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A multiproduct single machine economic production quantity model for an imperfect production system under warehouse construction cost
Int. J. Production Economics 169 (2015) 203–214
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Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe
A multiproduct single machine economic production quantity model for an imperfect production system under warehouse construction cost Seyed Hamid Reza Pasandideh a,1, Seyed Taghi Akhavan Niaki b,2, Amir Hossein Nobil c,3, Leopoldo Eduardo Cárdenas-Barrón d,n a
Department of Industrial Engineering, Faculty of Engineering, Kharazmi University, Tehran, Iran Department of Industrial Engineering, Sharif University of Technology, Tehran, Iran c Faculty of Mechanical and Industrial Engineering, Islamic Azad University, Qazvin Branch, Qazvin, Iran d School of Engineering and Sciences Tecnológico de Monterrey, E. Garza Sada 2501 Sur, C.P. 64849 Monterrey, Nuevo León, Mexico b
art ic l e i nf o
a b s t r a c t
Article history: Received 10 March 2015 Accepted 6 August 2015 Available online 15 August 2015
This paper develops an economic production quantity (EPQ) inventory model for a multiproduct singlemachine lot-sizing problem with nonconforming items including scrap and rework, where reworks are classified into several groups based on failure severity. In this inventory model, shortage is allowed and is backordered. The aim is to determine the optimal period length, the lot size, and the allowable shortage of each product so as the total cost, including setup, production, warehouse construction, holding, shortage, reworking, and disposal is minimized. Besides, the available total budget is scarce and that there is a lower bound on the service level of each product. While the convexity is proved, an exact method is proposed to solve the nonlinear programming problem. A numerical example is provided at the end to demonstrate the applicability of the proposed solution procedure as well as to perform a sensitivity analysis. & 2015 Elsevier B.V. All rights reserved.
Keywords: Inventory control EPQ NLP model Nonconforming items Convex programming Exact solution
1. Introduction Harris (1913) proposed the classical economic order quantity (EOQ) inventory model. It is well-known that this inventory model has assumptions that are not practical in many real-world problems. For instance, it considers that there is no shortage of inventory, the parameters are deterministic, all items are perfect quality. For that reasons, several academicians and researchers have extended it; making it more applicable in real-world inventory problems. Hadley and Whitin (1963) not only provided a summary of the basic EOQ inventory models, but also extended the Harris (1913) inventory model to include shortage. Another extension is due to Abboud and Sfairy (1997) who developed an EOQ inventory model under the effect of time limited free back-orders. They assumed in their research that during a stock-out period the customers would be willing to wait for a limited time to take delivery of their order at no additional charge. In
n
Corresponding author. Tel.: þ 52 81 83284235; fax: þ 52 81 83284153. E-mail addresses: [email protected] (S.H.R. Pasandideh), [email protected] (S.T.A. Niaki), [email protected] (A.H. Nobil), [email protected] (L.E. Cárdenas-Barrón). 1 Tel.: þ98 21 88830891; fax: þ98 21 88329213. 2 Tel.: þ98 21 66165740; fax: þ98 21 66022702. 3 Tel.: þ98 9123390684; fax: þ98 2166022702. http://dx.doi.org/10.1016/j.ijpe.2015.08.004 0925-5273/& 2015 Elsevier B.V. All rights reserved.
this new variant of the EOQ inventory model, a fraction of the customers are typically lost and hence the lost opportunity costs are considered. Another unrealistic assumption of the classical EOQ inventory model is that all items are assumed to have a perfect quality. However, deteriorating items are of great importance in the inventory system of modern organizations. Deterioration is regularly defined as defective, scrapped, obsolescence, change, and perishable. Whitin (1953) was the first researcher who introduced deteriorating goods in terms of becoming old-fashioned after a specified period. A decade later, Ghare and Schrader (1963) presented an EOQ inventory model for exponentially decaying inventories with deteriorating items. Covert and Philip (1973) extended the EOQ inventory model for deteriorating items with survival time with a Weibull distribution. Later, Weiss (1982) analyzed an inventory system with deteriorating items. While all parameters are assumed to be deterministic in his work, the maintenance cost is assumed to be a nonlinear function of time. Eroglu and Ozdemir (2007) proposed an EOQ inventory model with backordered shortages and defective items including scrap items with the rate of θ and imperfect items with the rate of 1 θ which are sold at a discounted selling price as a single lot. Interested readers are referred to Goyal and Giri (2001) who presented a review of deteriorating inventory control literature since 1990s, and Bakker et al. (2012) who provided another review on deteriorating inventory
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literature from 2001 to 2012. Recently, Skouri et al. (2014) developed a single-echelon inventory installation under the EOQ strategy with backorders and rejection of defective supply batches to study the effects of supply quality on system cost. When the items are manufactured in a production system (instead of purchasing) then the so-called economic production quantity (EPQ) inventory model can be used to determine the optimal production quantity. The classical EPQ inventory model was first introduced in 1918 by Taft (1918), in which all the parameters are assumed deterministic and that shortage is not allowed. At the end of the fifties, Rogers (1958) extended the EPQ inventory model to include multiple items. Since then, many extensions of this inventory model have been proposed to relax its constraining assumptions. For instance, Hayek and Salameh (2001) investigated the effect of imperfect quality items on the EPQ inventory model. Their model is derived based on reworking of imperfect quality items produced. Afterwards, Chiu et al. (2007) developed an inventory model that determines the optimal run time in an EPQ model with rework, scrap, and random machine breakdowns. Li et al. (2008) evaluated the effect of a postponement strategy on a manufacturer in a two-echelon supply chain that utilizes EPQ-based inventory models with planned backorders. After, Cárdenas-Barrón (2009) developed an EPQ inventory model with planned backorders to determine the optimal