorance of these inspection errors may result in distortion of supply chain performance (Khan et al., 2012; Hsu and Hsu 2013a; Hsu and Hsu 2013b; Jauhari et al., 2016; Pal and Mahapatra 2017; Khan et al., 2017b). The financial
4
Journal Pre-proof
outcomes of this human characteristic are a vital facet of modelling quality in SC coordination (Khan et al., 2012). A quite extensive number of mathematical models have been developed to address this operational factor, though only a few studied the issue in the JELS settings. For instance, Khan et al. (2014) incorporated learning in production and inspection errors in a two-stage SC, which follows an equal lot size policy. Khan
of
et al. (2017a) addressed the issue of defective items and inspection error in a supplier-buyer SC and assumed that the buyer has the option of either repairing the defective items via a local manufacturer or replacing them by purchasing new items from a local seller. Alfares and Attia (2017) developed a more complex model that
pro
coordinates inventories between a single supplier and multiple buyers. The authors formulate the quality issue of the production system at the supplier and inspections errors at the buyers. In addition to the quality issue, their model assumes that the supplier agrees with the buyers to follow a vendor-managed inventory (VMI)consignment stock (CS) policy. One of the few studies that employed statistical characteristics of uncertainty
re-
is the one of Sarkar et al. (2015), who developed a three-stage model (supplier-manufacturer-retailer) assuming that demand depends on price and stock level. 2.3. Setup cost reduction
urn al P
As discussed before, implementing a cooperative SC inventory policy may result in a higher SC efficiency. If the parties involved determine to collaborate in a long run, it may be more lucrative to invest in their collaboration to increase the turnover (Glock, 2012a). One of the standard practices seen in the industry is the investment of buyers in the supplier’s manufacturing process, aiming at decreasing the costs associated with the production process at the supplier (Panwar et al., 2015). The literature started to include this SC practice in inventory models in the early 80s. However, it appeared for the first time in the JELS literature in early 2000. To model the effect of the investment, studies mostly developed investment functions and included them in the JELS model. As an example, Sarkar et al. (2015) studied an inventory-production model with a continuousreview policy where setup cost assumed to reduce because of the improvement in the production process. In a similar study, Sarkar et al. (2014) studied the effect of setup cost reduction and variable backorder costs on an inventory-production inventory model. While the literature mostly has studied setup cost reduction in singlestage models, only a few have studied the problem in the context of the JELS models. Notable contributions
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
are the ones of Shu and Zhou (2014) and Sarkar et al. (2015). 2.4. Variable lead-time
One of the first studies that addressed variable lead-time was due to Liao and Shyu (1991). They developed an inventory model assuming that lead-time can be decomposed into several linear components with different piece-wise linear continuous crashing cost, where each component of the crashing costs may be reduced by considering normally distributed lead-time demand. Later, Ben-Daya and Raouf (1994) extended Liao and
5
Journal Pre-proof
Shyu’s (1991) in two steps. In the first one, they added order quantity as a decision variable to Liao and Shyu’s model, while in the latter one, they defined an exponential relationship between lead-time and total crashing cost. Ouyang et al. (1996) developed two models based on normally distributed lead-time and distributionfree lead-time (with known mean and standard deviation) and calculated the expected value of additional
of
information by using the concept of Porteus (1986). They proved for distribution free lead-time case; the industry manager has to pay less than one percent of the total cost. Moon and Choi (1988) extended their model by considering the reorder point as a decision variable. Ben-Daya and Hariga (2003) extended the same
pro
concept with lead-time crashing costs depending on lot size.
Another research in this area is one of Pan and Yang (2002), who assumed that the vendor might reduce lead-time upon the request of the buyer. Hoque and Goyal (2006) and Hoque (2007) considered the simultaneous effect of batch sizes and lead-time reduction on inventory replenishment decisions of a supplier-buyer
re-
relationship. Pan and Lo (2008) discussed a supply chain model with lead-time crashing costs under learning effects. Glock (2012b) assumed that lead-time could be reduced by either growing the production rate or reducing setup and transportation time. Finally, Mou et al. (2017) investigated the concurrent impact of investment in quality improvement and reduction in lead-times on a supplier-buyer joint lot-size decision. Even though for
urn al P
simplicity of problems lead-time in the relevant literature is assumed to follow a known probability distribution when it is uncertain, it might be quite difficult in some circumferences to specify the exact distribution. An alternative way preferred by researchers and practitioners is to apply the distribution-free approach (DFA), a method that helps to calculate the total cost of SC without having data available on the lead-time demand distribution (Sarkar et al., 2015, Tiwari et al., 2018). One of the first researchers who studied this approach was Moon and Gallego (1994) who introduced the DFA to model a continuous-review inventory model with a service level constraint. Later, Chu et al. (2005) applied the DFA in a continuous-review inventory model with a variable lead-time and backorders, where they assumed the reorder point is stochastic. Modak and Kelle (2019) employed DFA to formulate a dual-channel SC where the customer has the option to buy from either online or from stores.
2.5. Synthesis of the research streams
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Table 1 looks closely into the details of the studies that address defective items in the JELS models context and compare them with the contribution of this study. The columns in Table 1 illustrate the different characteristics of the models in the literature. As the table illustrates, the closet study to ours is the one of Sarkar and Giri (2018). However, Sarkar and Giri’s (2018) model has a few distinct differences with the study at hand. First of all, their model assumes that lead-time demand only follows a normal distribution, while in this study, in addition to the normal distribution, we inveterate the case where lead-time may not have a known distribution. Second, their model does not take inspection error into account, which routinely happens in many
6
Journal Pre-proof
industries. It is clear that the decision support models that aim to add a contribution to this research area should be as realistic as possible. This paper takes a step in making the model in the literature more realistic by integrating the research streams given in Table 1 to investigate their concurrent effect on the inventory replenishment policy of a two-level SC.
Setup cost Variable reduction lead-time
√ √ √
√
√ √ √ √ √ √ √ √ √ √ √ √
√
√ √
√
√
pro
√
urn al P
Dey (2019) Mandal & Giri (2019) Br¨auer & Buscher (2019) Gutgutia & Jha (2018) Sarkar & Giri (2018) Das et al. (2017) Giri et al. (2017) Khan et al. (2017b) Dey (2017) Khan et al. (2014) Lin & Lin (2014) Hsu and Hsu (2013c) Ouyang et al. (2013) Lin et al. (2012) Su (2012) Khan et al. (2012) Pal et al. (2012) Tiwari et al. (2018) This paper
Defective Error in quality items inspection
√ √
re-
Author (s)
of
Table 1: Comparison of the present model with some related works in the literature
√ √ √
√ √
Stochastic lead-time demand
Backorders
√ √ √
√ √
√
√ √
√
√ √ √ √
√ √
√
3. Model description and formulation
In this section, we present the assumptions and notations used in formulating the model. 3.1. Assumptions
We assume the following assumptions to develop our model: 1. The SC includes single vendor and single buyer who make decision about a single product.
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
√ 2. The reorder point (r) is defined as r = DL + kσ L, where DL = the expected demand during lead-time, √ and kσ L is safety stock (see for instance Ouyang et al. (1996), Ouyang and Wu (1997), and Ho(2009)). 3. The inventory is continuously reviewed and whenever it drops to the reorder point r, the buyer places an order of size Q, which is delivered in m equal-size shipments. 4. Each lot contains a percentage of defective items, therefore each lot is screened immediately at a fixed rate s, which is greater than the demand rate D.
7
Journal Pre-proof
5. The buyer is interested in investing an initial amount of money on the vendor’s manufacturing to reduce setup cost. The capital investment that the buyer invests, Is (S), is a logarithmic function of the setup
IS (S) = B ln
where B =
1 θ
S0 S
0 < S ≤ S0
,
of
cost S, i.e.,
with θ being the percentage decrease in S per dollar increase in Is (S), and S0 is Vendor’s
pro
original setup cost. It may be noted that the investment function is convex for all S with the restriction (See for instance Porteus (1985), Sarkar et al. (2015), Sarkar & Giri (2018a, 2018b)). 6. The lead-time L consists of n mutually independent components. The i-th component has a minimum duration ui in days, normal duration vi in days, and a crashing cost per day ci . Further, we rearrange
re-
ci as c1 ≤ c2 ≤ c3 ........ ≤ cn . Then, it is clear that lead-time reduction should first occur on component 1 (because it has the minimum unit crashing cost), and then component 2, and so on (See for instance Porteus (1985), Sarkar et al. (2015), Sarkar & Giri (2018a, 2018b)). Pn
j=1
vj denote the maximum duration of lead-time and Li as the length of lead-time with
urn al P
7. Let L0 =
components 1, 2, ...., i, crashed to their minimum duration, then Li can be expressed as (see for instance Ouyang et al. (1996), Annadurai and Uthayakumar (2010), and Priyam and Uthayakumar (2014))
Li =
n X j=1
vj −
i X j=1
(vj − uj )
where i = 1, 2, 3, ..., n and the lead-time crashing cost, C(L), per cycle is given by
C(L) = ci (Li−1 − L) +
i−1 X j=1
cj (vj − uj )
8. The buyer offers a discount for backordered items to his customer and the backorder ratio β is considered as a variable, which is proportional to the backorder price discount (πx ) offered by the buyer. Thus,
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
β = β0 πx /π0 , 0 ≤ β0 < 1 and 0 ≤ πx ≤ π0 . The buyer’s makes decisions in order to obtain profits. If the backorder price discount πx is greater than the marginal profit π0 , then the buyer may decide not to offer the discount as it is not a profitable option (see for instance Pan et al. (2004), and Lin (2009)).
8
Journal Pre-proof
3.2. Notations We use the following notations to develop the model: Decision variables buyer’s order quantity (units)
r
buyer’s reorder point (unit time)
L
length of the lead-time (unit time)
m
number of lots delivered from the seller to the buyer in one production cycle, a positive integer
S
vendor’s setup cost ($/order)
πx
backorder price discount ($/unit)
k
safety factor
pro
Parameters
buyer’s constant demand (unit/unit time)
A
buyer’s ordering cost ($/order)
hv
vendor’s holding cost ($/unit/unit time)
s
screening rate
Cs
screening cost ($/unit)
µ µ0 P m1
urn al P
θ
re-
D
C0
of
Q
scraping cost per unit ($/unit) annual fractional cost of the capital investment($/unit time) cost of falsely accepting a defective product ($/unit) cost of falsely rejecting a non-defective product ($/unit) production rate (units/unit time) a random variable showing probability of a Type I error (classifying a non-defective product as defective)
m2
a random variable showing probability of a Type II error (classifying a defective product as non-defective)
y ye C(L) β β0 σ
percentage of defective items that were supplied by the vendor to the buyer
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
the percentage of defective products observed by the buyer through screening
lead-time crashing cost function backorder ratio upper bound of the backorder ratio standard deviation of the lead-time demand
9
Journal Pre-proof
i-th component of lead-time with ui as minimum duration (unit time), i = 1, 2, ...., n
vi
i-th component of lead-time with vi as normal duration (unit time), i = 1, 2, ...., n
ci
i-th component of lead-time with ci as crashing cost per unit time ($/unit time), i = 1, 2, 3, ...n
X
lead-time demand which has a distribution function, f with finite mean DL √ and standard deviation σ L
E(x)
mathematical expectation of x
x+
max {x, 0}
E(X − r)+
the expected shortage quantity at the end of the cycle (units)
ET Cb ()
buyer’s expected total cost per unit time($/unit time)
ET Cv ()
vendor’s expected total cost per unit time ($/unit time)
JET C()
the joint expected total cost per unit time ($/unit time)
3.3. Model formulation
re-
pro
of
ui
This study considers a case in which a buyer and a vendor try to cooperatively determine the inventory lot-size to minimize their SC total cost. The buyer places an order of Q items to the vendor. Due to a high
urn al P
setup cost, the vendor produces mQ items in one setup with a constant production rate of P and delivers it in multiple shipments. Each delivered lot by the vendor contains y percent of defective items. Thus, on the arrival of each shipment, the buyer inspects each received lot at a constant screening rate of s to remove defective items. However, the inspection is not error-free and two types (Type I & Type II) of inspection errors may occur during the inspection at the buyer. The type I error occurs by classifying non-defective products as defective products while the type II errors by classifying defective products as non-defective products. Therefore, the fraction of defective items observed by the inspectors would be ye = (1 − y)m1 + (1 − m2 )y, where (1 − y)m1 and (1 − m2 )y are the percentages attributed to Type I and Type II inspection errors, respectively. Due to the random nature of the defective rate and inspection process, it is assumed that y, m1 , m2 are random variables and independent of each other. Hence, the expected value of observed defective items at the buyer is E[ye ] = (1 − E[y])E[m1 ] + E[y](1 − E[m2 ]). To simplify the notation, E[ye ] is abbreviated as Ye in this paper.
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Therefore, the buyer’s expected ordering cycle length is
Q(1−Ye ) . D
3.3.1. Average cost form buyer’s perspective The buyer picks out the continuous review policy to review his/her inventory and places an order of size Q whenever the non-defective inventory falls to the reorder point r. The buyer receives the order quantity after a variable lead-time L. As the lead-time demand is stochastic, shortages might take place when the on-hand stock is not sufficient to fulfill the demand. Hence, our model accounts for shortage cost in the total cost of SC. Shortage occurs when X > r,. Then, the buyer’s expected shortage quantity at the end of the cycle is E(X −r)+ .
10
Journal Pre-proof
Hence, the expected backorder quantity is βE(X − r)+ . Consequently, the expected loss sales per ordering cycle is (1 − β)E(X − r)+ and the expected stock-out cost per ordering cycle is [πx β + π0 (1 − β)] E(X − r)+ . In addition, at the beginning of each cycle the buyer’s expected net inventory is the safety stock r −DL plus the previous cycle lost sales (1 − β)E(X − r)+ . As a result, the expected net inventory of non-defective items per
of
unit time immediately after the successive order is Q(1−Ye )+r−DL+(1−β)E(X −r)+ . Therefore, the average inventory of non-defective items over the cycle can be calculated from
Q 2 (1 − Ye ) + r − DL + (1 − β)E(X
− r)+ .
pro
The buyer’s total holding cost of non-defective and defective items per unit time can now be calculated as h i DQYe e) hb Q(1−Y + s(1−Y + r − DL + (1 − β)E(X − r)+ . 2 e)
Due to inspection error, the buyer incurs misclassification costs, which are, the cost of falsely rejected
non-defective items and falsely accepted defective items. Another fact need to be considered here is that the cost of false acceptance could sometimes be very high and even fatal, particularly in some intricate industries
greater than that of µ0 i.e., µ >> µ0 .
re-
such as automotive, aerospace, and nuclear. To accommodate this fact, the value of µ is chosen to be much
The misclassification cost for Type-I error (i.e., the cost of false rejected non-defective items) is Cr = µ0 Q(1 − E[y])E[m1 ]
urn al P
The misclassification cost for Type-II error (i.e., the cost of false accepted defective items) is Ca = µQE[y]E[m2 ] Hence, the total misclassification cost per unit time is M C =
D 1−Ye {µE[y]E[m2 ]
+ µ0 (1 − E[y])E[m1 ]}
Further, the correctly identified defective items, i.e, QE(y)(1 − E[m2 ]) incur scraping cost for the buyer. Hence, the buyer’s scrapping cost per unit time is SC =
Co D 1−Ye {E[y](1
− E[m2 ])}.
According to the discussion above, the buyer’s expected total cost per unit time is
ET Cb (Q, r, β, L)
=
Ordering cost + holding cost + backorder cost + misclaficassion cost + inspection cost
+
lead-time crashing cost + scraping cost AD Q(1 − Ye ) DQYe + = + hb + + r − DL + (1 − β) E(X − r) Q(1 − Ye ) 2 s(1 − Ye ) D µDE[y]E[m2 ] µ0 D(1 − E[y])E[m1 ] + [πx β + π0 (1 − β)] E(X − r)+ + + Q(1 − Ye ) 1 − Ye 1 − Ye Cs D DC(L) Co DE[y](1 − E[m2 ]) + + + 1 − Ye Q(1 − Ye ) 1 − Ye
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
(1)
As per assumption (8), the buyer offers backorder price discount on stock-out items to his customers. Therefore
11
Journal Pre-proof
including β = β0 πx /π0 , stock-out cost, modifying buyer’s cost function in (1) becomes
= + +
of
ET Cb (Q, r, β, L)
AD Q(1 − Ye ) DQYe + + hb + + r − DL + (1 − β) E(X − r) Q(1 − Ye ) 2 s(1 − Ye ) D β0 πx2 µDE[y]E[m2 ] µ0 D(1 − E[y])E[m1 ] + π 0 − β0 π x + + Q(1 − Ye ) π0 1 − Ye 1 − Ye Cs D DC(L) Co DE[y](1 − E[m2 ]) + + 1 − Ye Q(1 − Ye ) 1 − Ye
pro
3.3.2. Average cost form vendor’s perspective
(2)
On the other hand, the vendor’s expected total cost per unit time is
ET Cv (m)
setup cost + holding cost
SD mQ(1−Ye ) .
re-
Vendor’s setup cost per unit time =
=
Furthermore, the vendor’s average inventory is the difference of the vendor’s and the buyer’s accumulated inventories (see Figure 1). That is,
urn al P
=
2 D Q Q m2 Q2 Q + (m − 1) − (1 + 2 + ... + (m − 1)) mQ − mQ P D 2P D Q D (m − 1) − (m − 2) 2 P
Therefore, the vendor’s holding cost per unit time is
hv Q D = (m − 1) − (m − 2) 2(1 − Ye ) P
Thus, the vendor’s expected total cost per unit time is
hv Q D SD ET Cv (m) = + (m − 1) − (m − 2) mQ(1 − Ye ) 2(1 − Ye ) P
(3)
Now, building upon (3), we intend to explore the effect of investment in setup reduction on the total
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
cost and coordination policy. This is only possible when the setup cost is no longer considered to be one of the fixed parameters, but one of the decision variables. Additional investment is a viable strategy taken by buyers to reduce supplier’s setup costs. A prevalent assumption by research studies is that once the buyer invests in the supplier’s manufacturing process, the set cost reduces over time. The reduction mostly takes a logarithmic form. For example, an investment of $200 can reduce the original setup cost $100 by ten percent to $90. Again for the next setup, an investment of the same amount can reduce the setup cost $90 to $81, and so on.
12
pro reurn al P
Figure 1: The inventory pattern for the buyer and the vendor
Taking the buyer’s investment, the vendor’s expected total cost per unit time becomes
ET Cv (m) = B ln
S0 S
SD hv Q D + + (m − 1) − (m − 2) mQ(1 − Ye ) 2(1 − Ye ) P
(4)
The expected joint total cost of the SC is the summation of the buyer’s expected cost given by (2) and the vendor’s expected cost given by (4), i.e.,
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
of
Journal Pre-proof
JET C(Q, S, r, πx , L, m)
=
+
+ +
" D S Cs D Q(1 − Ye ) DQYe + A+ + C(L) + + θB ln + hb Q(1 − Ye ) m 1 − Ye 2 s(1 − Ye ) # β0 π x D β0 πx2 + r − DL + 1 − E(X − r) + + π0 − β0 πx E(X − r)+ π0 Q(1 − Ye ) π0 Co DE[y](1 − E[m2 ]) hv Q D µDE[y]E[m2 ] + (m − 1) − (m − 2) + 1 − Ye 2(1 − Ye ) P 1 − Ye µ0 D(1 − E[y])E[m1 ] (5) 1 − Ye
S0 S
13
Journal Pre-proof
Let, M (πx ) =
+ π0 − β0 πx , H(m) = hv (m − 1) − (m − 2) D P , then the above problem takes the form
2 β0 π x π0
of
" S0 D S Cs D DQYe Q(1 − Ye ) JET C(Q, S, r, πx , L, m) = θB ln + A+ + C(L) + + hb + S Q(1 − Ye ) m 1 − Ye 2 s(1 − Ye ) # " # β0 π x Co DE[y](1 − E[m2 ]) DM (πx ) + r − DL + hb 1 − E(X − r)+ + + π0 Q(1 − Ye ) 1 − Ye µDE[y]E[m2 ] µ0 D(1 − E[y])E[m1 ] QH(m) + + 2(1 − Ye ) 1 − Ye 1 − Ye
(6)
pro
+
of
4. Solution procedure
In the next section, we will develop two different solution approaches based on distribution function of
re-
the lead-time demand. 4.1. Lead-time demand follows normal distribution
Assuming that the lead-time demand is normally distributed, the buyer’s expected shortage quantity can
+
E(X − r) =
where f (x) =
Z
urn al P
be calculated as
√1 e− σ 2π
∞
r
(x − r)f (x)dx,
(x−µ)2 2σ 2
, µ and σ being the mean and standard deviation, respectively. √ Therefore, for a demand with mean DL and standard deviation σ L during the lead-time, the expected shortage quantity is given by
+
E(X − r) =
We have k =
r−DL √ , σ L
Z
r
∞
(x − r) √
and further assuming z =
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
x−DL √ , σ L
1 −1 √ e 2 2πσ L
x−DL √ σ L
2
dx,
above equation takes the following form:
√ Z E(X − r)+ = σ L
∞
k
(z − k)φ(z)dz,
where φ(z) is the standard normal probability density function. R∞ Assuming Ψ(k) = k (z − k)φ(z)dz, we have
√ E(X − r)+ = σ LΨ(k),
14
(7)
Journal Pre-proof
Using (7), the problem (6) can be reformulated as
+ + subject to
S0 S
of
=
" D S Cs D DQYe Q(1 − Ye ) θB ln + A+ + C(L) + + hb + Q(1 − Ye ) m 1 − Ye 2 s(1 − Ye ) # " # √ √ Co DE[y](1 − E[m2 ]) β0 π x DM (πx ) kσ L + hb 1 − + σ LΨ(k) + π0 Q(1 − Ye ) 1 − Ye
QH(m) µDE[y]E[m2 ] µ0 D(1 − E[y])E[m1 + + 2(1 − Ye ) 1 − Ye 1 − Ye
0 < S ≤ S0
pro
JET C N (Q, S, k, πx , L, m)
0 ≤ πx ≤ π0
(8)
The aim is to minimize JET C N (Q, S, r, πx , L, m) with respect to six decision variables and two constraints.
re-
The objective function derived in (8) is a constrained non-linear programming problem. In order to solve this problem, we initially relax the constraints 0 < S ≤ S0 and 0 ≤ πx ≤ π0 . For a given value of m, the first order partial derivatives of JET C N (S, Q, k, πx , L, m) with respect to S, Q, k, πx , and L, respectively, are
urn al P
as follows: ∂JET C N ∂S ∂JET C N ∂Q
=
=
+
and
∂JET C N ∂k
=
∂JET C N ∂πx
=
∂JET C N ∂L
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
=
−
θB D + S mQ(1 − Ye ) √ D S − 2 A+ + C(L) + M (πx )σ LΨ(k) Q (1 − Ye ) m h (1 − Y ) H(m) DYe i e hb + + 2 s(1 − Ye ) 2(1 − Ye ) √ √ β0 π x DM (πx ) hb σ L − hb 1 − + σ L[1 − Φ(k)] π0 Q(1 − Ye ) D 2β0 πx hb β0 √ − β0 − σ LΨ(k) Q(1 − Ye ) π0 π0 −
hb kσL−1/2 σL−1/2 Ψ(k) β0 π x DM πx + hb 1 − + 2 2 π0 Q(1 − Ye ) Dmi Q(1 − Ye )
(9)
(10) (11) (12)
(13)
For a fixed value of S, Q, k, πx , m, the joint total cost when lead-time demand follows normal distribution JET C N (Q, S, k, πx , L, m) is concave in L as ∂ 2 JET C N 1 1 β0 π x DM (πx ) −3/2 = − hb kσL − hb 1 − + σL−3/2 Ψ(k) < 0 ∂L2 4 4 π0 Q(1 − Ye )
(14)
Therefore, for a fixed (S, Q, k, πx , m), the minimum total expected cost will occur at the end points of
15
Journal Pre-proof
the interval [Li , Li−1 ]. On the other hand, for fixed m and L ∈ [Li , Li−1 ], by equating (9) − (12) to zero, we obtain
(15)
(17)
=
1−
πx
=
hb Q(1 − Ye ) π0 + 2D 2
S
=
mθBQ(1 − Ye ) D
and
of
hb Q(1 − Ye ) Q(1 − Ye )hb 1 − βπ0 π0 x + DM (πx )
Φ(k)
pro
Q =
v √ u u 2D A + S + C(L) + M (πx )σ LΨ(k) m t e hb (1 − Ye )2 + 2DY + H(m) s
(16)
(18)
Therefore, for fixed m and L ∈ [Li , Li−1 ], when the constraints 0 < S ≤ S0 and 0 ≤ πx ≤ π0 are
re-
relaxed, from (15) − (18), we obtain the optimal values of Q, k, πx , S (we denote these values by (S ∗ , Q∗ , k ∗ , πx∗ ) such that the total expected cost is minimum). The following proposition guaranties that, for fixed m and L ∈ [Li , Li−1 ], when the constraints 0 < S ≤ S0 and 0 ≤ πx ≤ π0 are relaxed, the point (S ∗ , Q∗ , k ∗ , πx∗ ) is the
urn al P
optimal solution at which the expected total cost per unit time is minimum.
Proposition 1: For fixed m and L ∈ [Li , Li−1 ], the Hessian Matrix for JET C N (S, Q, k, πx , L, m) is positive definite at point (S ∗ , Q∗ , k ∗ , πx∗ ) obtained from (9) − (12). Proof: See Appendix.
The optimal value of m can be obtained when
JET C N (m∗ − 1) ≥ JET C N (m∗ ) ≤ JET C N (m∗ + 1)
(19)
Let us consider the two constraints 0 < S ≤ S0 and 0 ≤ πx ≤ π0 . From (17) and (18), it is clear that S ∗ and πx∗ are positive. Also, if S ∗ < S0 and πx∗ < π0 , then (Q∗ , S ∗ , πx∗ ) is an interior optimal solution for given m and L ∈ [Li , Li−1 ]. However, if S ∗ > S0 , then no investment will be made for reducing the setup cost, and if
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
πx∗ > π0 , then no discount will be offered from the buyer’s side. Therefore, we set S ∗ = S0 and πx∗ = π0 . the following iterative algorithm is developed to find the optimal values for S, Q, k, πx , L, and m. Algorithm 1 Step 1 Set m = 1. Step 2 For each Li , i = 0, 1, 2, ..., n, perform 2a to 2e. 2a Set πxi1 = 0, Si1 = 0, and ki1 = 0 (implies Ψ(ki1 ) = 0.39894).
16
Journal Pre-proof
2b Substitute πxi1 , Si1 , and Ψ(ki1 ) into (15), evaluate Qi1 . 2c Utilize Qi1 to obtain the value of Φ(ki2 ) from (16), hence find ki2 by checking the normal table and evaluate Ψ(ki2 ).
of
2d Utilize Qi1 , determine πxi2 and Si2 from (17)and(18), respectively. 2e Repeat 2a to 2d until no changes occur in the values of Qi , ki , πxi , and Si . Denote the solution by
pro
(Qˆi , kˆi , π ˆxi , Sˆi ). Step 3 For each i = 0, 1, ..., n, compare π ˆxi with π0 and Sˆi with S0 .
3a If π ˆxi < π0 and Sˆi < S0 , then the solution found in Step 2 is optimal for a given Li . Denote this solution by (Q∗ , k ∗ , πx∗ , S ∗ ). Go to Step 5.
re-
∗ 3b If π ˆxi ≥ π0 and Sˆi < S0 , then for a given Li , set πxi = π0 and evaluate new (Qˆi , kˆi , Sˆi ) from (15), (16),
and (18) by the same procedure as described in Step 2. If Sˆi < S0 , then the optimal solution is (Q∗ , k ∗ , πx∗ , S ∗ ) = (Qˆi , kˆi , π0 , Sˆi ) and go to Step 5. Otherwise, go to Step 4.
urn al P
3c If Sˆi ≥ S0 and π ˆxi < π0 , then for given Li , set S ∗ = S0 and evaluate new (Qˆi , kˆi , π ˆxi ) from (15), (16), and (17) by the same procedure as described in Step 2. If π ˆxi < π0 , then the near optimal solution is (Q∗ , k ∗ , πx∗ , S ∗ ) = (Qˆi , kˆi , πˆx , S0 ) and go to Step 5. Otherwise, go to Step 4. 3d If π ˆxi ≥ π0 and Sˆi ≥ S0 , go to Step 4.
∗ Step 4 For the given Li set πxi = π0 and Si∗ = S0 , utilize (15) and (16) to obtain the corresponding optimal
solution (Q∗i , ki∗ ) by using Step 2.
∗ ∗ Step 5 Find JET C N (Q∗i , ki∗ , πxi , Si∗ , Li , m) and mini=0,1,2,...,n JET C N (Q∗i , ki∗ , πxi , Si∗ , Li , m). ∗ ∗ ∗ ∗ , πx(m) , S(m) , L∗(m) , m) = mini=0,1,2,...,n JET C N (Q∗i , ki∗ , πxi Step 5a If JET C N (Q∗(m) , k(m) , Si∗ , Li , m), then ∗ ∗ ∗ (Q∗(m) , k(m) , πx(m) , S(m) , L∗(m) ) is the optimal solution for a fixed m.
Step 6 Set m = m + 1.
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
∗ ∗ ∗ ∗ ∗ ∗ If JET C N (Q∗(m) , k(m) , πx(m) , S(m) , L(m) , m) ≤ JET C N (Q∗(m−1) , k(m−1) , πx(m−1) , S(m−1) , L(m−1) , m − 1),
then go to Step 6. Otherwise go to Step 7. ∗ ∗ ∗ Step 7 Set (Q∗ , k ∗ , πx∗ , S ∗ , L∗ , m) = (Q∗m−1 , km−1 , πx(m−1) , Sm−1 , L∗m−1 , m), then (Q∗ , k ∗ , πx∗ , S ∗ , L∗ , m) is the
optimal solution. √ After substituting the value of k ∗ and L∗ , the reorder point can be obtained as r∗ = DL + k ∗ σ L∗ .
17
Journal Pre-proof
4.2. DFA model In the previous section, it was assumed that the lead-time demand follows a normal distribution. However in many real-world SC problems, the information about the form of the probability distribution of the lead-time demand may not be known. In this subsection, the assumption about the normal distribution of the lead-time
of
demand is relaxed and we only assume that the density function of the lead-time demand belongs to the class √ Ω with finite mean DL and standard deviation σ L. As the probability distribution of the lead-time demand
pro
X is unknown, it is not possible to obtain the exact value of E(X − r)+ . Therefore, the min-max DFA is used to obtain the optimal solutions of this problem (see for instance Gallego and Moon (1993), Ouyang et al. (1996), Ouyang and Wu (1997), and Chu et al. (2005)). Now the problem is to find
Min MaxF ∈Ω
JET C DF (S, Q, k, πx , L, m) 0 < S ≤ S0
re-
subject to
0 ≤ πx ≤ π0
(20)
urn al P
To solve this problem, the result derived by Gallego and Moon (1993) is used.
Proposition 2: For any F ∈ Ω
E(X − r)+ ≤
1 p 2 { σ L + (r − DL)2 − (r − DL)} 2
(21)
Moreover, the upper bound is tight by Gallego and Moon (1993). √ Substituting r = DL + kσ L into (21), one can find the following inequality
E(X − r)+ ≤
1 √ p σ L( 1 + k 2 − k) 2
Now, using model (6) and inequality (22), the problem in (20) can reduced to
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
18
(22)
Journal Pre-proof
D + Q(1 − Ye )
S Cs D A+ + C(L) + m 1 − Ye # √ + kσ L
= θB ln " DQYe Q(1 − Ye ) + hb + 2 s(1 − Ye ) " # β0 π x DM (πx ) 1 √ p + hb 1 − 1 + k2 − k + σ L π0 Q(1 − Ye ) 2 + +
subject to
S0 S
of
(S, Q, k, πx , L, m)
QH(m) Co DE[y](1 − E[m2 ]) µDE[y]E[m2 ] + + 2(1 − Ye ) 1 − Ye 1 − Ye µ0 D(1 − E[y])E[m1 ] 1 − Ye
pro
Min JET C
DF
0 < S ≤ S0
(23)
re-
0 ≤ πx ≤ π0
To solve this problem, we follow the approach as in Section 5.1. The partial derivatives of JET C DF (Q, k, πx , S, L) with respect to Q, k, πx , S, and L, for the given value of m will be
=
+
and
θB D + S mQ(1 − Ye ) 1 √ p D S − 2 A+ + C(L) + M (πx ) σ L( 1 + k 2 − k) Q (1 − Ye ) m 2 h (1 − Y ) i H(m) DYe e hb + + 2 s(1 − Ye ) 2(1 − Ye ) √ k β0 π x DM (πx ) 1 √ hb σ L + hb 1 − σ L √ −1 + π0 Q(1 − Ye ) 2 1 + k2 D 2β0 πx h b β0 1 √ p − β0 − σ L( 1 + k 2 − k) Q(1 − Ye ) π0 π0 2
= −
urn al P
∂JET C DF ∂S ∂JET C DF ∂Q
∂JET C DF ∂k
=
∂JET C DF ∂πx
=
∂JET C DF ∂L
=
1 1 β0 π x DM (πx ) hb kσL−1/2 + hb 1 − + 2 4 π0 Q(1 − Ye ) p Dmi ( 1 + k 2 − k)σL−1/2 − Q(1 − Ye )
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Similar to the model discussed in section 5.1, for the fixed values of S, Q, k, πx , the joint total cost when lead-time follows DFA, JET C DF (S, Q, k, πx , L) is convex in L, as
∂JET C DF ∂L2
=
1 1 β0 π x DM (πx ) − hb kσL−3/2 − hb 1 − + 4 8 π0 Q(1 − Ye ) p −3/2 ( 1 + k 2 − k)σL <0
19
(24)
Journal Pre-proof
Therefore, the minimum total expected annual cost will occur at the end point of the interval [Li , Li−1 ]. Now, for fixed m and L ∈ [Li , Li−1 ], by equating the first order partial derivatives of JET C DF (S, Q, k, πx , L, m) with respect to Q, S, k, πx to zero, we can obtain
k 1 + k2
=
πx
=
S
=
of
√
√ √ + C(L) + M (πx ) 12 σ L( 1 + k 2 − k) e hb (1 − Ye )2 + 2DY + H(m) s 2hb 1− DM (πx ) β0 π x hb 1 − π0 + Q(1−Y e) S m
hb Q(1 − Ye ) π0 + 2D 2
and
(25) (26)
(27)
(28)
re-
mθBQ(1 − Ye ) D
pro
Q =
v u u 2D A + t
Proposition 3: For fixed m and L ∈ [Li , Li−1 ], the Hessian Matrix for JET C DF (Q, k, πx , S, L, m) is positive definite at the point (Q∗∗ , k ∗∗ , πx∗∗ , S ∗∗ ) obtained from (25) − (28). Proof: The proof is similar to Proposition 1.
urn al P
The optimal value of m can be obtained when
JET C DF (m∗∗ − 1) ≥ JET C DF (m∗∗ ) ≤ JET C DF (m∗∗ + 1)
(29)
The following iterative algorithm is developed to find the optimal values for S, Q, k, πx , L, and m. Algorithm 2 Step 1 Set m = 1.
Step 2 For each Li , i = 0, 1, 2, ..., n, perform 2a to 2e. 2a Set πxi1 = 0, Si1 = 0, and ki1 = 0.
2b Substitute πxi1 , Si1 , and ki1 into (25), and evaluate Qi1 .
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
2c Utilize Qi1 to obtain the value of ki1 , πxi and Si from (26), (27), and (28). 2d Utilize Qi1 determine πxi2 and Si2 from (17)and(18), respectively. 2e Repeat the steps 2b to 2d until no changes occur in the values of Qi , ki , πxi , and Si . Denote the solution by (Q¯i , k¯i , π ¯xi , S¯i ). ∗∗ Step 3 For each i = 0, 1, ..., n, compare πxi with π0 and Si∗∗ with S0 .
20
Journal Pre-proof
3a If π ¯xi < π0 and S¯i < S0 then the solution found in Step 2 is optimal for given Li . Denote this solution by (Q∗∗ , k ∗∗ , πx∗∗ , S ∗∗ ). Go to Step 5. ∗ 3b If π ¯xi ≥ π0 and S¯i < S0 , then for a given Li , set πxi = π0 and evaluate new (Q¯i , k¯i , S¯i ) from (25), (26),
and (28) by the same procedure as described in Step 2. If S¯i < S0 , then the optimal solution is
of
(Q∗∗ , k ∗∗ , πx∗∗ , S ∗∗ ) = (Q¯i , k¯i , π0 , S¯i ) and go to Step 5. Otherwise, go to Step 4.
3c If S¯i ≥ S0 and π ¯xi < π0 , then for a given Li , set S ∗ = S0 and evaluate new (Q¯i , k¯i , π ¯xi ) from (25), (26),
pro
and (27) by the same procedure as presented in Step 2. If π ¯xi < π0 , then the near optimal solution is (Q∗∗ , k ∗∗ , πx∗∗ , S ∗∗ ) = (Q¯i , k¯i , π¯x , S0 ), and go to Step 5. Otherwise, go to Step 4. 3d If π ¯xi ≥ π0 and S¯i ≥ S0 , go to Step 4.
solution (Q∗i , ki∗ ) by using Step 2.
re-
∗ Step 4 For the given Li set πxi = π0 and Si∗ = S0 , utilize (25) and (26) to obtain the corresponding optimal
∗∗ ∗∗ ∗∗ ∗∗ DF ∗∗ ∗∗ ∗∗ ∗∗ Step 5 Find JET C DF (Q∗∗ (Q∗∗ i , ki , πxi , Si , Li , m) and mini=0,1,2,...,n JET C i , ki , πxi , Si , Li , m). ∗∗ ∗∗ ∗∗ ∗∗ DF ∗∗ ∗∗ ∗∗ ∗∗ Step 5a If JET C DF (Q∗∗ (Q∗∗ i , ki , πxi , Si , Li , m), then (m) , k(m) , πx(m) , S(m) , L(m) , m) = mini=0,1,2,...,n JET C
urn al P
∗∗ ∗∗ ∗∗ ∗∗ (Q∗∗ (m) , k(m) , πx(m) , S(m) , L(m) ) is the optimal solution for a fixed m.
Step 6 Set m = m + 1.
∗∗ ∗∗ ∗∗ ∗∗ DF ∗∗ ∗∗ ∗∗ If JET C DF (Q∗∗ (Q∗∗ (m) , k(m) , πx(m) , S(m) , L(m) , m) ≤ JET C (m−1) , k(m−1) , πx(m−1) , S(m−1) , L(m−1)∗∗ , m−
1), then go to Step 6. Otherwise, go to Step 7.
∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ Step 7 Set (Q∗∗ , k ∗∗ , πx∗∗ , S ∗∗ , L∗∗ , m) = (Q∗∗ m−1 , km−1 , πx(m−1) , Sm−1 , Lm−1 , m), then (Q , k , πx , S , L , m)
is the optimal solution.
√ After substituting the value of k ∗∗ and L∗∗ , the reorder point can be obtained as r∗∗ = DL∗∗ +k ∗∗ σ L∗∗ .
5. Numerical example and sensitivity analysis 5.1. Numerical Example
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
This section presents a detailed numerical example to shed light on the coordination policy of the proposed SC. To derive the optimal solutions, all the calculation are done with the help of Mathematica 9.0 software. The following data set is used for analysis, which is similar to the one used in Khan et al. (2014) and Pan et al. (2004): D = 1000 units/year, P = 3200 units/year, A = $25/order, hb = $3/unit/year, hv =$1.3/unit/year, s= 175200, Cs = $0.5/unit, C0 = $0.5/unit, π0 = $150/unit, µ =$200/unit, µ0 =$50/unit, σ = 7 units per week, where 1 year = 52 weeks, θ = 0.1 per dollar per year. Furthermore the lead-time demand has three components with the data shown in Table 2 (see Ouyang et al.(1999)) as well as the summarized lead-time components
21
Journal Pre-proof
information is given in Table 3. Similar to Khan et al. (2014), we assume that the fraction of defectives and the percentage of inspection errors (Type I and Type II), as the following:
f (m2 ) =
1 0.04−0 ,
0 ≤ y ≤ 0.04
0,
otherwise
of
f (m1 ) =
1 0.04−0 ,
0 ≤ m1 ≤ 0.04
0,
otherwise
1 0.04−0 ,
0,
pro
f (y) =
0 ≤ m2 ≤ 0.04 otherwise
According to the distribution functions, the expected value of the defective products and Type I and Type
re-
II errors are E(y) = 0.02, E(m1 ) = 0.02 and E(m2 ) = 0.02. Now, we solve the cases for which the upper bounds of the backorder ratio β0 = 0.2, 0.5, 0.8, and B = 5800. In Salameh and Jaber (2000) and Goyal et al. (2003) model, they considered the setup cost S0 = $400 per production run. Therefore, we solve the Logarithmic investment case for which the initial setup cost S0 = 400. For the three given values of β0 , we use
urn al P
the proposed algorithm 2 to find the optimal solution of the DFA model. Instead of assuming any particular distribution function for lead-time demand, we only consider the DFA for numerical illustration. The results of the numerical study are shown in Table 4.
Table 2: Lead-time components with data (See Ouyang et al. (1999))
Lead-time component i 1 2 3
Normal duration bi (days) 20 20 16
Minimum duration ai (days) 6 6 9
Unit crashing cost ci ($/days) 0.4 1.2 5.0
Table 3: Summarized lead-time data
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Lead-time (week) 8 6 4 3
C(L) 0 5.6 22.4 57.4
From Table 4, it can be concluded that for β0 = 0.2 the minimum total expected annual cost is equal to $3530, which occurs when the lead-time is equal to four weeks, and the lead-time component is i=2. The
optimal buyer’s order quantity and vendor’s setup cost in this case are equal to $325 units and $724 per order respectively, with the optimal safety factor equal to 6.12. The optimal total expected annual cost for β0 = 0.5 and β0 = 0.8 are respectively $3509, and $3486. The results clearly show that the optimal total expected
22
Journal Pre-proof
Table 4: Results of the solution procedure for DFA β0 m
i = 0, Li = 8 πx
Q
k
S
πx
Q
k
i = 2, Li = 4
JET C DF
S
πx
Q
k
i = 3, Li = 3 JET C DF
S
πx
Q
k
JET C DF
0.2 1 306 75.79 550 4.68
3888
299 75.77 536 4.746
3823
295 75.76 530 4.78
3764
306
75.79 550 4.69
3778
2 500 75.65 448 5.20
3716
487 75.63 437 5.27
3645
483 75.62 433 5.29
3585
505
75.65 453 5.17
3708
3 638 75.55 382 5.64
3673
622 75.54 372 5.72
3597
619 75.53 370 5.73
3536
651
75.56 389 5.59
3567
4 745 75.48 334 6.04
3672
727 75.47 326 6.11
3592
of
S
i = 1, Li = 6
JET C DF
724 75.47 325 6.12
3530
766
75.50 344 5.95
3570
3689
812 75.42 292 6.47
3606
811 75.42 291 6.47
3543
862
75.44 309 6.28
3592
3864
296 75.76 531 4.57
3802
293 75.76 526 4.59
3748
304
75.79 546 4.50
3764
2 494 75.64 444 5.01
3690
483 75.62 433 5.07
3623
479 75.62 430 5.09
3566
502
75.65 450 4.97
3592
3 631 75.54 378 5.44
3645
616 75.53 369 5.50
3573
614 75.53 367 5.52
3516
647
75.56 387 5.37
3550
4 737 75.48 330 5.82
3642
720 75.46 323 5.89
3566
719 75.46 322 5.89
3509
761
75.49 341 5.72
3552
pro
5 832 75.43 299 6.39 0.5 1 303 75.78 544 4.51
3658
804 75.42 288 6.24
3579
805 75.42 289 6.23
3521
857
75.44 307 6.04
3573
3840
293 75.76 526 4.38
3781
291 75.75 522 4.40
3730
302
75.78 543 4.31
3749
2 489 75.63 439 4.81
3663
478 75.62 429 4.86
3599
475 75.61 426 4.88
3547
498
75.64 447 4.76
3776
3 624 75.54 373 5.22
3616
610 75.52 365 5.28
3548
609 75.52 364 5.29
3495
642.45 75.55 384 5.15
3533
4 728 75.47 326 5.59
3611
712 75.46 319 5.66
3539
712 75.46 320 5.65
3486
756
75.49 339 5.48
3534
5 812 75.42 292 5.92
3625
795 75.41 285 5.99
3550
798 75.41 286 5.98
3497
851
75.44 305 5.78
3554
re-
5 822 75.42 295 6.16 0.8 1 300 75.78 538 4.33
annual cost decreases if the value of β0 increases. This is, due to the fact that giving more discount attract
urn al P
customers. Table 5 illustrate the summary of Table 4.
Table 5: The optimal solutions for DFA
β0 0.2 0.5 0.8
m? 4 4 4
Investment with price discount (proposed model) S? πx? k? L? Q? JET C DF (.) 724 75.47 6.12 4 325 3530 719 75.46 5.89 4 322 3509 712 75.46 5.65 4 320 3486
Fixed setup cost and no price discount (S = S0 = 400 & πx = π0 = 150) m? k? L? Q? JET C DF (.) 4 6.819 4 276.526 3607.53 4 6.813 4 276.568 3607.07 4 6.807 4 276.609 3606.61
Savings (%) 2.20 2.79 3.44
The total misclassification cost is equal to $1103.25, which is around 31% of the total cost. Among the total misclassification cost, the cost of false acceptance is equal to $83.27, and the cost of falsely rejection is $1019.98. Note that the cost of false acceptance is much higher than the cost of false rejection, because falsely
accepted imperfect quality products may hamper goodwill of the firm. The total misclassification cost mainly depends on three factors: (i) the percentage of defective items supplied by the vendor to the buyer, (ii) the
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
expected value of type-I inspection error, and (iii) the expected value of type-II inspection error. 5.2. Sensitivity Analysis
This subsection discusses the results of sensitivity analysis on some of parameters of the model. It can be observed from table 6, when the ordering cost (A) decreases, the joint expected total cost also decreases. This result is not unexpected because in practice when the buyer controls his ordering cost, the total inventory cost automatically decreases. When the buyer’s or vendor’s holding cost (hb and hv ) increases the joint expected cost also increases. This shows the effect of holding cost on decision-making, and the result
23
Journal Pre-proof
Table 6: Sensitivity analysis on other parameters
C0
µ0
σ
S 684 703 722 739 832 747 682 631 712 712 712 712 712 712 712 712 610 693 770 842
JET C 3436 3470 3502 3534 3103 3367 3600 3813 3481 3492 3501 3512 2874 3282 3690 4098 3131 3418 3688 3945
Parameter
B
hv
Changes 3000 4500 6000 7500 0.5 1 1.5 2 0.25 0.75 1.25 1.75 75 150 225 300 0.005 0.015 0.025 0.035
Cs
µ
E(y)
Q 251 287 325 364 436 354 301 264 320 320 320 320 320 320 320 320 319 319 320 320
of
hb
Q 307 315 324 332 373 335 306 283 320 320 320 320 320 320 320 320 274 311 345 378
pro
A
Changes 10 20 30 40 1.5 2.5 3.5 4.5 0.25 0.75 1.25 1.75 20 40 60 80 2 6 10 14
re-
Parameter
S 290 496 749 1050 972 789 670 587 712 712 712 712 712 712 712 712 721 715 709 703
JET C 3533 3539 3474 3354 3120 3362 3563 3737 3226 3747 4267 4787 3434 3466 3497 3528 3393 3455 3518 3582
urn al P
is widely acceptable also. So, in this case, the small-sized shipment should be beneficial for both players, when their holding cost is high. With the increase in the value of expected percentage of defective items that were supplied by vendor to buyer E(y), the optimal order quantity (Q), and the joint expected total cost (JET C) increases, but the optimal vendor’s set up cost (S) decreases. The parameters C0 , Cs , hb and hv has no impact on the optimal order quantity (Q) and the optimal vendor’s set up cost (S). The vendor’s optimal set up cost is highly sensitive to B, hv and σ. The joint expected total cost is highly sensitive to hb , hv , Cs , µ0 , σ whereas less sensitive to the parameters A, B, µ, C0 and E(y).
5.2.1. Effect of demand and order quantity on joint expected total cost of the entire SC In this part of the paper, we investigate the joint effect of demand and the buyer’s order quantity on the expected total cost of the SC. Figure 3 depicts their joint effect on the total expected cost of the SC. Note that Q, is a decision variable, and Figure 2 illustrates that JET C function is convex in Q. The second partial
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
derivative of JET C with respect to Q can be calculated as follows: √ √ d2 JET C DF D(2(A + C(L) + S/m) + ( 1 + k 2 − k) L(π0 − β0 πx + (β0 πx2 )/π0 )σ) = >0 dQ2 ((1 − E(m1 ) − (1 − E(m1 ) − E(m2 ))E(y))Q3 ) Thus, the analytical method also justifies convexity of the JET C function in Q. Note from Figure 3 that as demand increases, the optimal value of Q increases accordingly. From equation (25), the following result
24
pro
Figure 2: Effect of demand and order quantity on the JET C
can be driven:
re-
P s hv (m − 1) + hb (1 − Ye )2 dQ∗ >0 = dD 2D P s hv H(m) + hb (1 − Ye )2 + 2Dhb P Ye
Which justifies what is shown in figure 2. Thus, we have the following proposition.
urn al P
Proposition 4: The joint expected total cost of the SC is a convex function of Q. A growth in SC demand will increase the buyer’s order quantity, which will eventually increases the total cost of the SC. In order to investigate the effect of demand growth on the SC, different values of demand were tested on the model, while the production rate was kept identical. Table 7, indicates the result of the analysis. Looking at Table 7, it can be understood that under a fixed production rate, varying demand up to four times more will only change the optimal order quantity up to 1.59 times more than its original value. Table 7: Effect of the demand on the optimal order quantity (Q) and the JET C DF
D P Q∗ JET C DF
500 3200 264 2097
750 3200 293 2837
1000 3200 320 3486
1250 3200 345 4082
1500 3200 371 4641
2000 3200 422 5685
3000 3200 534 7582
Figure 3 helps to gain further insights about the model’s behavior for the case where demand and production
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
of
Journal Pre-proof
rates change simultaneously. Before going further into the detail of Figure 3, let us first examine analytically whether or not the JET C is a decreasing function in P . Doing so gives: dJET C DF Dhv (m − 2)Q = dP (2(1 − E(m1 ) − (1 − E(m1 ) − E(m2 ))E(y))P 2
We may expect that 1 > [E(m1 ) + (1 − E(m1 ) − E(m2 ))E(y)] because E(m1 ), E(m2 ) and E(y) are quite small. Under this assumption,
dJET C DF dP
> 0 if m > 2. Otherwise, the JET C DF will become a decreasing function
25
pro
Figure 3: Effect of production rate and demand rate on the JET C DF
re-
of P . Thus, we can conclude the following proposition.
Proposition 5: The joint expected total cost of the SC will be a increasing function of P if (i) 1 > [E(m1 ) + (1 − E(m1 ) − E(m2 ))E(y)] and m > 2
urn al P
(ii)1 < [E(m1 ) + (1 − E(m1 ) − E(m2 ))E(y)] and m < 2 Otherwise, it will become decreasing function of P .
As the capital investment to reduce the vendor’s setup cost depends on the two crucial parameters B and S0 , we now, turn the analysis to investigate the impact of B and S0 on JET C. Figure 4 presents the effect of the parameters B and S0 on the JET C. Furthermore, by taking the first derivative of JET C with respect to B and S0 , we get
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
of
Journal Pre-proof
Figure 4: Effect of B and S0 on the JET C
dJET C DF S0 = τ log dB S
26
Journal Pre-proof
and dJET C DF Bτ = dS0 S0 Clearly, dJET C DF /dS0 > 0, indicating that the JET C increases with an increase in the initial set up cost of
of
the vendor. On the other hand, dJET C DF /dB ≥ 0 if S ≤ S0 , which indicates the same relation between B and JET C. Thus, we can say joint expected total cost will increase with the increasing value of B. Moreover, notice that,
Clearly,
dJET C DF dhv
√
√ √ √ 1 + k 2 πx σ L + π0 (1 − E(m1 ) − (1 − E(m1 ) + E(m2 ))E(y))Q + k + 1 + k 2 σ L 2π0
> 0 and
dJET C DF dhb
> 0. That is, the joint expected total cost of the SC increases with
re-
β0 k − dJET C DF = dhb
pro
((m − 1)P − (m − 2)D)Q dJET C DF = dhv (2(1 − E(m1 ) − (1 − E(m1 ) + E(m2 ))E(y))P )
increasing value of the holding cost of both the buyer and the vendor. Figure 5 depicts the effect of the holding
urn al P
costs on JET C.
Figure 5: Effect of holding costs on JET C
Following the similar approach as DFA and by using algorithm 1, the optimal solution for the case of normal
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
distribution is obtained and presented in table 8. Table 8: The optimal solutions for the normal distribution case
β0
0.2 0.5 0.8
m? 8 8 8
Q? 155.82 155.84 155.87
(Optimal decision ) S? k? πx? 694.70 2.7325 71.9971 694.77 2.7051 75.2246 694.89 2.6751 75.2246
27
L? 8 8 8
JET C N (.) 3006 3004 3003
Journal Pre-proof
6. Conclusion This paper studied a two-echelon supply chain inventory model for a situation where the batches shipped from the vendor to the buyer contain defective items. The buyer uses the continuous review policy to replenish inventories in which delivery lead-time is variable, and the reorder point is stochastic. The buyer is willing to
of
reduce the vendor’s set up cost by making an initial investment in the vendor’s production facilities. The buyer faces a shortage but offers a discount on the backordered items to avoid lost sales. One human factor is brought
pro
into the picture in this study, i.e., inspection error. The inspector at the buyer’s side misclassifies inspected items, which result in committing Type I & Type II inspection errors. The join expected total cost of the SC was formulated for the case under study. Two different scenarios for the lead-time demand are considered. In the first one, the lead-time demand assumed to follow the normal distribution while in the latter one, the DFA is taken. This model proposed an optimal policy for an integrated vendor-buyer supply chain system if the
re-
lead-time is dependent on production lot size, setup time and the way to reduce the setup cost. Using this policy, any industry manager may decide exactly which portion of time within transportation time and setup time they would like to reduce. This study proposed a price discount strategy by the buyer during the shortage period to secure customer demand and the right investment amount to reduce order processing cost. Several
urn al P
insights were derived by analyzing the model numerically and theoretically, such as: (i) when the percentage of defectives in the lot (E(y)) is high, it is recommended for the buyer to procure more and more in order to satisfy his customer’s demand with good quality of items. Since, with increase in the number of defective items yields into higher number of returned back items from the end-user, which will be not good for his business; (ii) Type I error has a more profound effect on the total cost of the SC, which confirms the findings of previous studies such as Khan et al., (2014). Offering a discount by the buyer was found to decrease the total cost of the SC, yet the buyer should decrease the order quantity to realize more savings. The joint effect of setup cost reduction and price discount was found to be influential in decreasing the expected total cost of the supply chain. (iii) the small-sized shipment should be beneficial for both players when their holding cost is high. The model can be extended along with several directions. An immediate extension of this research would be to include the effect of learning in quality to decrease the probability of Type I and Type II errors. If this
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
is the case, a more interesting extension would be to investigate whether learning in screening is worth an investment. One can also extend this work by assuming that the buyer sources from multiple suppliers. In this case, it would be interesting to study whether it is reasonable for the buyer to invest in reducing the setup cost of all suppliers or only the suppliers who have higher setup costs. The interesting part of this extension would be to take into account the limited budget of the buyer for investment that may incite the suppliers to compete for the investment. Another factor that may affect this setting is to study the relation between the level of supplier’s competition and its impact on the buyer’s selling price, and also on the variation of the backorder
28
Journal Pre-proof
threshold that the buyer would like to offer to its customers.
Acknowledgement The research of the first author has been supported by the NRF Singapore (Grant NRF-RSS2016-004). The
of
authors are grateful to the editor-in-chief and two reviewers for their constructive comments and invaluable contributions to enhance the presentation of this paper.
pro
References
Alfares, H.K., & Attia, A.M., 2017. A supply chain model with vendor-managed inventory, consignment, and quality inspection errors. International Journal of Production Research 55, 5706-5727. Annadurai, K., & Uthayakumar, R. 2010. Ordering cost reduction in probabilistic inventory model with
agement, 5(6), 403-410.
re-
controllable lead time and a service level. International Journal of Management Science and Engineering Man-
Arshinder, K., Kanda, A., & Deshmukh, S., 2011. A review on supply chain coordination: coordination mechanisms, managing uncertainty and research directions. Supply chain coordination under uncertainty. Springer,
urn al P
pp. 39-82.
Br¨auer, I., & Buscher, U. 2018, September. Pricing and Ordering Decisions in a JELS-Model for Items with Imperfect Quality. In International Conference on Intelligent Systems in Production Engineering and Maintenance (pp. 244-253). Springer, Cham.
´ Behzadi, G., OSullivan, M. J., Olsen, T. L., & Zhang, A. 2018. Agribusiness supply chain risk management: A review of quantitative decision models. Omega 79, 21-42. Ben-Daya, M. A., & Raouf, A., 1994. Inventory models involving lead time as a decision variable. Journal of the Operational Research Society, 45(5), 579-582.
Ben-Daya, M., & Hariga, M., 2003. Lead-time reduction in a stochastic inventory system with learning consideration. International Journal of Production Research, 41(3), 571-579. Bijvank, M., & Vis, I.F., 2011. Lost-sales inventory theory: A review. European Journal of Operational Re-
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
search 215, 1-13.
Chen, Z., & Sarker, B., 2010. Multi-vendor integrated procurement-production system under shared transportation and just-in-time delivery system. Journal of the Operational Research Society 61, 1654-1666. Chu, P., Yang, K. L., & Chen, P. S. 2005. Improved inventory models with service level and lead time. Computers & Operations Research, 32(2), 285-296. Das, B.C., Das, B., & Mondal, S.K., 2017. An integrated production-inventory model with defective item dependent stochastic credit period. Computers & Industrial Engineering 110, 255-263.
29
Journal Pre-proof
David, I., & Eben-Chaime, M., 2008. How accurate is the integrated vendor continuous model? International Journal of Production Economics 114, 805-810. De Giovanni, P., Karray, & S., Mart´ın-Herr´ an, G. 2019. Vendor Management Inventory with consignment contracts and the benefits of cooperative advertising. European Journal of Operational Research 272(2), 465-480.
of
De Treville, S., Shapiro, R.D., & Hameri, A.-P., 2004. From supply chain to demand chain: the role of lead time reduction in improving demand chain performance. Journal of Operations Management 21, 613-627. Deros, B.M., Peng, C.Y., Ab Rahman, M.N., Ismail, A.R., & Sulong, A.B., 2008. Assessing acceptance sam-
pro
pling application in manufacturing electrical and electronic products. Journal of Achievements in Materials and Manufacturing Engineering 31, 622-628.
Dey, O., 2017. A fuzzy random integrated inventory model with imperfect production under optimal vendor investment. Operational Research, 1-15. https://doi.org/10.1007/s12351-016-0286-1.
re-
Dey, O. 2019. A fuzzy random integrated inventory model with imperfect production under optimal vendor investment. Operational Research, 19(1), 101-115.
Dominguez, R., Cannella, S., Barbosa-P´ovoa, A. P., & Framinan, J. M. 2018. Information sharing in supply chains with heterogeneous retailers. Omega, 79, 116-132
urn al P
Eben-Chaime, M., 2004. The effect of discreteness in vendor-buyer relationships. IIE Transactions 36, 583-589. Ertogral, K., Darwish, M., & Ben-Daya, M., 2007. Production and shipment lot sizing in a vendor supply chain with transportation cost. European Journal of Operational Research 176, 1592-1606. Gallego, G., & Moon, I. 1993. The distribution free newsboy problem: review and extensions. Journal of the Operational Research Society, 44(8), 825-834.
Giri, B., Chakraborty, A., & Maiti, T., 2017. Consignment stock policy with unequal shipments and process unreliability for a two-level supply chain. International Journal of Production Research 55, 2489-2505. Glock, C.H., 2012a. The joint economic lot size problem: A review. International Journal of Production Economics 135, 671-686.
Glock, C.H., 2012b. Lead time reduction strategies in a single-vendor single-buyer integrated inventory model with lot size-dependent lead times and stochastic demand. International Journal of Production Economics 136, 37-44.
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Glock, C.H., Grosse, E.H., Neumann, W.P., & Sgarbossa, F., 2017. Human factors in industrial and logistic system design. Darmstadt Technical University, Department of Business Administration, Economics and Law, Institute for Business Studies (BWL) Goyal, S., 1977. An integrated inventory model for a single supplier-single customer problem. The International Journal of Production Research 15, 107-111. Goyal, S. K., Huang, C. K., & Chen, K. C. 2003. A simple integrated production policy of an imperfect item
30
Journal Pre-proof
for vendor and buyer. Production Planning & Control, 14(7), 596-602. Grosse, E.H., Glock, C.H., Jaber, M.Y., & Neumann, W.P., 2015. Incorporating human factors in order picking planning models: framework and research opportunities. International Journal of Production Research 53, 695-717.
of
Gutgutia, A., & Jha, J. K. 2018. A closed-form solution for the distribution free continuous review integrated inventory model. Operational Research, 18(1), 159-186.
Heydari, J., Govindan, K., & Jafari, A., 2017. Reverse and closed loop supply chain coordination by consider-
pro
ing government role. Transportation Research Part D: Transport and Environment 52, 379-398. Ho, C. H. 2009. A minimax distribution free procedure for an integrated inventory model with defective goods and stochastic lead time demand. International Journal of Information and Management Sciences, 20(1), 161171.
re-
Hoque, M., 2007. An alternative model for integrated vendor inventory under controllable lead time and its heuristic solution. International Journal of Systems Science 38, 501-509. Hoque, M.A., & Goyal, S.K., 2006. A heuristic solution procedure for an integrated inventory system under controllable lead-time with equal or unequal sized batch shipments between a vendor and a buyer. International
urn al P
Journal of Production Economics 102, 217-225.
Hsu, J. T., & Hsu, L. F., 2013a. Two EPQ models with imperfect production processes, inspection errors, planned backorders, and sales returns. Computers & Industrial Engineering, 64(1), 389-402. Hsu, J. T., & Hsu, L. F., 2013b. An EOQ model with imperfect quality items, inspection errors, shortage backordering, and sales returns. International Journal of Production Economics, 143(1), 162-170. Hsu, J.-T., & Hsu, L.-F., 2013c. An integrated vendor inventory model with imperfect items and planned back orders. The International Journal of Advanced Manufacturing Technology 68, 2121-2132. Jauhari, W. A., Sofiana, A., Kurdhi, N. A., & Laksono, P. W. (2016). An integrated inventory model for supplier-manufacturer-retailer system with imperfect quality and inspection errors. International Journal of Logistics Systems and Management, 24(3), 383-407.
Jawad, H., Jaber, M. Y., & Nuwayhid, R. Y. 2018. Improving supply chain sustainability using exergy analysis. European Journal of Operational Research, 269(1), 258-271.
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Kazemi, N., Olugu, E.U., Abdul-Rashid, S.H., & Ghazilla, R.A.B.R., 2015. Development of a fuzzy economic order quantity model for imperfect quality items using the learning effect on fuzzy parameters. Journal of Intelligent & Fuzzy Systems 28, 2377-2389. Khan, M., Jaber, M., & Guiffrida, A., 2012. The effect of human factors on the performance of a two level supply chain. International Journal of Production Research 50, 517-533. Khan, M., Jaber, M.Y., & Ahmad, A.-R., 2014. An integrated supply chain model with errors in quality
31
Journal Pre-proof
inspection and learning in production. Omega 42, 16-24. Khan, M., Ahmad, A.-R., & Hussain, M., 2017a. Integrated decision models for a vendor supply chain with inspection errors and purchase and repair options. The International Journal of Advanced Manufacturing Technology, 1-8. https://doi.org/10.1007/s00170-017-1137-9.
of
Khan, M., Hussain, M., & C´ardenas-Barr´on, L.E., 2017b. Learning and screening errors in an EPQ inventory model for supply chains with stochastic lead time demands. International Journal of Production Research 55, 4816-4832.
pro
Krafft, M., & Mantrala, M.K., 2006. Retailing in the 21st Century. Springer
Kurniawan, R., Zailani, S.H., Iranmanesh, M., & Rajagopal, P., 2017. The effects of vulnerability mitigation strategies on supply chain effectiveness: risk culture as moderator. Supply Chain Management: An International Journal 22, 1-15.
Marketing Management 62, 1-16.
re-
Lambert, D.M., & Enz, M.G., 2017. Issues in supply chain management: Progress and potential. Industrial
Liao, C. J., & Shyu, C. H., 1991. An analytical determination of lead time with normal demand. International Journal of Operations & Production Management, 11(9), 72-78.
urn al P
Lin, Y. J. 2009. An integrated vendor-buyer inventory model with backorder price discount and effective investment to reduce ordering cost. Computers & Industrial Engineering, 56(4), 1597-1606. Lin, H.-J., & Lin, Y.-J., 2014. Supply chain coordination with defective items and quantity discount. International Journal of Systems Science 45, 2529-2538.
Lin, Y.-J., Ouyang, L.-Y., & Dang, Y.-F., 2012. A joint optimal ordering and delivery policy for an integrated supplier inventory model with trade credit and defective items. Applied Mathematics and Computation 218, 7498-7514.
Mandal, P., & Giri, B. C. 2019. A two-warehouse integrated inventory model with imperfect production process under stock-dependent demand and quantity discount offer. International Journal of Systems Science: Operations & Logistics, 6(1), 15-26.
Marchi, B., Ries, J., Zanoni, S., & Glock, C., 2016. A joint economic lot size model with financial collaboration and uncertain investment opportunity. International Journal of Production Economics 176, 170-182.
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Modak, N. M., & Kelle, P. 2019. Managing a dual-channel supply chain under price and delivery-time dependent stochastic demand. European Journal of Operational Research, 272(1), 147-161. Mokhtari, H., & Rezvan, M.T., 2017. A single-supplier, multi-buyer, multi-product VMI production-inventory system under partial backordering. Operational Research, 1-21. https://doi.org/10.1007/s12351-017-0311-z. Moon, I., & Gallego, G. 1994. Distribution free procedures for some inventory models. Journal of the Operational Research Society, 45(6), 651-658.
32
Journal Pre-proof
Moon, I., & Choi, S. 1998. TECHNICAL NOTEA note on lead time and distributional assumptions in continuous review inventory models. Computers & Operations Research, 25(11), 1007-1012. Mou, Q., Cheng, Y., & Liao, H., 2017. A note on lead time reduction strategies in a single-vendor-single-buyer integrated inventory model with lot size-dependent lead times and stochastic demand. International Journal
of
of Production Economics 193, 827-831.
Ouyang, L. Y., Yeh, N. C., & Wu, K. S. 1996. Mixture inventory model with backorders and lost sales for variable lead time. Journal of the Operational Research Society, 47(6), 829-832.
pro
Ouyang, L. Y., & Wu, K. S. 1997. Mixture inventory model involving variable lead time with a service level constraint. Computers & Operations Research, 24(9), 875-882.
Ouyang, L. Y., Chen, C. K., & Chang, H. C. 1999. Lead time and ordering cost reductions in continuous review inventory systems with partial backorders. Journal of the Operational Research Society, 50(12), 1272-1279.
re-
Ouyang, L.-Y., Chen, L.-Y., & Yang, C.-T., 2013. Impacts of collaborative investment and inspection policies on the integrated inventory model with defective items. International Journal of Production Research 51, 5789-5802.
Pal, B., Sana, S.S., & Chaudhuri, K., 2012. Three-layer supply chain production-inventory model for rework-
urn al P
able items. Applied Mathematics and Computation 219, 530-543.
Pal, S., & Mahapatra, G. S. (2017). A manufacturing-oriented supply chain model for imperfect quality with inspection errors, stochastic demand under rework and shortages. Computers & Industrial Engineering, 106, 299-314.
Pan, J.C.-H., & Yang, J.-S., 2002. A study of an integrated inventory with controllable lead time. International Journal of Production Research 40, 1263-1273.
Pan, J. C. H., & Lo, M. C. (2008). The learning effect on setup cost reduction for mixture inventory models with variable lead time. Asia-Pacific Journal of Operational Research, 25(04), 513-529. Pan, J. C. H., Lo, M. C., & Hsiao, Y. C. 2004. Optimal reorder point inventory models with variable lead time and backorder discount considerations. European Journal of Operational Research, 158(2), 488-505. Panwar, A., Nepal, B.P., Jain, R., & Rathore, A.P.S., 2015. On the adoption of lean manufacturing principles in process industries. Production Planning & Control 26, 564-587.
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Porteus, E. L. 1985. Investing in reduced setups in the EOQ model. Management Science, 31(8), 998-1010. Salameh, M. K., & Jaber, M. Y. 2000. Economic production quantity model for items with imperfect quality. International Journal of Production Economics, 64(1-3), 59-64. Sainathan, A., & Groenevelt, H. 2018. Vendor Managed Inventory Contracts-Coordinating the Supply Chain while looking from the Vendor’s Perspective. European Journal of Operational Research 272(1), 249-260. Sarkar, B., & Moon, I. 2014. Improved quality, setup cost reduction, and variable backorder costs in an im-
33
Journal Pre-proof
perfect production process. International Journal of Production Economics, 155, 204-213. Sarkar, B., Chaudhuri, K., & Moon, I., 2015. Manufacturing setup cost reduction and quality improvement for the distribution free continuous-review inventory model with a service level constraint. Journal of Manufacturing Systems 34, 74-82.
of
Sarkar, S., & Giri, B. C. (2018a). Stochastic supply chain model with imperfect production and controllable defective rate. International Journal of Systems Science: Operations & Logistics, 1-14.
Sarkar, S., & Giri, B. C. (2018b). A vendor-buyer integrated inventory system with variable lead time and
pro
uncertain market demand. Operational Research. doi:10.1007/s12351-018-0418-x
Scholten, K., & Fynes, B., 2017. Risk and uncertainty management for sustainable supply chains, Sustainable supply chains. Springer, pp. 413-436.
Shu, H., & Zhou, X., 2014. An optimal policy for a single-vendor and a single-buyer integrated system with
re-
setup cost reduction and process-quality improvement. International Journal of Systems Science 45, 1242-1252. Su, C.-H., 2012. Optimal replenishment policy for an integrated inventory system with defective items and allowable shortage under trade credit. International Journal of Production Economics 139, 247-256. Tiwari, S., Sana, S.S., & Sarkar, S., 2018. Joint economic lot sizing model with stochastic demand and control-
urn al P
lable lead-time by reducing ordering cost and setup cost. Revista de la Real Academia de Ciencias Exactas, F´ısicas y Naturales. Serie A. Matem´aticas 112, 1075-1099. Tiwari, S., Daryanto, Y., & Wee, H.M., 2018. Sustainable inventory management with deteriorating and imperfect quality items considering carbon emission. Journal of Cleaner Production 192, 281-292. Wee, H. M., Yu, J., & Chen, M. C. 2007. Optimal inventory model for items with imperfect quality and shortage backordering. Omega, 35(1), 7-11.
Wee, H. M., & Widyadana, G. A. 2013. A production model for deteriorating items with stochastic preventive maintenance time and rework process with FIFO rule. Omega, 41(6), 941-954. Zavanella, L., & Zanoni, S., 2009. A one-vendor multi-buyer integrated production-inventory model: The consignment stock case. International Journal of Production Economics 118, 225-232.
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
34
Journal Pre-proof
Appendix Proof of proposition 1. For given value of L, the Hessian matrix H is ∂ 2 JET C N (·) ∗ ∂Q∗ ∂πx
∂ 2 JET C N (·) ∂Q∗ ∂k∗
∂ 2 JET C N (·) ∗ ∂Q∗ ∂πx
∂ 2 JET C N (·) ∗2 ∂πx
∂ 2 JET C N (·) ∗ ∂k ∗ ∂πx
∂ 2 JET C N (·) ∂k∗ ∂Q∗
∂ 2 JET C N (·) ∗ ∂k∗ ∂πx
∂ 2 JET C N (·) ∂k∗ 2
∂ 2 JET C N (·) ∂S ∗ ∂Q∗
∂ 2 JET C N (·) ∗ ∂S ∗ ∂πx
∂ 2 JET C N (·) ∂S ∗ ∂k∗
where JET C N (·) = JET C N (Q∗ , k ∗ , πx∗ , L∗ , S ∗ )
= =
∂ 2 JET C N ∂Q∗ ∂k ∗
=
∂ 2 JET C N ∂Q∗ ∂S ∗
=
∂ 2 JET C N ∂k ∗ ∂πx∗
=
∂ 2 JET C N ∂k ∗ ∂S ∗ 2 ∂ JET C N ∂πx∗ ∂S ∗
∂ 2 JET C N (·) ∗ ∂S ∗ ∂πx
∂ 2 JET C N (·) ∂k∗ ∂S ∗
∂ 2 JET C N (·) ∂S ∗ 2
√ S∗ 2D ∗ ∗ LΨ(k )M (π ) + A + F + C(L) + σ x m Q∗ 3 (1 − Ye ) ∗ √ DM (πx ) σ Lφ(k ∗ ) = hb G(πx∗ ) + ∗ Q (1 − Ye ) √ 2Dβ0 = LΨ(k ∗ ) σ Q∗ (1 − Ye )π0 =
urn al P
∂ 2 JET C N ∂k ∗ 2 ∂ 2 JET C N ∂πx∗ 2 ∂ 2 JET C N ∂S ∗ 2 2 ∂ JET C N ∂Q∗ ∂πx∗
re-
∂ 2 JET C N ∂Q∗ 2
∂ 2 JET C N (·) ∂Q∗ ∂S ∗
of
H=
∂ 2 JET C N (·) ∂Q∗ 2
pro
=
=
θB S∗2 √ ∂ 2 JET C N Dσ LΨ(k ∗ ) 2β0 πx∗ = ∗2 β0 − ∂πx∗ ∂Q∗ π0 Q (1 − Ye ) √ ∂ 2 JET C N Dσ LM (πx∗ ) = [1 − Φ(k ∗ )] ∗ ∗ ∂k ∂Q Q∗ 2 (1 − Ye ) ∂ 2 JET C N D =− 2 ∗ ∂S ∗ ∂Q∗ mQ (1 − Ye ) √ ∂ 2 JET C N hb β0 D 2β0 πx∗ = − ∗ − β0 σ L[1 − Φ(k ∗ )] ∗ ∗ ∂πx ∂k π0 Q (1 − Ye ) π0
∂ 2 JET C N =0 ∂S ∗ ∂k ∗ ∂ 2 JET C N =0 ∂S ∗ ∂πx∗
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
The first principal minor of H is √ S∗ 2D ∗ ∗ |H11 | = ∗ 3 A + F + C(L) + σ LΨ(k )M (πx ) + >0 m Q (1 − Ye )
35
Journal Pre-proof
The second principal minor of H is
of
# √ " ∗ ∗ √ 2D S LΨ(k ) 2Dβ σ 0 |H22 | = A + F + C(L) + σ LΨ(k ∗ )M (πx∗ ) + m Q∗ (1 − Ye )π0 Q∗ 3 (1 − Ye ) ( ) 2h √ i2 2β0 πx∗ D2 ∗ − σ LΨ(k − β ) 0 π0 Q∗ 4 (1 − Ye )2 h i2 √ D2 β σ √LΨ(k ∗ ) 0 S∗ 4D2 β0 σ LΨ(k ∗ ) A + F + C(L) + (4 − β0 ) = + m Q∗ 4 π0 (1 − Ye )2 Q∗ 4 (1 − Ye )2
pro
> 0
The third principal minor of H is
re-
h i2 D2 β σ √LΨ(k ∗ ) 4D2 β σ √LΨ(k ∗ ) ∗ 0 S 0 |H33 | = A + F + C(L) + + (4 − β ) 0 4 m Q∗ 4 (1 − Ye )2 Q∗ π0 (1 − Ye )2
+
> = =
urn al P
=
√ √ DM (πx∗ ) 2Dβ0 σ LΨ(k ∗ ) hb G(πx∗ ) + ∗ σ Lφ(k ∗ ) − Q (1 − Ye ) Q∗ (1 − Ye )π0 " # √ 2 Dσ LM (πx∗ ) ∗ [1 − Φ(k )] Q∗ 2 (1 − Ye ) √ √ 4D2 β0 σ LΨ(k ∗ ) S∗ DM (πx∗ ) ∗ A + F + C(L) + G(π ) + Lφ(k ∗ ) h σ b x m Q∗ (1 − Ye ) Q∗ 4 π0 (1 − Ye )2 i2 i2 h √ h √ D2 β0 σ LΨ(k ∗ ) D2 β0 σ LΨ(k ∗ ) √ (4 − β0 ) hb G(πx∗ )σ Lφ(k ∗ ) + (4 − β0 ) Q∗ 4 (1 − Ye )2 Q∗ 4 (1 − Ye )2 " #2 √ √ √ DM (πx∗ )σ Lφ(k ∗ ) 2Dβ0 σ LΨ(k ∗ ) Dσ LM (πx∗ ) − [1 − Φ(k ∗ )] Q∗ (1 − Ye ) Q∗ (1 − Ye )π0 Q∗ 2 (1 − Ye ) i2 h √ √ D3 β0 σ LΨ(k ∗ ) √ β0 σ LΨ(k ∗ ) 2D3 σ 2 L[M (πx∗ )]2 ∗ ∗ ∗ 2 (4 − β0 ) M (πx )σ Lφ(k ) − [1 − Φ(k )] π0 Q∗ 5 (1 − Ye )3 Q∗ 5 (1 − Ye )3 √ i D3 σ 2 Lσ LΨ(k ∗ )M (πx∗ ) h β0 β0 Ψ(k ∗ ) (4 − β0 ) φ(k ∗ ) − 2 [1 − Φ(k ∗ )]2 M (πx∗ ) 5 ∗ 3 π0 Q (1 − Ye ) √ 2 ∗ 3 2 ∗ ∗ D σ Lσ LΨ(k )M (πx ) β0 π x ∗ ∗ ∗ ∗ 2 β0 Ψ(k ) (4 − β0 ) φ(k ) − 2(1 − Φ(k )) (πx − π0 ) + β0 π02 Q∗ 5 (1 − Ye )3 √ D3 σ 2 Lσ LΨ(k ∗ )M (πx∗ ) ∗ ∗ ∗ 2 β Ψ(k ) (4 − β ) φ(k ) − 2β (1 − Φ(k )) 0 0 0 5 Q∗ (1 − Ye )3 √ 3 2 D σ Lσ LΨ(k ∗ )M (πx∗ ) β0 (2 − β0 )Ψ(k ∗ )φ(k ∗ ) + 2β0 Ψ(k ∗ )φ(k ∗ ) − 2β0 (1 − Φ(k ∗ ))2 5 ∗ 3 Q (1 − Ye ) √ 3 2 2D σ Lσ Lβ0 Ψ(k ∗ )M (πx∗ ) ∗ ∗ ∗ 2 Ψ(k )φ(k ) − (1 − Φ(k )) 5 Q∗ (1 − Ye )3
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
> =
>
> 0,
36
Journal Pre-proof
The fourth principal minor determinant of H is
=
> =
of
pro
=
re-
|H44 | =
" √ D θB DM (πx∗ ) 2Dβ0 σ LΨ(k ∗ ) D ∗ |H33 | + hb G(πx ) + ∗ − Q∗ (1 − Ye )π0 mQ∗ 2 (1 − Ye ) Q (1 − Ye ) S ∗2 mQ∗ 2 (1 − Ye ) # 2 √ √ hb β0 D 2β0 πx∗ D ∗ ∗ 2 (σ L[1 − Φ(k )]) σ Lφ(k ) + − ∗ − β0 π0 Q (1 − Ye ) π0 mQ∗ 2 (1 − Ye ) " √ 2 2Dβ0 σ LΨ(k ∗ ) θB D DM (πx∗ ) ∗ − |H | + h G(π ) + 33 b x Q∗ (1 − Ye )π0 Q∗ (1 − Ye ) S ∗2 mQ∗ 2 (1 − Ye ) # 2 √ √ hb β0 D 2β0 πx∗ ∗ ∗ 2 − ∗ − β0 (σ L[1 − Φ(k )]) σ Lφ(k ) + π0 Q (1 − Ye ) π0 2 " 2 θB D hb β0 Dβ0 ∗ |H | + − (2π − β ) 33 0 x π0 π0 Q∗ (1 − Ye ) S ∗2 mQ∗ 2 (1 − Ye ) # √ 2Dβ0 σ 2 LΨ(k ∗ )φ(k ∗ )h ∗ 2 (σ L[1 − Φ(k )]) − ∗ Q (1 − Ye )(1 − Φ(k ∗ ))π0 # 2 " D 2Dβ0 σ 2 LΨ(k ∗ )φ(k ∗ )h θB |H33 | − Q∗ (1 − Ye )(1 − Φ(k ∗ ))π0 S ∗2 mQ∗ 2 (1 − Ye ) # 2 " D θB 2Dβ0 σ 2 LΨ(k ∗ )φ(k ∗ )h |H33 | + Q∗ (1 − Ye )(Φ(k ∗ ) − 1)π0 S ∗2 mQ∗ 2 (1 − Ye )
urn al P
> 0
As the principal minors of the Hessian Matrix are positive, the Hessian Matrix H is positive definite at the point (Q∗ , k ∗ , πx∗ , S ∗ ). Hence the total expected annual cost function is a global minimum at that point.
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
37
*Credit Author Statement
Journal Pre-proof Author Contributions Section Dear editor We inform you the contributor roles: Sunil Tiwari: Conceptualization, Methodology, Investigation, Writing - Original Draft, Writing - Review & Editing, Validation
Nikunja Mohan Modak: Methodology, Validation
of
Nima Kazemi: Conceptualization, Methodology, Writing - Review & Editing
Leopoldo Eduardo Cárdenas-Barrón: Conceptualization, Methodology, Investigation,
Jo
urn al P
re-
Sumon Sarkar: Methodology, Validation
pro
Writing - Original Draft, Writing - Review & Editing, Supervision
PII: DOI: Reference:
S0925-5273(20)30044-X https://doi.org/10.1016/j.ijpe.2020.107643 PROECO 107643
To appear in:
International Journal of Production Economics
Received date : 24 April 2019 Revised date : 13 January 2020 Accepted date : 14 January 2020 Please cite this article as: S. Tiwari, N. Kazemi, N.M. Modak et al., The effect of human errors on an integrated stochastic supply chain model with setup cost reduction and backorder price discount. International Journal of Production Economics (2020), doi: https://doi.org/10.1016/j.ijpe.2020.107643. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier B.V.
*Title page including author details Click here to download Title page including author details: Title Page.pdf
Journal Pre-proof The effect of human errors on an integrated stochastic supply chain model with setup cost reduction and backorder price discount Sunil Tiwari a Department of Industrial Systems Engineering and Management, National University of Singapore, 1 Engineering Drive 2, 117576, Singapore
Nima Kazemi b
Centre for Transportation and Logistics, Massachusetts Institute of Technology, Cambridge, MA
pro
b
of
Email: [email protected]; [email protected]
02142, USA
Email: [email protected] Nikunja Mohan Modak c
Palpara Vidyamandir, Chakdaha, West Bengal, India
re-
c
Email: [email protected] Leopoldo Eduardo Cárdenas-Barrón d* Department of Industrial and Systems Engineering, School of Engineering and Sciences,
urn al P
d
Tecnológico de Monterrey, E. Garza Sada 2501 Sur, C.P. 64849, Monterrey, Nuevo León, México Email: [email protected] Sumon Sarkar e
e
Department of Mathematics, Jadavpur University, Kolkata - 700032, India Email: [email protected]
Jo
a
Journal Pre-proof *Manuscript Click here to view linked References
The effect of human errors on an integrated stochastic supply chain model with setup cost reduction and backorder price discount Sunil Tiwari1 , Nima Kazemi1 , Nikunja Mohan Modak1 , Leopoldo Eduardo C´ ardenas-Barr´ on1 , Sumon Sarkar1 a Department
pro
of
of Industrial Systems Engineering and Management, National University of Singapore, 1 Engineering Drive 2, 117576, Singapore b Centre for Transportation and Logistics, Massachusetts Institute of Technology, Cambridge, MA 02142, USA c Palpara Vidyamandir, Chakdaha, West Bengal, India d Department of Industrial and Systems Engineering, School of Engineering and Sciences, Tecnol´ ogico de Monterrey, E. Garza Sada 2501 Sur, C.P. 64849, Monterrey, Nuevo Le´ on, M´ exico e Department of Mathematics, Jadavpur University, Kolkata - 700032, India
Abstract
This study develops an integrated vendor-buyer inventory model for a supply chain where quality issues and
re-
human error affect its coordination. Each lot shipped to the buyer contains defective items, with the rate, which randomly changes from a lot to the other. The buyer inspects every shipped lot to segregate defective items. However, the inspection process at the buyer’s end goes wrong in the classification of defective and nondefective items. On the other hand, the buyer may run out of inventory, but in order to avoid lost sales, he/she
urn al P
offers a price discount on the backlogged items to his/her customers. Due to the vendor-buyer relationship, the buyer invests in reducing the setup cost of the vendor. Supply chain’s lead-time is considered variable, and two models are developed based on the probability distribution of the lead-time demand. In the first model, it is assumed that lead-time demand follows a normal distribution, while in the latter one, it does not follow any certain distribution. Two models are developed to determine the joint optimal decision variables that minimize the total cost of the supply chain. Two iterative algorithms are developed to obtain the optimal solution for both models. A set of numerical analysis and sensitivity analysis are conducted to gain insights. Keywords: Defective items, setup cost reduction, backorder price discount, inspection error, variable lead-time, distribution-free approach.
1. Introduction
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Supply chain management (SCM) encompasses different entities that make various strategic, tactical, and operational decisions at different stages, aiming at creating a competitive advantage for their business (Lambert and Enz, 2017). SCM is thus concerned with the coordination of material, information, and money along with a network of companies whose objective is to achieve better performance (Marchi et al., 2016). The growing structure of supply chains due to the involvement of various entities makes them more dynamic, yet so vulnerable ∗ Corresponding
author
Preprint submitted to Elsevier
January 13, 2020
Journal Pre-proof
to external risk and vulnerability factors (Kurniawan et al., 2017; Scholten and Fynes, 2017, Behzadi et al., 2018; Dominguez et al., 2018). This calls for effective SCM coordination and involvement of all parties in decision making to ensure that the whole supply chain (SC) works toward a joint objective (Glock, 2012a). One of the favorable research streams in SC is a coordination of inventory replenishment decisions among SC
of
members, which is the topic that has shown rapid growth over the recent years (Arshinder et al., 2011, Heydari et al., 2017). In inventory management literature, this type of model is recognized as the joint economic lot size (JELS). The so-called JELS models simulate today’s business practices (e.g., automotive, apparel, grocery)
pro
where there exists a long relationship between buyers and suppliers. The underlying assumption in the JELS models is that the buyer and supplier tend to make a collaborative decision on their inventory replenishment (Glock, 2012a). As joint inventory replenishment is a prevalent practice in SCs, there is a growing necessity to develop the decision support models that could help to gain insight into deriving the optimum policies for
re-
such cases.
Product quality is of high strategic importance to the buyer-supplier relationship and is one of the significant factors that both sides consider in their joint inventory replenishment. To ensure a high quality of buying products, the buying companies mostly carry out an inspection to ensure the high quality of incoming raw
urn al P
materials or products (Khan et al., 2017a). Even though developing new technologies could have led to the automation of quality inspection, many companies are still relying on human inspectors. When a human operator is involved in SC processes, his/her characteristic will affect the performance of the SC. Thus, ignoring human characteristics in SC operations may lead to making decisions that are sub-optimal and have financial burdens (Glock et al., 2017).
Lead-time is another concern for coordinating inventories between a buyer and a supplier. Setting up lead-time strategies may have several direct and indirect impacts on an SC, for instance, on inventory costs or customer service, among others (Deros et al., 2008). A prominent example of lead-time reduction has observed in a Japanese car manufacturing company where just-in-time production has led to a lead-time reduction, which consequently has increased efficiency and competitiveness of the company. The lead-time reduction is particularly significant when the SC encounters an uncertain demand, as high lead-time increases the risk of stock-out (De Treville et al., 2004). In addition to lead-time reduction, this research is also motivated by the
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
practice of the buyers who invest money to reduce the setup cost of the supplier company. Reducing setup costs of manufacturing companies could improve their operational performance that would positively influence the performance of the entire SC. The last issue that this paper is taking into account is the stock-out situation. Customer behavior is appeared to be different in stock-out cases, depending on the type of industry and the nature of the business (Bijvank and Vis, 2011). While some prefer to source their product from another supplier, the others choose
2
Journal Pre-proof
to wait until the order is ready for shipment (Krafft and Mantrala, 2006). However, in some certain industries, such as fashionable goods, hi-fi equipment, cosmetics, and apparel, customers are ready to wait for backorders up to a certain period (Krafft and Mantrala, 2006). In such cases, buyers try to convince their customers by offering a price discount on the backordered items.
of
Our Contribution:
The purpose of this paper is to merge four research streams, namely the JELS model, human error in inspection, variable lead-time, set up cost reduction into a single research frame. Thus, this paper presents
pro
a vendor-buyer integrated inventory model with quality issues and variable lead-time in which the buyer is willing to invest in the vendor’s manufacturing facility to improve the quality. Human errors in the inspection are considered at the buyer side to represent a real industrial setting where human error is unavoidable. By considering these four operational elements, this paper investigates how these elements affect buyer-supplier
re-
on optimal inventory replenishment. As the literature review in the next section shows, this is the first paper in the literature that addresses these operational factors concurrently in a JELS model. Besides that, this paper investigates two states for lead-time demand. In the first case, we assume that demand follows a normal distribution. In the second one, we assume that the distribution of lead-time demand is known. Considering
urn al P
the assumption and settings of the model, we try to answer the following questions: 1. What would be the optimal ordering policy for an industry if demand during replenishment lead-time is stochastic?
2. What would be the appropriate price discount strategy, if adopted by the buyer during the shortage period to secure customer demand?
3. What would be the exact investment amount to reduce setup costs? 4. What is the impact of misclassification errors on decision-making? 5. Which methodology is the best to obtain the global minimum solution and what steps a manager will adopt when lead-time demand distribution is unknown?
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
The remaining parts of the paper are organized as follows. Section 2 discusses the relevant literature and the research gap that this study tries to cover. The next section provides the notations, assumptions, and the details of the mathematical model. Section 4 presents the solution algorithms developed to optimize the mathematical model. Section 5 illustrates the numerical study, and finally, Section 6 provides the conclusions and future research suggestions.
3
Journal Pre-proof
2. Literature Review The literature review section investigates the four research streams which (i.e., the JELS model, inspection errors in quality, setup cost reduction and lead-time variation) that build the main structure of the paper at hand. We will initially discuss the four different research streams individually. At the end of this section, we
of
will discuss the research streams together and synthesize them to specify the research gap and position this study in the literature.
pro
2.1. The JELS model
The JELS models are one of the mathematical models in the area of SC that illustrate how inventory replenishment decision impacts the profit of the entire SC. The literature on the JELS models has advanced over the years since the seminal paper of Goyal (1977), who developed a model with the simple assumption
re-
that the production quantity at the supplier and the order quantity at the buyer are equal. Subsequently, many researchers have tried to relax the simplistic assumption used Goyal (1977) by working on the structured delivery between supplier and buyer (Eben-Chaime, 2004, David and Eben-Chaime, 2008, Ertogral et al., 2007). The model of Goyal (1977) was also extended to the SCs that include more than one company in each
urn al P
stage, or multiple companies in every stage (Zavanella and Zanoni, 2009, Mokhtari and Rezvan, 2017, Chen and Sarker, 2010). Recent advancement of the JELS model include permissible delay in payments (Aljazzar et al., 2016), vendor-managed inventory (De Giovanni et al., 2019, Sainathan and Groenevelt, 2019), imperfect quality (Salameh and Jaber 2000, Wee et al., 2007, Wee and Widyadana 2013), environmental issues (Tiwari et al., 2018), and sustainability (Jawad et al., 2018). 2.2. Inspection errors in quality
Human error is regarded as one of the factors influencing the system’s performance in logistics and inventory systems (Grosse et al., 2015, Kazemi et al., 2015). Even though companies tend to automate their logistics processes, many of them, particularly small-sizes, still prefer to operate manually (Grosse et al., 2015). Thanks to the factors such as work environment, task characteristics, and worker’s capability, a manual task might sometimes be performed erroneously. When it comes to the quality inspection process, it may not be error-free
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
as it is predominantly a manual task. An operator engaged in inspecting the shipped or produced batches might sort perfect quality product as a defective item and vice versa. The first type of error is known as Type I, while the latter is called Type II. (i) Type I error - rejecting a non-defective item as defective and (ii) Type II error - accepting a defective item as non-defective. These misclassification errors have an obvious effect on the inventory holding cost, warranty cost and more importantly, on the customer’s goodwill. Hence, ignorance of these inspection errors may result in distortion of supply chain performance (Khan et al., 2012; Hsu and Hsu 2013a; Hsu and Hsu 2013b; Jauhari et al., 2016; Pal and Mahapatra 2017; Khan et al., 2017b). The financial
4
Journal Pre-proof
outcomes of this human characteristic are a vital facet of modelling quality in SC coordination (Khan et al., 2012). A quite extensive number of mathematical models have been developed to address this operational factor, though only a few studied the issue in the JELS settings. For instance, Khan et al. (2014) incorporated learning in production and inspection errors in a two-stage SC, which follows an equal lot size policy. Khan
of
et al. (2017a) addressed the issue of defective items and inspection error in a supplier-buyer SC and assumed that the buyer has the option of either repairing the defective items via a local manufacturer or replacing them by purchasing new items from a local seller. Alfares and Attia (2017) developed a more complex model that
pro
coordinates inventories between a single supplier and multiple buyers. The authors formulate the quality issue of the production system at the supplier and inspections errors at the buyers. In addition to the quality issue, their model assumes that the supplier agrees with the buyers to follow a vendor-managed inventory (VMI)consignment stock (CS) policy. One of the few studies that employed statistical characteristics of uncertainty
re-
is the one of Sarkar et al. (2015), who developed a three-stage model (supplier-manufacturer-retailer) assuming that demand depends on price and stock level. 2.3. Setup cost reduction
urn al P
As discussed before, implementing a cooperative SC inventory policy may result in a higher SC efficiency. If the parties involved determine to collaborate in a long run, it may be more lucrative to invest in their collaboration to increase the turnover (Glock, 2012a). One of the standard practices seen in the industry is the investment of buyers in the supplier’s manufacturing process, aiming at decreasing the costs associated with the production process at the supplier (Panwar et al., 2015). The literature started to include this SC practice in inventory models in the early 80s. However, it appeared for the first time in the JELS literature in early 2000. To model the effect of the investment, studies mostly developed investment functions and included them in the JELS model. As an example, Sarkar et al. (2015) studied an inventory-production model with a continuousreview policy where setup cost assumed to reduce because of the improvement in the production process. In a similar study, Sarkar et al. (2014) studied the effect of setup cost reduction and variable backorder costs on an inventory-production inventory model. While the literature mostly has studied setup cost reduction in singlestage models, only a few have studied the problem in the context of the JELS models. Notable contributions
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
are the ones of Shu and Zhou (2014) and Sarkar et al. (2015). 2.4. Variable lead-time
One of the first studies that addressed variable lead-time was due to Liao and Shyu (1991). They developed an inventory model assuming that lead-time can be decomposed into several linear components with different piece-wise linear continuous crashing cost, where each component of the crashing costs may be reduced by considering normally distributed lead-time demand. Later, Ben-Daya and Raouf (1994) extended Liao and
5
Journal Pre-proof
Shyu’s (1991) in two steps. In the first one, they added order quantity as a decision variable to Liao and Shyu’s model, while in the latter one, they defined an exponential relationship between lead-time and total crashing cost. Ouyang et al. (1996) developed two models based on normally distributed lead-time and distributionfree lead-time (with known mean and standard deviation) and calculated the expected value of additional
of
information by using the concept of Porteus (1986). They proved for distribution free lead-time case; the industry manager has to pay less than one percent of the total cost. Moon and Choi (1988) extended their model by considering the reorder point as a decision variable. Ben-Daya and Hariga (2003) extended the same
pro
concept with lead-time crashing costs depending on lot size.
Another research in this area is one of Pan and Yang (2002), who assumed that the vendor might reduce lead-time upon the request of the buyer. Hoque and Goyal (2006) and Hoque (2007) considered the simultaneous effect of batch sizes and lead-time reduction on inventory replenishment decisions of a supplier-buyer
re-
relationship. Pan and Lo (2008) discussed a supply chain model with lead-time crashing costs under learning effects. Glock (2012b) assumed that lead-time could be reduced by either growing the production rate or reducing setup and transportation time. Finally, Mou et al. (2017) investigated the concurrent impact of investment in quality improvement and reduction in lead-times on a supplier-buyer joint lot-size decision. Even though for
urn al P
simplicity of problems lead-time in the relevant literature is assumed to follow a known probability distribution when it is uncertain, it might be quite difficult in some circumferences to specify the exact distribution. An alternative way preferred by researchers and practitioners is to apply the distribution-free approach (DFA), a method that helps to calculate the total cost of SC without having data available on the lead-time demand distribution (Sarkar et al., 2015, Tiwari et al., 2018). One of the first researchers who studied this approach was Moon and Gallego (1994) who introduced the DFA to model a continuous-review inventory model with a service level constraint. Later, Chu et al. (2005) applied the DFA in a continuous-review inventory model with a variable lead-time and backorders, where they assumed the reorder point is stochastic. Modak and Kelle (2019) employed DFA to formulate a dual-channel SC where the customer has the option to buy from either online or from stores.
2.5. Synthesis of the research streams
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Table 1 looks closely into the details of the studies that address defective items in the JELS models context and compare them with the contribution of this study. The columns in Table 1 illustrate the different characteristics of the models in the literature. As the table illustrates, the closet study to ours is the one of Sarkar and Giri (2018). However, Sarkar and Giri’s (2018) model has a few distinct differences with the study at hand. First of all, their model assumes that lead-time demand only follows a normal distribution, while in this study, in addition to the normal distribution, we inveterate the case where lead-time may not have a known distribution. Second, their model does not take inspection error into account, which routinely happens in many
6
Journal Pre-proof
industries. It is clear that the decision support models that aim to add a contribution to this research area should be as realistic as possible. This paper takes a step in making the model in the literature more realistic by integrating the research streams given in Table 1 to investigate their concurrent effect on the inventory replenishment policy of a two-level SC.
Setup cost Variable reduction lead-time
√ √ √
√
√ √ √ √ √ √ √ √ √ √ √ √
√
√ √
√
√
pro
√
urn al P
Dey (2019) Mandal & Giri (2019) Br¨auer & Buscher (2019) Gutgutia & Jha (2018) Sarkar & Giri (2018) Das et al. (2017) Giri et al. (2017) Khan et al. (2017b) Dey (2017) Khan et al. (2014) Lin & Lin (2014) Hsu and Hsu (2013c) Ouyang et al. (2013) Lin et al. (2012) Su (2012) Khan et al. (2012) Pal et al. (2012) Tiwari et al. (2018) This paper
Defective Error in quality items inspection
√ √
re-
Author (s)
of
Table 1: Comparison of the present model with some related works in the literature
√ √ √
√ √
Stochastic lead-time demand
Backorders
√ √ √
√ √
√
√ √
√
√ √ √ √
√ √
√
3. Model description and formulation
In this section, we present the assumptions and notations used in formulating the model. 3.1. Assumptions
We assume the following assumptions to develop our model: 1. The SC includes single vendor and single buyer who make decision about a single product.
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
√ 2. The reorder point (r) is defined as r = DL + kσ L, where DL = the expected demand during lead-time, √ and kσ L is safety stock (see for instance Ouyang et al. (1996), Ouyang and Wu (1997), and Ho(2009)). 3. The inventory is continuously reviewed and whenever it drops to the reorder point r, the buyer places an order of size Q, which is delivered in m equal-size shipments. 4. Each lot contains a percentage of defective items, therefore each lot is screened immediately at a fixed rate s, which is greater than the demand rate D.
7
Journal Pre-proof
5. The buyer is interested in investing an initial amount of money on the vendor’s manufacturing to reduce setup cost. The capital investment that the buyer invests, Is (S), is a logarithmic function of the setup
IS (S) = B ln
where B =
1 θ
S0 S
0 < S ≤ S0
,
of
cost S, i.e.,
with θ being the percentage decrease in S per dollar increase in Is (S), and S0 is Vendor’s
pro
original setup cost. It may be noted that the investment function is convex for all S with the restriction (See for instance Porteus (1985), Sarkar et al. (2015), Sarkar & Giri (2018a, 2018b)). 6. The lead-time L consists of n mutually independent components. The i-th component has a minimum duration ui in days, normal duration vi in days, and a crashing cost per day ci . Further, we rearrange
re-
ci as c1 ≤ c2 ≤ c3 ........ ≤ cn . Then, it is clear that lead-time reduction should first occur on component 1 (because it has the minimum unit crashing cost), and then component 2, and so on (See for instance Porteus (1985), Sarkar et al. (2015), Sarkar & Giri (2018a, 2018b)). Pn
j=1
vj denote the maximum duration of lead-time and Li as the length of lead-time with
urn al P
7. Let L0 =
components 1, 2, ...., i, crashed to their minimum duration, then Li can be expressed as (see for instance Ouyang et al. (1996), Annadurai and Uthayakumar (2010), and Priyam and Uthayakumar (2014))
Li =
n X j=1
vj −
i X j=1
(vj − uj )
where i = 1, 2, 3, ..., n and the lead-time crashing cost, C(L), per cycle is given by
C(L) = ci (Li−1 − L) +
i−1 X j=1
cj (vj − uj )
8. The buyer offers a discount for backordered items to his customer and the backorder ratio β is considered as a variable, which is proportional to the backorder price discount (πx ) offered by the buyer. Thus,
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
β = β0 πx /π0 , 0 ≤ β0 < 1 and 0 ≤ πx ≤ π0 . The buyer’s makes decisions in order to obtain profits. If the backorder price discount πx is greater than the marginal profit π0 , then the buyer may decide not to offer the discount as it is not a profitable option (see for instance Pan et al. (2004), and Lin (2009)).
8
Journal Pre-proof
3.2. Notations We use the following notations to develop the model: Decision variables buyer’s order quantity (units)
r
buyer’s reorder point (unit time)
L
length of the lead-time (unit time)
m
number of lots delivered from the seller to the buyer in one production cycle, a positive integer
S
vendor’s setup cost ($/order)
πx
backorder price discount ($/unit)
k
safety factor
pro
Parameters
buyer’s constant demand (unit/unit time)
A
buyer’s ordering cost ($/order)
hv
vendor’s holding cost ($/unit/unit time)
s
screening rate
Cs
screening cost ($/unit)
µ µ0 P m1
urn al P
θ
re-
D
C0
of
Q
scraping cost per unit ($/unit) annual fractional cost of the capital investment($/unit time) cost of falsely accepting a defective product ($/unit) cost of falsely rejecting a non-defective product ($/unit) production rate (units/unit time) a random variable showing probability of a Type I error (classifying a non-defective product as defective)
m2
a random variable showing probability of a Type II error (classifying a defective product as non-defective)
y ye C(L) β β0 σ
percentage of defective items that were supplied by the vendor to the buyer
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
the percentage of defective products observed by the buyer through screening
lead-time crashing cost function backorder ratio upper bound of the backorder ratio standard deviation of the lead-time demand
9
Journal Pre-proof
i-th component of lead-time with ui as minimum duration (unit time), i = 1, 2, ...., n
vi
i-th component of lead-time with vi as normal duration (unit time), i = 1, 2, ...., n
ci
i-th component of lead-time with ci as crashing cost per unit time ($/unit time), i = 1, 2, 3, ...n
X
lead-time demand which has a distribution function, f with finite mean DL √ and standard deviation σ L
E(x)
mathematical expectation of x
x+
max {x, 0}
E(X − r)+
the expected shortage quantity at the end of the cycle (units)
ET Cb ()
buyer’s expected total cost per unit time($/unit time)
ET Cv ()
vendor’s expected total cost per unit time ($/unit time)
JET C()
the joint expected total cost per unit time ($/unit time)
3.3. Model formulation
re-
pro
of
ui
This study considers a case in which a buyer and a vendor try to cooperatively determine the inventory lot-size to minimize their SC total cost. The buyer places an order of Q items to the vendor. Due to a high
urn al P
setup cost, the vendor produces mQ items in one setup with a constant production rate of P and delivers it in multiple shipments. Each delivered lot by the vendor contains y percent of defective items. Thus, on the arrival of each shipment, the buyer inspects each received lot at a constant screening rate of s to remove defective items. However, the inspection is not error-free and two types (Type I & Type II) of inspection errors may occur during the inspection at the buyer. The type I error occurs by classifying non-defective products as defective products while the type II errors by classifying defective products as non-defective products. Therefore, the fraction of defective items observed by the inspectors would be ye = (1 − y)m1 + (1 − m2 )y, where (1 − y)m1 and (1 − m2 )y are the percentages attributed to Type I and Type II inspection errors, respectively. Due to the random nature of the defective rate and inspection process, it is assumed that y, m1 , m2 are random variables and independent of each other. Hence, the expected value of observed defective items at the buyer is E[ye ] = (1 − E[y])E[m1 ] + E[y](1 − E[m2 ]). To simplify the notation, E[ye ] is abbreviated as Ye in this paper.
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Therefore, the buyer’s expected ordering cycle length is
Q(1−Ye ) . D
3.3.1. Average cost form buyer’s perspective The buyer picks out the continuous review policy to review his/her inventory and places an order of size Q whenever the non-defective inventory falls to the reorder point r. The buyer receives the order quantity after a variable lead-time L. As the lead-time demand is stochastic, shortages might take place when the on-hand stock is not sufficient to fulfill the demand. Hence, our model accounts for shortage cost in the total cost of SC. Shortage occurs when X > r,. Then, the buyer’s expected shortage quantity at the end of the cycle is E(X −r)+ .
10
Journal Pre-proof
Hence, the expected backorder quantity is βE(X − r)+ . Consequently, the expected loss sales per ordering cycle is (1 − β)E(X − r)+ and the expected stock-out cost per ordering cycle is [πx β + π0 (1 − β)] E(X − r)+ . In addition, at the beginning of each cycle the buyer’s expected net inventory is the safety stock r −DL plus the previous cycle lost sales (1 − β)E(X − r)+ . As a result, the expected net inventory of non-defective items per
of
unit time immediately after the successive order is Q(1−Ye )+r−DL+(1−β)E(X −r)+ . Therefore, the average inventory of non-defective items over the cycle can be calculated from
Q 2 (1 − Ye ) + r − DL + (1 − β)E(X
− r)+ .
pro
The buyer’s total holding cost of non-defective and defective items per unit time can now be calculated as h i DQYe e) hb Q(1−Y + s(1−Y + r − DL + (1 − β)E(X − r)+ . 2 e)
Due to inspection error, the buyer incurs misclassification costs, which are, the cost of falsely rejected
non-defective items and falsely accepted defective items. Another fact need to be considered here is that the cost of false acceptance could sometimes be very high and even fatal, particularly in some intricate industries
greater than that of µ0 i.e., µ >> µ0 .
re-
such as automotive, aerospace, and nuclear. To accommodate this fact, the value of µ is chosen to be much
The misclassification cost for Type-I error (i.e., the cost of false rejected non-defective items) is Cr = µ0 Q(1 − E[y])E[m1 ]
urn al P
The misclassification cost for Type-II error (i.e., the cost of false accepted defective items) is Ca = µQE[y]E[m2 ] Hence, the total misclassification cost per unit time is M C =
D 1−Ye {µE[y]E[m2 ]
+ µ0 (1 − E[y])E[m1 ]}
Further, the correctly identified defective items, i.e, QE(y)(1 − E[m2 ]) incur scraping cost for the buyer. Hence, the buyer’s scrapping cost per unit time is SC =
Co D 1−Ye {E[y](1
− E[m2 ])}.
According to the discussion above, the buyer’s expected total cost per unit time is
ET Cb (Q, r, β, L)
=
Ordering cost + holding cost + backorder cost + misclaficassion cost + inspection cost
+
lead-time crashing cost + scraping cost AD Q(1 − Ye ) DQYe + = + hb + + r − DL + (1 − β) E(X − r) Q(1 − Ye ) 2 s(1 − Ye ) D µDE[y]E[m2 ] µ0 D(1 − E[y])E[m1 ] + [πx β + π0 (1 − β)] E(X − r)+ + + Q(1 − Ye ) 1 − Ye 1 − Ye Cs D DC(L) Co DE[y](1 − E[m2 ]) + + + 1 − Ye Q(1 − Ye ) 1 − Ye
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
(1)
As per assumption (8), the buyer offers backorder price discount on stock-out items to his customers. Therefore
11
Journal Pre-proof
including β = β0 πx /π0 , stock-out cost, modifying buyer’s cost function in (1) becomes
= + +
of
ET Cb (Q, r, β, L)
AD Q(1 − Ye ) DQYe + + hb + + r − DL + (1 − β) E(X − r) Q(1 − Ye ) 2 s(1 − Ye ) D β0 πx2 µDE[y]E[m2 ] µ0 D(1 − E[y])E[m1 ] + π 0 − β0 π x + + Q(1 − Ye ) π0 1 − Ye 1 − Ye Cs D DC(L) Co DE[y](1 − E[m2 ]) + + 1 − Ye Q(1 − Ye ) 1 − Ye
pro
3.3.2. Average cost form vendor’s perspective
(2)
On the other hand, the vendor’s expected total cost per unit time is
ET Cv (m)
setup cost + holding cost
SD mQ(1−Ye ) .
re-
Vendor’s setup cost per unit time =
=
Furthermore, the vendor’s average inventory is the difference of the vendor’s and the buyer’s accumulated inventories (see Figure 1). That is,
urn al P
=
2 D Q Q m2 Q2 Q + (m − 1) − (1 + 2 + ... + (m − 1)) mQ − mQ P D 2P D Q D (m − 1) − (m − 2) 2 P
Therefore, the vendor’s holding cost per unit time is
hv Q D = (m − 1) − (m − 2) 2(1 − Ye ) P
Thus, the vendor’s expected total cost per unit time is
hv Q D SD ET Cv (m) = + (m − 1) − (m − 2) mQ(1 − Ye ) 2(1 − Ye ) P
(3)
Now, building upon (3), we intend to explore the effect of investment in setup reduction on the total
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
cost and coordination policy. This is only possible when the setup cost is no longer considered to be one of the fixed parameters, but one of the decision variables. Additional investment is a viable strategy taken by buyers to reduce supplier’s setup costs. A prevalent assumption by research studies is that once the buyer invests in the supplier’s manufacturing process, the set cost reduces over time. The reduction mostly takes a logarithmic form. For example, an investment of $200 can reduce the original setup cost $100 by ten percent to $90. Again for the next setup, an investment of the same amount can reduce the setup cost $90 to $81, and so on.
12
pro reurn al P
Figure 1: The inventory pattern for the buyer and the vendor
Taking the buyer’s investment, the vendor’s expected total cost per unit time becomes
ET Cv (m) = B ln
S0 S
SD hv Q D + + (m − 1) − (m − 2) mQ(1 − Ye ) 2(1 − Ye ) P
(4)
The expected joint total cost of the SC is the summation of the buyer’s expected cost given by (2) and the vendor’s expected cost given by (4), i.e.,
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
of
Journal Pre-proof
JET C(Q, S, r, πx , L, m)
=
+
+ +
" D S Cs D Q(1 − Ye ) DQYe + A+ + C(L) + + θB ln + hb Q(1 − Ye ) m 1 − Ye 2 s(1 − Ye ) # β0 π x D β0 πx2 + r − DL + 1 − E(X − r) + + π0 − β0 πx E(X − r)+ π0 Q(1 − Ye ) π0 Co DE[y](1 − E[m2 ]) hv Q D µDE[y]E[m2 ] + (m − 1) − (m − 2) + 1 − Ye 2(1 − Ye ) P 1 − Ye µ0 D(1 − E[y])E[m1 ] (5) 1 − Ye
S0 S
13
Journal Pre-proof
Let, M (πx ) =
+ π0 − β0 πx , H(m) = hv (m − 1) − (m − 2) D P , then the above problem takes the form
2 β0 π x π0
of
" S0 D S Cs D DQYe Q(1 − Ye ) JET C(Q, S, r, πx , L, m) = θB ln + A+ + C(L) + + hb + S Q(1 − Ye ) m 1 − Ye 2 s(1 − Ye ) # " # β0 π x Co DE[y](1 − E[m2 ]) DM (πx ) + r − DL + hb 1 − E(X − r)+ + + π0 Q(1 − Ye ) 1 − Ye µDE[y]E[m2 ] µ0 D(1 − E[y])E[m1 ] QH(m) + + 2(1 − Ye ) 1 − Ye 1 − Ye
(6)
pro
+
of
4. Solution procedure
In the next section, we will develop two different solution approaches based on distribution function of
re-
the lead-time demand. 4.1. Lead-time demand follows normal distribution
Assuming that the lead-time demand is normally distributed, the buyer’s expected shortage quantity can
+
E(X − r) =
where f (x) =
Z
urn al P
be calculated as
√1 e− σ 2π
∞
r
(x − r)f (x)dx,
(x−µ)2 2σ 2
, µ and σ being the mean and standard deviation, respectively. √ Therefore, for a demand with mean DL and standard deviation σ L during the lead-time, the expected shortage quantity is given by
+
E(X − r) =
We have k =
r−DL √ , σ L
Z
r
∞
(x − r) √
and further assuming z =
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
x−DL √ , σ L
1 −1 √ e 2 2πσ L
x−DL √ σ L
2
dx,
above equation takes the following form:
√ Z E(X − r)+ = σ L
∞
k
(z − k)φ(z)dz,
where φ(z) is the standard normal probability density function. R∞ Assuming Ψ(k) = k (z − k)φ(z)dz, we have
√ E(X − r)+ = σ LΨ(k),
14
(7)
Journal Pre-proof
Using (7), the problem (6) can be reformulated as
+ + subject to
S0 S
of
=
" D S Cs D DQYe Q(1 − Ye ) θB ln + A+ + C(L) + + hb + Q(1 − Ye ) m 1 − Ye 2 s(1 − Ye ) # " # √ √ Co DE[y](1 − E[m2 ]) β0 π x DM (πx ) kσ L + hb 1 − + σ LΨ(k) + π0 Q(1 − Ye ) 1 − Ye
QH(m) µDE[y]E[m2 ] µ0 D(1 − E[y])E[m1 + + 2(1 − Ye ) 1 − Ye 1 − Ye
0 < S ≤ S0
pro
JET C N (Q, S, k, πx , L, m)
0 ≤ πx ≤ π0
(8)
The aim is to minimize JET C N (Q, S, r, πx , L, m) with respect to six decision variables and two constraints.
re-
The objective function derived in (8) is a constrained non-linear programming problem. In order to solve this problem, we initially relax the constraints 0 < S ≤ S0 and 0 ≤ πx ≤ π0 . For a given value of m, the first order partial derivatives of JET C N (S, Q, k, πx , L, m) with respect to S, Q, k, πx , and L, respectively, are
urn al P
as follows: ∂JET C N ∂S ∂JET C N ∂Q
=
=
+
and
∂JET C N ∂k
=
∂JET C N ∂πx
=
∂JET C N ∂L
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
=
−
θB D + S mQ(1 − Ye ) √ D S − 2 A+ + C(L) + M (πx )σ LΨ(k) Q (1 − Ye ) m h (1 − Y ) H(m) DYe i e hb + + 2 s(1 − Ye ) 2(1 − Ye ) √ √ β0 π x DM (πx ) hb σ L − hb 1 − + σ L[1 − Φ(k)] π0 Q(1 − Ye ) D 2β0 πx hb β0 √ − β0 − σ LΨ(k) Q(1 − Ye ) π0 π0 −
hb kσL−1/2 σL−1/2 Ψ(k) β0 π x DM πx + hb 1 − + 2 2 π0 Q(1 − Ye ) Dmi Q(1 − Ye )
(9)
(10) (11) (12)
(13)
For a fixed value of S, Q, k, πx , m, the joint total cost when lead-time demand follows normal distribution JET C N (Q, S, k, πx , L, m) is concave in L as ∂ 2 JET C N 1 1 β0 π x DM (πx ) −3/2 = − hb kσL − hb 1 − + σL−3/2 Ψ(k) < 0 ∂L2 4 4 π0 Q(1 − Ye )
(14)
Therefore, for a fixed (S, Q, k, πx , m), the minimum total expected cost will occur at the end points of
15
Journal Pre-proof
the interval [Li , Li−1 ]. On the other hand, for fixed m and L ∈ [Li , Li−1 ], by equating (9) − (12) to zero, we obtain
(15)
(17)
=
1−
πx
=
hb Q(1 − Ye ) π0 + 2D 2
S
=
mθBQ(1 − Ye ) D
and
of
hb Q(1 − Ye ) Q(1 − Ye )hb 1 − βπ0 π0 x + DM (πx )
Φ(k)
pro
Q =
v √ u u 2D A + S + C(L) + M (πx )σ LΨ(k) m t e hb (1 − Ye )2 + 2DY + H(m) s
(16)
(18)
Therefore, for fixed m and L ∈ [Li , Li−1 ], when the constraints 0 < S ≤ S0 and 0 ≤ πx ≤ π0 are
re-
relaxed, from (15) − (18), we obtain the optimal values of Q, k, πx , S (we denote these values by (S ∗ , Q∗ , k ∗ , πx∗ ) such that the total expected cost is minimum). The following proposition guaranties that, for fixed m and L ∈ [Li , Li−1 ], when the constraints 0 < S ≤ S0 and 0 ≤ πx ≤ π0 are relaxed, the point (S ∗ , Q∗ , k ∗ , πx∗ ) is the
urn al P
optimal solution at which the expected total cost per unit time is minimum.
Proposition 1: For fixed m and L ∈ [Li , Li−1 ], the Hessian Matrix for JET C N (S, Q, k, πx , L, m) is positive definite at point (S ∗ , Q∗ , k ∗ , πx∗ ) obtained from (9) − (12). Proof: See Appendix.
The optimal value of m can be obtained when
JET C N (m∗ − 1) ≥ JET C N (m∗ ) ≤ JET C N (m∗ + 1)
(19)
Let us consider the two constraints 0 < S ≤ S0 and 0 ≤ πx ≤ π0 . From (17) and (18), it is clear that S ∗ and πx∗ are positive. Also, if S ∗ < S0 and πx∗ < π0 , then (Q∗ , S ∗ , πx∗ ) is an interior optimal solution for given m and L ∈ [Li , Li−1 ]. However, if S ∗ > S0 , then no investment will be made for reducing the setup cost, and if
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
πx∗ > π0 , then no discount will be offered from the buyer’s side. Therefore, we set S ∗ = S0 and πx∗ = π0 . the following iterative algorithm is developed to find the optimal values for S, Q, k, πx , L, and m. Algorithm 1 Step 1 Set m = 1. Step 2 For each Li , i = 0, 1, 2, ..., n, perform 2a to 2e. 2a Set πxi1 = 0, Si1 = 0, and ki1 = 0 (implies Ψ(ki1 ) = 0.39894).
16
Journal Pre-proof
2b Substitute πxi1 , Si1 , and Ψ(ki1 ) into (15), evaluate Qi1 . 2c Utilize Qi1 to obtain the value of Φ(ki2 ) from (16), hence find ki2 by checking the normal table and evaluate Ψ(ki2 ).
of
2d Utilize Qi1 , determine πxi2 and Si2 from (17)and(18), respectively. 2e Repeat 2a to 2d until no changes occur in the values of Qi , ki , πxi , and Si . Denote the solution by
pro
(Qˆi , kˆi , π ˆxi , Sˆi ). Step 3 For each i = 0, 1, ..., n, compare π ˆxi with π0 and Sˆi with S0 .
3a If π ˆxi < π0 and Sˆi < S0 , then the solution found in Step 2 is optimal for a given Li . Denote this solution by (Q∗ , k ∗ , πx∗ , S ∗ ). Go to Step 5.
re-
∗ 3b If π ˆxi ≥ π0 and Sˆi < S0 , then for a given Li , set πxi = π0 and evaluate new (Qˆi , kˆi , Sˆi ) from (15), (16),
and (18) by the same procedure as described in Step 2. If Sˆi < S0 , then the optimal solution is (Q∗ , k ∗ , πx∗ , S ∗ ) = (Qˆi , kˆi , π0 , Sˆi ) and go to Step 5. Otherwise, go to Step 4.
urn al P
3c If Sˆi ≥ S0 and π ˆxi < π0 , then for given Li , set S ∗ = S0 and evaluate new (Qˆi , kˆi , π ˆxi ) from (15), (16), and (17) by the same procedure as described in Step 2. If π ˆxi < π0 , then the near optimal solution is (Q∗ , k ∗ , πx∗ , S ∗ ) = (Qˆi , kˆi , πˆx , S0 ) and go to Step 5. Otherwise, go to Step 4. 3d If π ˆxi ≥ π0 and Sˆi ≥ S0 , go to Step 4.
∗ Step 4 For the given Li set πxi = π0 and Si∗ = S0 , utilize (15) and (16) to obtain the corresponding optimal
solution (Q∗i , ki∗ ) by using Step 2.
∗ ∗ Step 5 Find JET C N (Q∗i , ki∗ , πxi , Si∗ , Li , m) and mini=0,1,2,...,n JET C N (Q∗i , ki∗ , πxi , Si∗ , Li , m). ∗ ∗ ∗ ∗ , πx(m) , S(m) , L∗(m) , m) = mini=0,1,2,...,n JET C N (Q∗i , ki∗ , πxi Step 5a If JET C N (Q∗(m) , k(m) , Si∗ , Li , m), then ∗ ∗ ∗ (Q∗(m) , k(m) , πx(m) , S(m) , L∗(m) ) is the optimal solution for a fixed m.
Step 6 Set m = m + 1.
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
∗ ∗ ∗ ∗ ∗ ∗ If JET C N (Q∗(m) , k(m) , πx(m) , S(m) , L(m) , m) ≤ JET C N (Q∗(m−1) , k(m−1) , πx(m−1) , S(m−1) , L(m−1) , m − 1),
then go to Step 6. Otherwise go to Step 7. ∗ ∗ ∗ Step 7 Set (Q∗ , k ∗ , πx∗ , S ∗ , L∗ , m) = (Q∗m−1 , km−1 , πx(m−1) , Sm−1 , L∗m−1 , m), then (Q∗ , k ∗ , πx∗ , S ∗ , L∗ , m) is the
optimal solution. √ After substituting the value of k ∗ and L∗ , the reorder point can be obtained as r∗ = DL + k ∗ σ L∗ .
17
Journal Pre-proof
4.2. DFA model In the previous section, it was assumed that the lead-time demand follows a normal distribution. However in many real-world SC problems, the information about the form of the probability distribution of the lead-time demand may not be known. In this subsection, the assumption about the normal distribution of the lead-time
of
demand is relaxed and we only assume that the density function of the lead-time demand belongs to the class √ Ω with finite mean DL and standard deviation σ L. As the probability distribution of the lead-time demand
pro
X is unknown, it is not possible to obtain the exact value of E(X − r)+ . Therefore, the min-max DFA is used to obtain the optimal solutions of this problem (see for instance Gallego and Moon (1993), Ouyang et al. (1996), Ouyang and Wu (1997), and Chu et al. (2005)). Now the problem is to find
Min MaxF ∈Ω
JET C DF (S, Q, k, πx , L, m) 0 < S ≤ S0
re-
subject to
0 ≤ πx ≤ π0
(20)
urn al P
To solve this problem, the result derived by Gallego and Moon (1993) is used.
Proposition 2: For any F ∈ Ω
E(X − r)+ ≤
1 p 2 { σ L + (r − DL)2 − (r − DL)} 2
(21)
Moreover, the upper bound is tight by Gallego and Moon (1993). √ Substituting r = DL + kσ L into (21), one can find the following inequality
E(X − r)+ ≤
1 √ p σ L( 1 + k 2 − k) 2
Now, using model (6) and inequality (22), the problem in (20) can reduced to
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
18
(22)
Journal Pre-proof
D + Q(1 − Ye )
S Cs D A+ + C(L) + m 1 − Ye # √ + kσ L
= θB ln " DQYe Q(1 − Ye ) + hb + 2 s(1 − Ye ) " # β0 π x DM (πx ) 1 √ p + hb 1 − 1 + k2 − k + σ L π0 Q(1 − Ye ) 2 + +
subject to
S0 S
of
(S, Q, k, πx , L, m)
QH(m) Co DE[y](1 − E[m2 ]) µDE[y]E[m2 ] + + 2(1 − Ye ) 1 − Ye 1 − Ye µ0 D(1 − E[y])E[m1 ] 1 − Ye
pro
Min JET C
DF
0 < S ≤ S0
(23)
re-
0 ≤ πx ≤ π0
To solve this problem, we follow the approach as in Section 5.1. The partial derivatives of JET C DF (Q, k, πx , S, L) with respect to Q, k, πx , S, and L, for the given value of m will be
=
+
and
θB D + S mQ(1 − Ye ) 1 √ p D S − 2 A+ + C(L) + M (πx ) σ L( 1 + k 2 − k) Q (1 − Ye ) m 2 h (1 − Y ) i H(m) DYe e hb + + 2 s(1 − Ye ) 2(1 − Ye ) √ k β0 π x DM (πx ) 1 √ hb σ L + hb 1 − σ L √ −1 + π0 Q(1 − Ye ) 2 1 + k2 D 2β0 πx h b β0 1 √ p − β0 − σ L( 1 + k 2 − k) Q(1 − Ye ) π0 π0 2
= −
urn al P
∂JET C DF ∂S ∂JET C DF ∂Q
∂JET C DF ∂k
=
∂JET C DF ∂πx
=
∂JET C DF ∂L
=
1 1 β0 π x DM (πx ) hb kσL−1/2 + hb 1 − + 2 4 π0 Q(1 − Ye ) p Dmi ( 1 + k 2 − k)σL−1/2 − Q(1 − Ye )
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Similar to the model discussed in section 5.1, for the fixed values of S, Q, k, πx , the joint total cost when lead-time follows DFA, JET C DF (S, Q, k, πx , L) is convex in L, as
∂JET C DF ∂L2
=
1 1 β0 π x DM (πx ) − hb kσL−3/2 − hb 1 − + 4 8 π0 Q(1 − Ye ) p −3/2 ( 1 + k 2 − k)σL <0
19
(24)
Journal Pre-proof
Therefore, the minimum total expected annual cost will occur at the end point of the interval [Li , Li−1 ]. Now, for fixed m and L ∈ [Li , Li−1 ], by equating the first order partial derivatives of JET C DF (S, Q, k, πx , L, m) with respect to Q, S, k, πx to zero, we can obtain
k 1 + k2
=
πx
=
S
=
of
√
√ √ + C(L) + M (πx ) 12 σ L( 1 + k 2 − k) e hb (1 − Ye )2 + 2DY + H(m) s 2hb 1− DM (πx ) β0 π x hb 1 − π0 + Q(1−Y e) S m
hb Q(1 − Ye ) π0 + 2D 2
and
(25) (26)
(27)
(28)
re-
mθBQ(1 − Ye ) D
pro
Q =
v u u 2D A + t
Proposition 3: For fixed m and L ∈ [Li , Li−1 ], the Hessian Matrix for JET C DF (Q, k, πx , S, L, m) is positive definite at the point (Q∗∗ , k ∗∗ , πx∗∗ , S ∗∗ ) obtained from (25) − (28). Proof: The proof is similar to Proposition 1.
urn al P
The optimal value of m can be obtained when
JET C DF (m∗∗ − 1) ≥ JET C DF (m∗∗ ) ≤ JET C DF (m∗∗ + 1)
(29)
The following iterative algorithm is developed to find the optimal values for S, Q, k, πx , L, and m. Algorithm 2 Step 1 Set m = 1.
Step 2 For each Li , i = 0, 1, 2, ..., n, perform 2a to 2e. 2a Set πxi1 = 0, Si1 = 0, and ki1 = 0.
2b Substitute πxi1 , Si1 , and ki1 into (25), and evaluate Qi1 .
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
2c Utilize Qi1 to obtain the value of ki1 , πxi and Si from (26), (27), and (28). 2d Utilize Qi1 determine πxi2 and Si2 from (17)and(18), respectively. 2e Repeat the steps 2b to 2d until no changes occur in the values of Qi , ki , πxi , and Si . Denote the solution by (Q¯i , k¯i , π ¯xi , S¯i ). ∗∗ Step 3 For each i = 0, 1, ..., n, compare πxi with π0 and Si∗∗ with S0 .
20
Journal Pre-proof
3a If π ¯xi < π0 and S¯i < S0 then the solution found in Step 2 is optimal for given Li . Denote this solution by (Q∗∗ , k ∗∗ , πx∗∗ , S ∗∗ ). Go to Step 5. ∗ 3b If π ¯xi ≥ π0 and S¯i < S0 , then for a given Li , set πxi = π0 and evaluate new (Q¯i , k¯i , S¯i ) from (25), (26),
and (28) by the same procedure as described in Step 2. If S¯i < S0 , then the optimal solution is
of
(Q∗∗ , k ∗∗ , πx∗∗ , S ∗∗ ) = (Q¯i , k¯i , π0 , S¯i ) and go to Step 5. Otherwise, go to Step 4.
3c If S¯i ≥ S0 and π ¯xi < π0 , then for a given Li , set S ∗ = S0 and evaluate new (Q¯i , k¯i , π ¯xi ) from (25), (26),
pro
and (27) by the same procedure as presented in Step 2. If π ¯xi < π0 , then the near optimal solution is (Q∗∗ , k ∗∗ , πx∗∗ , S ∗∗ ) = (Q¯i , k¯i , π¯x , S0 ), and go to Step 5. Otherwise, go to Step 4. 3d If π ¯xi ≥ π0 and S¯i ≥ S0 , go to Step 4.
solution (Q∗i , ki∗ ) by using Step 2.
re-
∗ Step 4 For the given Li set πxi = π0 and Si∗ = S0 , utilize (25) and (26) to obtain the corresponding optimal
∗∗ ∗∗ ∗∗ ∗∗ DF ∗∗ ∗∗ ∗∗ ∗∗ Step 5 Find JET C DF (Q∗∗ (Q∗∗ i , ki , πxi , Si , Li , m) and mini=0,1,2,...,n JET C i , ki , πxi , Si , Li , m). ∗∗ ∗∗ ∗∗ ∗∗ DF ∗∗ ∗∗ ∗∗ ∗∗ Step 5a If JET C DF (Q∗∗ (Q∗∗ i , ki , πxi , Si , Li , m), then (m) , k(m) , πx(m) , S(m) , L(m) , m) = mini=0,1,2,...,n JET C
urn al P
∗∗ ∗∗ ∗∗ ∗∗ (Q∗∗ (m) , k(m) , πx(m) , S(m) , L(m) ) is the optimal solution for a fixed m.
Step 6 Set m = m + 1.
∗∗ ∗∗ ∗∗ ∗∗ DF ∗∗ ∗∗ ∗∗ If JET C DF (Q∗∗ (Q∗∗ (m) , k(m) , πx(m) , S(m) , L(m) , m) ≤ JET C (m−1) , k(m−1) , πx(m−1) , S(m−1) , L(m−1)∗∗ , m−
1), then go to Step 6. Otherwise, go to Step 7.
∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ Step 7 Set (Q∗∗ , k ∗∗ , πx∗∗ , S ∗∗ , L∗∗ , m) = (Q∗∗ m−1 , km−1 , πx(m−1) , Sm−1 , Lm−1 , m), then (Q , k , πx , S , L , m)
is the optimal solution.
√ After substituting the value of k ∗∗ and L∗∗ , the reorder point can be obtained as r∗∗ = DL∗∗ +k ∗∗ σ L∗∗ .
5. Numerical example and sensitivity analysis 5.1. Numerical Example
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
This section presents a detailed numerical example to shed light on the coordination policy of the proposed SC. To derive the optimal solutions, all the calculation are done with the help of Mathematica 9.0 software. The following data set is used for analysis, which is similar to the one used in Khan et al. (2014) and Pan et al. (2004): D = 1000 units/year, P = 3200 units/year, A = $25/order, hb = $3/unit/year, hv =$1.3/unit/year, s= 175200, Cs = $0.5/unit, C0 = $0.5/unit, π0 = $150/unit, µ =$200/unit, µ0 =$50/unit, σ = 7 units per week, where 1 year = 52 weeks, θ = 0.1 per dollar per year. Furthermore the lead-time demand has three components with the data shown in Table 2 (see Ouyang et al.(1999)) as well as the summarized lead-time components
21
Journal Pre-proof
information is given in Table 3. Similar to Khan et al. (2014), we assume that the fraction of defectives and the percentage of inspection errors (Type I and Type II), as the following:
f (m2 ) =
1 0.04−0 ,
0 ≤ y ≤ 0.04
0,
otherwise
of
f (m1 ) =
1 0.04−0 ,
0 ≤ m1 ≤ 0.04
0,
otherwise
1 0.04−0 ,
0,
pro
f (y) =
0 ≤ m2 ≤ 0.04 otherwise
According to the distribution functions, the expected value of the defective products and Type I and Type
re-
II errors are E(y) = 0.02, E(m1 ) = 0.02 and E(m2 ) = 0.02. Now, we solve the cases for which the upper bounds of the backorder ratio β0 = 0.2, 0.5, 0.8, and B = 5800. In Salameh and Jaber (2000) and Goyal et al. (2003) model, they considered the setup cost S0 = $400 per production run. Therefore, we solve the Logarithmic investment case for which the initial setup cost S0 = 400. For the three given values of β0 , we use
urn al P
the proposed algorithm 2 to find the optimal solution of the DFA model. Instead of assuming any particular distribution function for lead-time demand, we only consider the DFA for numerical illustration. The results of the numerical study are shown in Table 4.
Table 2: Lead-time components with data (See Ouyang et al. (1999))
Lead-time component i 1 2 3
Normal duration bi (days) 20 20 16
Minimum duration ai (days) 6 6 9
Unit crashing cost ci ($/days) 0.4 1.2 5.0
Table 3: Summarized lead-time data
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Lead-time (week) 8 6 4 3
C(L) 0 5.6 22.4 57.4
From Table 4, it can be concluded that for β0 = 0.2 the minimum total expected annual cost is equal to $3530, which occurs when the lead-time is equal to four weeks, and the lead-time component is i=2. The
optimal buyer’s order quantity and vendor’s setup cost in this case are equal to $325 units and $724 per order respectively, with the optimal safety factor equal to 6.12. The optimal total expected annual cost for β0 = 0.5 and β0 = 0.8 are respectively $3509, and $3486. The results clearly show that the optimal total expected
22
Journal Pre-proof
Table 4: Results of the solution procedure for DFA β0 m
i = 0, Li = 8 πx
Q
k
S
πx
Q
k
i = 2, Li = 4
JET C DF
S
πx
Q
k
i = 3, Li = 3 JET C DF
S
πx
Q
k
JET C DF
0.2 1 306 75.79 550 4.68
3888
299 75.77 536 4.746
3823
295 75.76 530 4.78
3764
306
75.79 550 4.69
3778
2 500 75.65 448 5.20
3716
487 75.63 437 5.27
3645
483 75.62 433 5.29
3585
505
75.65 453 5.17
3708
3 638 75.55 382 5.64
3673
622 75.54 372 5.72
3597
619 75.53 370 5.73
3536
651
75.56 389 5.59
3567
4 745 75.48 334 6.04
3672
727 75.47 326 6.11
3592
of
S
i = 1, Li = 6
JET C DF
724 75.47 325 6.12
3530
766
75.50 344 5.95
3570
3689
812 75.42 292 6.47
3606
811 75.42 291 6.47
3543
862
75.44 309 6.28
3592
3864
296 75.76 531 4.57
3802
293 75.76 526 4.59
3748
304
75.79 546 4.50
3764
2 494 75.64 444 5.01
3690
483 75.62 433 5.07
3623
479 75.62 430 5.09
3566
502
75.65 450 4.97
3592
3 631 75.54 378 5.44
3645
616 75.53 369 5.50
3573
614 75.53 367 5.52
3516
647
75.56 387 5.37
3550
4 737 75.48 330 5.82
3642
720 75.46 323 5.89
3566
719 75.46 322 5.89
3509
761
75.49 341 5.72
3552
pro
5 832 75.43 299 6.39 0.5 1 303 75.78 544 4.51
3658
804 75.42 288 6.24
3579
805 75.42 289 6.23
3521
857
75.44 307 6.04
3573
3840
293 75.76 526 4.38
3781
291 75.75 522 4.40
3730
302
75.78 543 4.31
3749
2 489 75.63 439 4.81
3663
478 75.62 429 4.86
3599
475 75.61 426 4.88
3547
498
75.64 447 4.76
3776
3 624 75.54 373 5.22
3616
610 75.52 365 5.28
3548
609 75.52 364 5.29
3495
642.45 75.55 384 5.15
3533
4 728 75.47 326 5.59
3611
712 75.46 319 5.66
3539
712 75.46 320 5.65
3486
756
75.49 339 5.48
3534
5 812 75.42 292 5.92
3625
795 75.41 285 5.99
3550
798 75.41 286 5.98
3497
851
75.44 305 5.78
3554
re-
5 822 75.42 295 6.16 0.8 1 300 75.78 538 4.33
annual cost decreases if the value of β0 increases. This is, due to the fact that giving more discount attract
urn al P
customers. Table 5 illustrate the summary of Table 4.
Table 5: The optimal solutions for DFA
β0 0.2 0.5 0.8
m? 4 4 4
Investment with price discount (proposed model) S? πx? k? L? Q? JET C DF (.) 724 75.47 6.12 4 325 3530 719 75.46 5.89 4 322 3509 712 75.46 5.65 4 320 3486
Fixed setup cost and no price discount (S = S0 = 400 & πx = π0 = 150) m? k? L? Q? JET C DF (.) 4 6.819 4 276.526 3607.53 4 6.813 4 276.568 3607.07 4 6.807 4 276.609 3606.61
Savings (%) 2.20 2.79 3.44
The total misclassification cost is equal to $1103.25, which is around 31% of the total cost. Among the total misclassification cost, the cost of false acceptance is equal to $83.27, and the cost of falsely rejection is $1019.98. Note that the cost of false acceptance is much higher than the cost of false rejection, because falsely
accepted imperfect quality products may hamper goodwill of the firm. The total misclassification cost mainly depends on three factors: (i) the percentage of defective items supplied by the vendor to the buyer, (ii) the
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
expected value of type-I inspection error, and (iii) the expected value of type-II inspection error. 5.2. Sensitivity Analysis
This subsection discusses the results of sensitivity analysis on some of parameters of the model. It can be observed from table 6, when the ordering cost (A) decreases, the joint expected total cost also decreases. This result is not unexpected because in practice when the buyer controls his ordering cost, the total inventory cost automatically decreases. When the buyer’s or vendor’s holding cost (hb and hv ) increases the joint expected cost also increases. This shows the effect of holding cost on decision-making, and the result
23
Journal Pre-proof
Table 6: Sensitivity analysis on other parameters
C0
µ0
σ
S 684 703 722 739 832 747 682 631 712 712 712 712 712 712 712 712 610 693 770 842
JET C 3436 3470 3502 3534 3103 3367 3600 3813 3481 3492 3501 3512 2874 3282 3690 4098 3131 3418 3688 3945
Parameter
B
hv
Changes 3000 4500 6000 7500 0.5 1 1.5 2 0.25 0.75 1.25 1.75 75 150 225 300 0.005 0.015 0.025 0.035
Cs
µ
E(y)
Q 251 287 325 364 436 354 301 264 320 320 320 320 320 320 320 320 319 319 320 320
of
hb
Q 307 315 324 332 373 335 306 283 320 320 320 320 320 320 320 320 274 311 345 378
pro
A
Changes 10 20 30 40 1.5 2.5 3.5 4.5 0.25 0.75 1.25 1.75 20 40 60 80 2 6 10 14
re-
Parameter
S 290 496 749 1050 972 789 670 587 712 712 712 712 712 712 712 712 721 715 709 703
JET C 3533 3539 3474 3354 3120 3362 3563 3737 3226 3747 4267 4787 3434 3466 3497 3528 3393 3455 3518 3582
urn al P
is widely acceptable also. So, in this case, the small-sized shipment should be beneficial for both players, when their holding cost is high. With the increase in the value of expected percentage of defective items that were supplied by vendor to buyer E(y), the optimal order quantity (Q), and the joint expected total cost (JET C) increases, but the optimal vendor’s set up cost (S) decreases. The parameters C0 , Cs , hb and hv has no impact on the optimal order quantity (Q) and the optimal vendor’s set up cost (S). The vendor’s optimal set up cost is highly sensitive to B, hv and σ. The joint expected total cost is highly sensitive to hb , hv , Cs , µ0 , σ whereas less sensitive to the parameters A, B, µ, C0 and E(y).
5.2.1. Effect of demand and order quantity on joint expected total cost of the entire SC In this part of the paper, we investigate the joint effect of demand and the buyer’s order quantity on the expected total cost of the SC. Figure 3 depicts their joint effect on the total expected cost of the SC. Note that Q, is a decision variable, and Figure 2 illustrates that JET C function is convex in Q. The second partial
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
derivative of JET C with respect to Q can be calculated as follows: √ √ d2 JET C DF D(2(A + C(L) + S/m) + ( 1 + k 2 − k) L(π0 − β0 πx + (β0 πx2 )/π0 )σ) = >0 dQ2 ((1 − E(m1 ) − (1 − E(m1 ) − E(m2 ))E(y))Q3 ) Thus, the analytical method also justifies convexity of the JET C function in Q. Note from Figure 3 that as demand increases, the optimal value of Q increases accordingly. From equation (25), the following result
24
pro
Figure 2: Effect of demand and order quantity on the JET C
can be driven:
re-
P s hv (m − 1) + hb (1 − Ye )2 dQ∗ >0 = dD 2D P s hv H(m) + hb (1 − Ye )2 + 2Dhb P Ye
Which justifies what is shown in figure 2. Thus, we have the following proposition.
urn al P
Proposition 4: The joint expected total cost of the SC is a convex function of Q. A growth in SC demand will increase the buyer’s order quantity, which will eventually increases the total cost of the SC. In order to investigate the effect of demand growth on the SC, different values of demand were tested on the model, while the production rate was kept identical. Table 7, indicates the result of the analysis. Looking at Table 7, it can be understood that under a fixed production rate, varying demand up to four times more will only change the optimal order quantity up to 1.59 times more than its original value. Table 7: Effect of the demand on the optimal order quantity (Q) and the JET C DF
D P Q∗ JET C DF
500 3200 264 2097
750 3200 293 2837
1000 3200 320 3486
1250 3200 345 4082
1500 3200 371 4641
2000 3200 422 5685
3000 3200 534 7582
Figure 3 helps to gain further insights about the model’s behavior for the case where demand and production
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
of
Journal Pre-proof
rates change simultaneously. Before going further into the detail of Figure 3, let us first examine analytically whether or not the JET C is a decreasing function in P . Doing so gives: dJET C DF Dhv (m − 2)Q = dP (2(1 − E(m1 ) − (1 − E(m1 ) − E(m2 ))E(y))P 2
We may expect that 1 > [E(m1 ) + (1 − E(m1 ) − E(m2 ))E(y)] because E(m1 ), E(m2 ) and E(y) are quite small. Under this assumption,
dJET C DF dP
> 0 if m > 2. Otherwise, the JET C DF will become a decreasing function
25
pro
Figure 3: Effect of production rate and demand rate on the JET C DF
re-
of P . Thus, we can conclude the following proposition.
Proposition 5: The joint expected total cost of the SC will be a increasing function of P if (i) 1 > [E(m1 ) + (1 − E(m1 ) − E(m2 ))E(y)] and m > 2
urn al P
(ii)1 < [E(m1 ) + (1 − E(m1 ) − E(m2 ))E(y)] and m < 2 Otherwise, it will become decreasing function of P .
As the capital investment to reduce the vendor’s setup cost depends on the two crucial parameters B and S0 , we now, turn the analysis to investigate the impact of B and S0 on JET C. Figure 4 presents the effect of the parameters B and S0 on the JET C. Furthermore, by taking the first derivative of JET C with respect to B and S0 , we get
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
of
Journal Pre-proof
Figure 4: Effect of B and S0 on the JET C
dJET C DF S0 = τ log dB S
26
Journal Pre-proof
and dJET C DF Bτ = dS0 S0 Clearly, dJET C DF /dS0 > 0, indicating that the JET C increases with an increase in the initial set up cost of
of
the vendor. On the other hand, dJET C DF /dB ≥ 0 if S ≤ S0 , which indicates the same relation between B and JET C. Thus, we can say joint expected total cost will increase with the increasing value of B. Moreover, notice that,
Clearly,
dJET C DF dhv
√
√ √ √ 1 + k 2 πx σ L + π0 (1 − E(m1 ) − (1 − E(m1 ) + E(m2 ))E(y))Q + k + 1 + k 2 σ L 2π0
> 0 and
dJET C DF dhb
> 0. That is, the joint expected total cost of the SC increases with
re-
β0 k − dJET C DF = dhb
pro
((m − 1)P − (m − 2)D)Q dJET C DF = dhv (2(1 − E(m1 ) − (1 − E(m1 ) + E(m2 ))E(y))P )
increasing value of the holding cost of both the buyer and the vendor. Figure 5 depicts the effect of the holding
urn al P
costs on JET C.
Figure 5: Effect of holding costs on JET C
Following the similar approach as DFA and by using algorithm 1, the optimal solution for the case of normal
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
distribution is obtained and presented in table 8. Table 8: The optimal solutions for the normal distribution case
β0
0.2 0.5 0.8
m? 8 8 8
Q? 155.82 155.84 155.87
(Optimal decision ) S? k? πx? 694.70 2.7325 71.9971 694.77 2.7051 75.2246 694.89 2.6751 75.2246
27
L? 8 8 8
JET C N (.) 3006 3004 3003
Journal Pre-proof
6. Conclusion This paper studied a two-echelon supply chain inventory model for a situation where the batches shipped from the vendor to the buyer contain defective items. The buyer uses the continuous review policy to replenish inventories in which delivery lead-time is variable, and the reorder point is stochastic. The buyer is willing to
of
reduce the vendor’s set up cost by making an initial investment in the vendor’s production facilities. The buyer faces a shortage but offers a discount on the backordered items to avoid lost sales. One human factor is brought
pro
into the picture in this study, i.e., inspection error. The inspector at the buyer’s side misclassifies inspected items, which result in committing Type I & Type II inspection errors. The join expected total cost of the SC was formulated for the case under study. Two different scenarios for the lead-time demand are considered. In the first one, the lead-time demand assumed to follow the normal distribution while in the latter one, the DFA is taken. This model proposed an optimal policy for an integrated vendor-buyer supply chain system if the
re-
lead-time is dependent on production lot size, setup time and the way to reduce the setup cost. Using this policy, any industry manager may decide exactly which portion of time within transportation time and setup time they would like to reduce. This study proposed a price discount strategy by the buyer during the shortage period to secure customer demand and the right investment amount to reduce order processing cost. Several
urn al P
insights were derived by analyzing the model numerically and theoretically, such as: (i) when the percentage of defectives in the lot (E(y)) is high, it is recommended for the buyer to procure more and more in order to satisfy his customer’s demand with good quality of items. Since, with increase in the number of defective items yields into higher number of returned back items from the end-user, which will be not good for his business; (ii) Type I error has a more profound effect on the total cost of the SC, which confirms the findings of previous studies such as Khan et al., (2014). Offering a discount by the buyer was found to decrease the total cost of the SC, yet the buyer should decrease the order quantity to realize more savings. The joint effect of setup cost reduction and price discount was found to be influential in decreasing the expected total cost of the supply chain. (iii) the small-sized shipment should be beneficial for both players when their holding cost is high. The model can be extended along with several directions. An immediate extension of this research would be to include the effect of learning in quality to decrease the probability of Type I and Type II errors. If this
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
is the case, a more interesting extension would be to investigate whether learning in screening is worth an investment. One can also extend this work by assuming that the buyer sources from multiple suppliers. In this case, it would be interesting to study whether it is reasonable for the buyer to invest in reducing the setup cost of all suppliers or only the suppliers who have higher setup costs. The interesting part of this extension would be to take into account the limited budget of the buyer for investment that may incite the suppliers to compete for the investment. Another factor that may affect this setting is to study the relation between the level of supplier’s competition and its impact on the buyer’s selling price, and also on the variation of the backorder
28
Journal Pre-proof
threshold that the buyer would like to offer to its customers.
Acknowledgement The research of the first author has been supported by the NRF Singapore (Grant NRF-RSS2016-004). The
of
authors are grateful to the editor-in-chief and two reviewers for their constructive comments and invaluable contributions to enhance the presentation of this paper.
pro
References
Alfares, H.K., & Attia, A.M., 2017. A supply chain model with vendor-managed inventory, consignment, and quality inspection errors. International Journal of Production Research 55, 5706-5727. Annadurai, K., & Uthayakumar, R. 2010. Ordering cost reduction in probabilistic inventory model with
agement, 5(6), 403-410.
re-
controllable lead time and a service level. International Journal of Management Science and Engineering Man-
Arshinder, K., Kanda, A., & Deshmukh, S., 2011. A review on supply chain coordination: coordination mechanisms, managing uncertainty and research directions. Supply chain coordination under uncertainty. Springer,
urn al P
pp. 39-82.
Br¨auer, I., & Buscher, U. 2018, September. Pricing and Ordering Decisions in a JELS-Model for Items with Imperfect Quality. In International Conference on Intelligent Systems in Production Engineering and Maintenance (pp. 244-253). Springer, Cham.
´ Behzadi, G., OSullivan, M. J., Olsen, T. L., & Zhang, A. 2018. Agribusiness supply chain risk management: A review of quantitative decision models. Omega 79, 21-42. Ben-Daya, M. A., & Raouf, A., 1994. Inventory models involving lead time as a decision variable. Journal of the Operational Research Society, 45(5), 579-582.
Ben-Daya, M., & Hariga, M., 2003. Lead-time reduction in a stochastic inventory system with learning consideration. International Journal of Production Research, 41(3), 571-579. Bijvank, M., & Vis, I.F., 2011. Lost-sales inventory theory: A review. European Journal of Operational Re-
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
search 215, 1-13.
Chen, Z., & Sarker, B., 2010. Multi-vendor integrated procurement-production system under shared transportation and just-in-time delivery system. Journal of the Operational Research Society 61, 1654-1666. Chu, P., Yang, K. L., & Chen, P. S. 2005. Improved inventory models with service level and lead time. Computers & Operations Research, 32(2), 285-296. Das, B.C., Das, B., & Mondal, S.K., 2017. An integrated production-inventory model with defective item dependent stochastic credit period. Computers & Industrial Engineering 110, 255-263.
29
Journal Pre-proof
David, I., & Eben-Chaime, M., 2008. How accurate is the integrated vendor continuous model? International Journal of Production Economics 114, 805-810. De Giovanni, P., Karray, & S., Mart´ın-Herr´ an, G. 2019. Vendor Management Inventory with consignment contracts and the benefits of cooperative advertising. European Journal of Operational Research 272(2), 465-480.
of
De Treville, S., Shapiro, R.D., & Hameri, A.-P., 2004. From supply chain to demand chain: the role of lead time reduction in improving demand chain performance. Journal of Operations Management 21, 613-627. Deros, B.M., Peng, C.Y., Ab Rahman, M.N., Ismail, A.R., & Sulong, A.B., 2008. Assessing acceptance sam-
pro
pling application in manufacturing electrical and electronic products. Journal of Achievements in Materials and Manufacturing Engineering 31, 622-628.
Dey, O., 2017. A fuzzy random integrated inventory model with imperfect production under optimal vendor investment. Operational Research, 1-15. https://doi.org/10.1007/s12351-016-0286-1.
re-
Dey, O. 2019. A fuzzy random integrated inventory model with imperfect production under optimal vendor investment. Operational Research, 19(1), 101-115.
Dominguez, R., Cannella, S., Barbosa-P´ovoa, A. P., & Framinan, J. M. 2018. Information sharing in supply chains with heterogeneous retailers. Omega, 79, 116-132
urn al P
Eben-Chaime, M., 2004. The effect of discreteness in vendor-buyer relationships. IIE Transactions 36, 583-589. Ertogral, K., Darwish, M., & Ben-Daya, M., 2007. Production and shipment lot sizing in a vendor supply chain with transportation cost. European Journal of Operational Research 176, 1592-1606. Gallego, G., & Moon, I. 1993. The distribution free newsboy problem: review and extensions. Journal of the Operational Research Society, 44(8), 825-834.
Giri, B., Chakraborty, A., & Maiti, T., 2017. Consignment stock policy with unequal shipments and process unreliability for a two-level supply chain. International Journal of Production Research 55, 2489-2505. Glock, C.H., 2012a. The joint economic lot size problem: A review. International Journal of Production Economics 135, 671-686.
Glock, C.H., 2012b. Lead time reduction strategies in a single-vendor single-buyer integrated inventory model with lot size-dependent lead times and stochastic demand. International Journal of Production Economics 136, 37-44.
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Glock, C.H., Grosse, E.H., Neumann, W.P., & Sgarbossa, F., 2017. Human factors in industrial and logistic system design. Darmstadt Technical University, Department of Business Administration, Economics and Law, Institute for Business Studies (BWL) Goyal, S., 1977. An integrated inventory model for a single supplier-single customer problem. The International Journal of Production Research 15, 107-111. Goyal, S. K., Huang, C. K., & Chen, K. C. 2003. A simple integrated production policy of an imperfect item
30
Journal Pre-proof
for vendor and buyer. Production Planning & Control, 14(7), 596-602. Grosse, E.H., Glock, C.H., Jaber, M.Y., & Neumann, W.P., 2015. Incorporating human factors in order picking planning models: framework and research opportunities. International Journal of Production Research 53, 695-717.
of
Gutgutia, A., & Jha, J. K. 2018. A closed-form solution for the distribution free continuous review integrated inventory model. Operational Research, 18(1), 159-186.
Heydari, J., Govindan, K., & Jafari, A., 2017. Reverse and closed loop supply chain coordination by consider-
pro
ing government role. Transportation Research Part D: Transport and Environment 52, 379-398. Ho, C. H. 2009. A minimax distribution free procedure for an integrated inventory model with defective goods and stochastic lead time demand. International Journal of Information and Management Sciences, 20(1), 161171.
re-
Hoque, M., 2007. An alternative model for integrated vendor inventory under controllable lead time and its heuristic solution. International Journal of Systems Science 38, 501-509. Hoque, M.A., & Goyal, S.K., 2006. A heuristic solution procedure for an integrated inventory system under controllable lead-time with equal or unequal sized batch shipments between a vendor and a buyer. International
urn al P
Journal of Production Economics 102, 217-225.
Hsu, J. T., & Hsu, L. F., 2013a. Two EPQ models with imperfect production processes, inspection errors, planned backorders, and sales returns. Computers & Industrial Engineering, 64(1), 389-402. Hsu, J. T., & Hsu, L. F., 2013b. An EOQ model with imperfect quality items, inspection errors, shortage backordering, and sales returns. International Journal of Production Economics, 143(1), 162-170. Hsu, J.-T., & Hsu, L.-F., 2013c. An integrated vendor inventory model with imperfect items and planned back orders. The International Journal of Advanced Manufacturing Technology 68, 2121-2132. Jauhari, W. A., Sofiana, A., Kurdhi, N. A., & Laksono, P. W. (2016). An integrated inventory model for supplier-manufacturer-retailer system with imperfect quality and inspection errors. International Journal of Logistics Systems and Management, 24(3), 383-407.
Jawad, H., Jaber, M. Y., & Nuwayhid, R. Y. 2018. Improving supply chain sustainability using exergy analysis. European Journal of Operational Research, 269(1), 258-271.
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Kazemi, N., Olugu, E.U., Abdul-Rashid, S.H., & Ghazilla, R.A.B.R., 2015. Development of a fuzzy economic order quantity model for imperfect quality items using the learning effect on fuzzy parameters. Journal of Intelligent & Fuzzy Systems 28, 2377-2389. Khan, M., Jaber, M., & Guiffrida, A., 2012. The effect of human factors on the performance of a two level supply chain. International Journal of Production Research 50, 517-533. Khan, M., Jaber, M.Y., & Ahmad, A.-R., 2014. An integrated supply chain model with errors in quality
31
Journal Pre-proof
inspection and learning in production. Omega 42, 16-24. Khan, M., Ahmad, A.-R., & Hussain, M., 2017a. Integrated decision models for a vendor supply chain with inspection errors and purchase and repair options. The International Journal of Advanced Manufacturing Technology, 1-8. https://doi.org/10.1007/s00170-017-1137-9.
of
Khan, M., Hussain, M., & C´ardenas-Barr´on, L.E., 2017b. Learning and screening errors in an EPQ inventory model for supply chains with stochastic lead time demands. International Journal of Production Research 55, 4816-4832.
pro
Krafft, M., & Mantrala, M.K., 2006. Retailing in the 21st Century. Springer
Kurniawan, R., Zailani, S.H., Iranmanesh, M., & Rajagopal, P., 2017. The effects of vulnerability mitigation strategies on supply chain effectiveness: risk culture as moderator. Supply Chain Management: An International Journal 22, 1-15.
Marketing Management 62, 1-16.
re-
Lambert, D.M., & Enz, M.G., 2017. Issues in supply chain management: Progress and potential. Industrial
Liao, C. J., & Shyu, C. H., 1991. An analytical determination of lead time with normal demand. International Journal of Operations & Production Management, 11(9), 72-78.
urn al P
Lin, Y. J. 2009. An integrated vendor-buyer inventory model with backorder price discount and effective investment to reduce ordering cost. Computers & Industrial Engineering, 56(4), 1597-1606. Lin, H.-J., & Lin, Y.-J., 2014. Supply chain coordination with defective items and quantity discount. International Journal of Systems Science 45, 2529-2538.
Lin, Y.-J., Ouyang, L.-Y., & Dang, Y.-F., 2012. A joint optimal ordering and delivery policy for an integrated supplier inventory model with trade credit and defective items. Applied Mathematics and Computation 218, 7498-7514.
Mandal, P., & Giri, B. C. 2019. A two-warehouse integrated inventory model with imperfect production process under stock-dependent demand and quantity discount offer. International Journal of Systems Science: Operations & Logistics, 6(1), 15-26.
Marchi, B., Ries, J., Zanoni, S., & Glock, C., 2016. A joint economic lot size model with financial collaboration and uncertain investment opportunity. International Journal of Production Economics 176, 170-182.
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Modak, N. M., & Kelle, P. 2019. Managing a dual-channel supply chain under price and delivery-time dependent stochastic demand. European Journal of Operational Research, 272(1), 147-161. Mokhtari, H., & Rezvan, M.T., 2017. A single-supplier, multi-buyer, multi-product VMI production-inventory system under partial backordering. Operational Research, 1-21. https://doi.org/10.1007/s12351-017-0311-z. Moon, I., & Gallego, G. 1994. Distribution free procedures for some inventory models. Journal of the Operational Research Society, 45(6), 651-658.
32
Journal Pre-proof
Moon, I., & Choi, S. 1998. TECHNICAL NOTEA note on lead time and distributional assumptions in continuous review inventory models. Computers & Operations Research, 25(11), 1007-1012. Mou, Q., Cheng, Y., & Liao, H., 2017. A note on lead time reduction strategies in a single-vendor-single-buyer integrated inventory model with lot size-dependent lead times and stochastic demand. International Journal
of
of Production Economics 193, 827-831.
Ouyang, L. Y., Yeh, N. C., & Wu, K. S. 1996. Mixture inventory model with backorders and lost sales for variable lead time. Journal of the Operational Research Society, 47(6), 829-832.
pro
Ouyang, L. Y., & Wu, K. S. 1997. Mixture inventory model involving variable lead time with a service level constraint. Computers & Operations Research, 24(9), 875-882.
Ouyang, L. Y., Chen, C. K., & Chang, H. C. 1999. Lead time and ordering cost reductions in continuous review inventory systems with partial backorders. Journal of the Operational Research Society, 50(12), 1272-1279.
re-
Ouyang, L.-Y., Chen, L.-Y., & Yang, C.-T., 2013. Impacts of collaborative investment and inspection policies on the integrated inventory model with defective items. International Journal of Production Research 51, 5789-5802.
Pal, B., Sana, S.S., & Chaudhuri, K., 2012. Three-layer supply chain production-inventory model for rework-
urn al P
able items. Applied Mathematics and Computation 219, 530-543.
Pal, S., & Mahapatra, G. S. (2017). A manufacturing-oriented supply chain model for imperfect quality with inspection errors, stochastic demand under rework and shortages. Computers & Industrial Engineering, 106, 299-314.
Pan, J.C.-H., & Yang, J.-S., 2002. A study of an integrated inventory with controllable lead time. International Journal of Production Research 40, 1263-1273.
Pan, J. C. H., & Lo, M. C. (2008). The learning effect on setup cost reduction for mixture inventory models with variable lead time. Asia-Pacific Journal of Operational Research, 25(04), 513-529. Pan, J. C. H., Lo, M. C., & Hsiao, Y. C. 2004. Optimal reorder point inventory models with variable lead time and backorder discount considerations. European Journal of Operational Research, 158(2), 488-505. Panwar, A., Nepal, B.P., Jain, R., & Rathore, A.P.S., 2015. On the adoption of lean manufacturing principles in process industries. Production Planning & Control 26, 564-587.
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Porteus, E. L. 1985. Investing in reduced setups in the EOQ model. Management Science, 31(8), 998-1010. Salameh, M. K., & Jaber, M. Y. 2000. Economic production quantity model for items with imperfect quality. International Journal of Production Economics, 64(1-3), 59-64. Sainathan, A., & Groenevelt, H. 2018. Vendor Managed Inventory Contracts-Coordinating the Supply Chain while looking from the Vendor’s Perspective. European Journal of Operational Research 272(1), 249-260. Sarkar, B., & Moon, I. 2014. Improved quality, setup cost reduction, and variable backorder costs in an im-
33
Journal Pre-proof
perfect production process. International Journal of Production Economics, 155, 204-213. Sarkar, B., Chaudhuri, K., & Moon, I., 2015. Manufacturing setup cost reduction and quality improvement for the distribution free continuous-review inventory model with a service level constraint. Journal of Manufacturing Systems 34, 74-82.
of
Sarkar, S., & Giri, B. C. (2018a). Stochastic supply chain model with imperfect production and controllable defective rate. International Journal of Systems Science: Operations & Logistics, 1-14.
Sarkar, S., & Giri, B. C. (2018b). A vendor-buyer integrated inventory system with variable lead time and
pro
uncertain market demand. Operational Research. doi:10.1007/s12351-018-0418-x
Scholten, K., & Fynes, B., 2017. Risk and uncertainty management for sustainable supply chains, Sustainable supply chains. Springer, pp. 413-436.
Shu, H., & Zhou, X., 2014. An optimal policy for a single-vendor and a single-buyer integrated system with
re-
setup cost reduction and process-quality improvement. International Journal of Systems Science 45, 1242-1252. Su, C.-H., 2012. Optimal replenishment policy for an integrated inventory system with defective items and allowable shortage under trade credit. International Journal of Production Economics 139, 247-256. Tiwari, S., Sana, S.S., & Sarkar, S., 2018. Joint economic lot sizing model with stochastic demand and control-
urn al P
lable lead-time by reducing ordering cost and setup cost. Revista de la Real Academia de Ciencias Exactas, F´ısicas y Naturales. Serie A. Matem´aticas 112, 1075-1099. Tiwari, S., Daryanto, Y., & Wee, H.M., 2018. Sustainable inventory management with deteriorating and imperfect quality items considering carbon emission. Journal of Cleaner Production 192, 281-292. Wee, H. M., Yu, J., & Chen, M. C. 2007. Optimal inventory model for items with imperfect quality and shortage backordering. Omega, 35(1), 7-11.
Wee, H. M., & Widyadana, G. A. 2013. A production model for deteriorating items with stochastic preventive maintenance time and rework process with FIFO rule. Omega, 41(6), 941-954. Zavanella, L., & Zanoni, S., 2009. A one-vendor multi-buyer integrated production-inventory model: The consignment stock case. International Journal of Production Economics 118, 225-232.
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
34
Journal Pre-proof
Appendix Proof of proposition 1. For given value of L, the Hessian matrix H is ∂ 2 JET C N (·) ∗ ∂Q∗ ∂πx
∂ 2 JET C N (·) ∂Q∗ ∂k∗
∂ 2 JET C N (·) ∗ ∂Q∗ ∂πx
∂ 2 JET C N (·) ∗2 ∂πx
∂ 2 JET C N (·) ∗ ∂k ∗ ∂πx
∂ 2 JET C N (·) ∂k∗ ∂Q∗
∂ 2 JET C N (·) ∗ ∂k∗ ∂πx
∂ 2 JET C N (·) ∂k∗ 2
∂ 2 JET C N (·) ∂S ∗ ∂Q∗
∂ 2 JET C N (·) ∗ ∂S ∗ ∂πx
∂ 2 JET C N (·) ∂S ∗ ∂k∗
where JET C N (·) = JET C N (Q∗ , k ∗ , πx∗ , L∗ , S ∗ )
= =
∂ 2 JET C N ∂Q∗ ∂k ∗
=
∂ 2 JET C N ∂Q∗ ∂S ∗
=
∂ 2 JET C N ∂k ∗ ∂πx∗
=
∂ 2 JET C N ∂k ∗ ∂S ∗ 2 ∂ JET C N ∂πx∗ ∂S ∗
∂ 2 JET C N (·) ∗ ∂S ∗ ∂πx
∂ 2 JET C N (·) ∂k∗ ∂S ∗
∂ 2 JET C N (·) ∂S ∗ 2
√ S∗ 2D ∗ ∗ LΨ(k )M (π ) + A + F + C(L) + σ x m Q∗ 3 (1 − Ye ) ∗ √ DM (πx ) σ Lφ(k ∗ ) = hb G(πx∗ ) + ∗ Q (1 − Ye ) √ 2Dβ0 = LΨ(k ∗ ) σ Q∗ (1 − Ye )π0 =
urn al P
∂ 2 JET C N ∂k ∗ 2 ∂ 2 JET C N ∂πx∗ 2 ∂ 2 JET C N ∂S ∗ 2 2 ∂ JET C N ∂Q∗ ∂πx∗
re-
∂ 2 JET C N ∂Q∗ 2
∂ 2 JET C N (·) ∂Q∗ ∂S ∗
of
H=
∂ 2 JET C N (·) ∂Q∗ 2
pro
=
=
θB S∗2 √ ∂ 2 JET C N Dσ LΨ(k ∗ ) 2β0 πx∗ = ∗2 β0 − ∂πx∗ ∂Q∗ π0 Q (1 − Ye ) √ ∂ 2 JET C N Dσ LM (πx∗ ) = [1 − Φ(k ∗ )] ∗ ∗ ∂k ∂Q Q∗ 2 (1 − Ye ) ∂ 2 JET C N D =− 2 ∗ ∂S ∗ ∂Q∗ mQ (1 − Ye ) √ ∂ 2 JET C N hb β0 D 2β0 πx∗ = − ∗ − β0 σ L[1 − Φ(k ∗ )] ∗ ∗ ∂πx ∂k π0 Q (1 − Ye ) π0
∂ 2 JET C N =0 ∂S ∗ ∂k ∗ ∂ 2 JET C N =0 ∂S ∗ ∂πx∗
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
The first principal minor of H is √ S∗ 2D ∗ ∗ |H11 | = ∗ 3 A + F + C(L) + σ LΨ(k )M (πx ) + >0 m Q (1 − Ye )
35
Journal Pre-proof
The second principal minor of H is
of
# √ " ∗ ∗ √ 2D S LΨ(k ) 2Dβ σ 0 |H22 | = A + F + C(L) + σ LΨ(k ∗ )M (πx∗ ) + m Q∗ (1 − Ye )π0 Q∗ 3 (1 − Ye ) ( ) 2h √ i2 2β0 πx∗ D2 ∗ − σ LΨ(k − β ) 0 π0 Q∗ 4 (1 − Ye )2 h i2 √ D2 β σ √LΨ(k ∗ ) 0 S∗ 4D2 β0 σ LΨ(k ∗ ) A + F + C(L) + (4 − β0 ) = + m Q∗ 4 π0 (1 − Ye )2 Q∗ 4 (1 − Ye )2
pro
> 0
The third principal minor of H is
re-
h i2 D2 β σ √LΨ(k ∗ ) 4D2 β σ √LΨ(k ∗ ) ∗ 0 S 0 |H33 | = A + F + C(L) + + (4 − β ) 0 4 m Q∗ 4 (1 − Ye )2 Q∗ π0 (1 − Ye )2
+
> = =
urn al P
=
√ √ DM (πx∗ ) 2Dβ0 σ LΨ(k ∗ ) hb G(πx∗ ) + ∗ σ Lφ(k ∗ ) − Q (1 − Ye ) Q∗ (1 − Ye )π0 " # √ 2 Dσ LM (πx∗ ) ∗ [1 − Φ(k )] Q∗ 2 (1 − Ye ) √ √ 4D2 β0 σ LΨ(k ∗ ) S∗ DM (πx∗ ) ∗ A + F + C(L) + G(π ) + Lφ(k ∗ ) h σ b x m Q∗ (1 − Ye ) Q∗ 4 π0 (1 − Ye )2 i2 i2 h √ h √ D2 β0 σ LΨ(k ∗ ) D2 β0 σ LΨ(k ∗ ) √ (4 − β0 ) hb G(πx∗ )σ Lφ(k ∗ ) + (4 − β0 ) Q∗ 4 (1 − Ye )2 Q∗ 4 (1 − Ye )2 " #2 √ √ √ DM (πx∗ )σ Lφ(k ∗ ) 2Dβ0 σ LΨ(k ∗ ) Dσ LM (πx∗ ) − [1 − Φ(k ∗ )] Q∗ (1 − Ye ) Q∗ (1 − Ye )π0 Q∗ 2 (1 − Ye ) i2 h √ √ D3 β0 σ LΨ(k ∗ ) √ β0 σ LΨ(k ∗ ) 2D3 σ 2 L[M (πx∗ )]2 ∗ ∗ ∗ 2 (4 − β0 ) M (πx )σ Lφ(k ) − [1 − Φ(k )] π0 Q∗ 5 (1 − Ye )3 Q∗ 5 (1 − Ye )3 √ i D3 σ 2 Lσ LΨ(k ∗ )M (πx∗ ) h β0 β0 Ψ(k ∗ ) (4 − β0 ) φ(k ∗ ) − 2 [1 − Φ(k ∗ )]2 M (πx∗ ) 5 ∗ 3 π0 Q (1 − Ye ) √ 2 ∗ 3 2 ∗ ∗ D σ Lσ LΨ(k )M (πx ) β0 π x ∗ ∗ ∗ ∗ 2 β0 Ψ(k ) (4 − β0 ) φ(k ) − 2(1 − Φ(k )) (πx − π0 ) + β0 π02 Q∗ 5 (1 − Ye )3 √ D3 σ 2 Lσ LΨ(k ∗ )M (πx∗ ) ∗ ∗ ∗ 2 β Ψ(k ) (4 − β ) φ(k ) − 2β (1 − Φ(k )) 0 0 0 5 Q∗ (1 − Ye )3 √ 3 2 D σ Lσ LΨ(k ∗ )M (πx∗ ) β0 (2 − β0 )Ψ(k ∗ )φ(k ∗ ) + 2β0 Ψ(k ∗ )φ(k ∗ ) − 2β0 (1 − Φ(k ∗ ))2 5 ∗ 3 Q (1 − Ye ) √ 3 2 2D σ Lσ Lβ0 Ψ(k ∗ )M (πx∗ ) ∗ ∗ ∗ 2 Ψ(k )φ(k ) − (1 − Φ(k )) 5 Q∗ (1 − Ye )3
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
> =
>
> 0,
36
Journal Pre-proof
The fourth principal minor determinant of H is
=
> =
of
pro
=
re-
|H44 | =
" √ D θB DM (πx∗ ) 2Dβ0 σ LΨ(k ∗ ) D ∗ |H33 | + hb G(πx ) + ∗ − Q∗ (1 − Ye )π0 mQ∗ 2 (1 − Ye ) Q (1 − Ye ) S ∗2 mQ∗ 2 (1 − Ye ) # 2 √ √ hb β0 D 2β0 πx∗ D ∗ ∗ 2 (σ L[1 − Φ(k )]) σ Lφ(k ) + − ∗ − β0 π0 Q (1 − Ye ) π0 mQ∗ 2 (1 − Ye ) " √ 2 2Dβ0 σ LΨ(k ∗ ) θB D DM (πx∗ ) ∗ − |H | + h G(π ) + 33 b x Q∗ (1 − Ye )π0 Q∗ (1 − Ye ) S ∗2 mQ∗ 2 (1 − Ye ) # 2 √ √ hb β0 D 2β0 πx∗ ∗ ∗ 2 − ∗ − β0 (σ L[1 − Φ(k )]) σ Lφ(k ) + π0 Q (1 − Ye ) π0 2 " 2 θB D hb β0 Dβ0 ∗ |H | + − (2π − β ) 33 0 x π0 π0 Q∗ (1 − Ye ) S ∗2 mQ∗ 2 (1 − Ye ) # √ 2Dβ0 σ 2 LΨ(k ∗ )φ(k ∗ )h ∗ 2 (σ L[1 − Φ(k )]) − ∗ Q (1 − Ye )(1 − Φ(k ∗ ))π0 # 2 " D 2Dβ0 σ 2 LΨ(k ∗ )φ(k ∗ )h θB |H33 | − Q∗ (1 − Ye )(1 − Φ(k ∗ ))π0 S ∗2 mQ∗ 2 (1 − Ye ) # 2 " D θB 2Dβ0 σ 2 LΨ(k ∗ )φ(k ∗ )h |H33 | + Q∗ (1 − Ye )(Φ(k ∗ ) − 1)π0 S ∗2 mQ∗ 2 (1 − Ye )
urn al P
> 0
As the principal minors of the Hessian Matrix are positive, the Hessian Matrix H is positive definite at the point (Q∗ , k ∗ , πx∗ , S ∗ ). Hence the total expected annual cost function is a global minimum at that point.
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
37
*Credit Author Statement
Journal Pre-proof Author Contributions Section Dear editor We inform you the contributor roles: Sunil Tiwari: Conceptualization, Methodology, Investigation, Writing - Original Draft, Writing - Review & Editing, Validation
Nikunja Mohan Modak: Methodology, Validation
of
Nima Kazemi: Conceptualization, Methodology, Writing - Review & Editing
Leopoldo Eduardo Cárdenas-Barrón: Conceptualization, Methodology, Investigation,
Jo
urn al P
re-
Sumon Sarkar: Methodology, Validation
pro
Writing - Original Draft, Writing - Review & Editing, Supervision