production quantity in a single-stage manufacturing system. Later, Taleizadeh et al. (2010) developed an EPQ model for a multiproduct single-machine production system with shortage, scrapped production rate, and service level constraint. In their model, shortage is permitted to occur in a mixture of lost sales and backorder. The objective of their research is to determine the permissible shortage level, the period length, and the lot sizes for each product such that the total inventory cost is minimized. Barzoki et al. (2011) presented an EPQ inventory model that considers work in process inventory and imperfect quality products. Sarkar and Moon (2011) introduced an EPQ model with random demand and imperfect quality items under the effect of inflation. Their inventory model assumes that the life cycle of an imperfect item follows a Weibull distribution. Moreover, Uthayakumar and Valliathal (2011) proposed an EPQ inventory model for Weibull deteriorating items in a fuzzy environment. They used triangular fuzzy numbers for costs' coefficients of holding, production, setup, shortage and lost opportunity. Afterward, Widyadana and Wee (2012) developed an EPQ model for deteriorating items with rework process and multiple production setups. They considered the (m,1) policy for one cycle in which the facility can produce items in m production setups and one rework setup. Tai (2013) proposed two single-period EPQ inventory models for deteriorating/imperfect items with rework. He assumes that imperfect quality items need to be reworked and that the perfect quality items might be deteriorating. The first inventory model is a single production-rework system and the second model is a system that considers several production plants and one rework plant. Besides, Taleizadeh et al. (2013a) developed a single-machine multiproduct EPQ inventory model with imperfect quality items that are reworked. The goal of their research is to minimize the total cost subject to budget and service level constraints. In addition, Taleizadeh et al. (2013b) presented an EPQ inventory model with random defective items, production capacity limitation and failure in repair. Their inventory model obtains the optimal period lengths, order quantities and backordered quantities. Recently, Dey and Giri (2014) developed a single vendor-single buyer EOQ inventory control model with imperfect production process. They assumed the scrapped rate to be an additional control parameter together with the number of shipments from the vendor to the buyer. Wee et al. (2014) proposed an EPQ model with the rework process and non-synchronized screening. They developed a solution procedure to determine the optimal solution. Moreover, Taleizadeh et al. (2014) derived an EPQ model with scrapped items, rework process, interrupted manufacturing process,
and backordering with the purpose of minimizing the expected total cost. Other relevant and recent works that address defective items and their rework are Wee et al. (2013), Sarkar et al. (2014), Taleizadeh et al. (2015) and Treviño-Garza et al. (2015). In an attempt to propose another variant of the EPQ inventory model that is more applicable in real-world inventory problems, a multiproduct single-machine EPQ inventory model for an imperfect production system is proposed in this paper. In this inventory model, each product is assumed in a combination of imperfect and perfect quality items, where imperfect items include scrap and rework. Also, a construction cost is considered to provide a warehouse space for each product. Moreover, there is a limited budget, and that the shortage is allowed in backorder form. The main objective is to determine the optimal period length, the lot size and the allowable shortage for each product so as the total cost, including setup, production, warehouse construction, holding, shortage, reworking, and disposal is minimized. While the convexity is proved, an exact method based on derivatives is proposed to solve the nonlinear optimization problem. The remainder of this paper is organized as follows. Section 2 defines the problem and the assumptions are explicitly stated too. Section 3 develops the mathematical optimization model. Section 4 presents the solution procedure. Finally, Section 5 solves a numerical example. Finally, Section 6 gives a conclusion and discuses some future research directions.
2. Problem definition and assumptions The single-machine imperfect production problem of this research assumes that perfect as well as imperfect quality products are produced at certain percentages on a single machine. Furthermore, all imperfect products are classified as reworked and scrapped. In this inventory problem, the annual constant production rate of item i in a regular production time, (P i ), is assumed to be greater than to the annual constant demand rate of product i (Di ), where the annual constant imperfect production rate is σ i P i : Mathematically speaking, ð1 σ i Þ P i 4 Di or ai ¼ ð1 σ i Þ P i Di 4 0. In addition, the σ i parameter considers two types of parameters, the proportion of re-workable products (αji ) and the proportion of scrap items (θi ). After termination of the regular production, scrapped items are disposed and the rework process starts with the v1i P i ; v2i P i ; :::; and vm i P i rates, where it is assumed no scrapped item is produced during the rework process. As the rework process of a product usually does not require more time compared to its corresponding regular production time, the rework rate is greater than or equal to the regular production rate for all products, i.e. vji Z1. As a result, the rework production rate vji P i of the product i is greater than or equal to the demand rate (Di ). In other words, vji P i 4 Di or yji ¼ vji P i Di 4 0. Additionally, the following conditions are assumed in to model the multiproduct single machine EPQ inventory problem. 1. All products are manufactured on a single machine. Thus, the production cycle length of all items is equal. In other words, T 1 ¼ T 2 ¼ ::: ¼ T n ¼ T: 2. Shortages are allowed and take the backorder form. 3. The production process includes perfect as well as imperfect quality items. 4. The imperfect quality items include reworked and scrapped items. 5. For each item, there are m types of failures that require rework. 6. The number of re-workable items with percentage rework rate of α1i is less than the quantity of re-workable items with the percentage rework rate of α2i and so on. In other words, m1 αm Z::: Z α1i . i Z αi 7. The rework rates (vji ) are proportions of the regular production rate.
S.H.R. Pasandideh et al. / Int. J. Production Economics 169 (2015) 203–214
205
σ i : the proportion of produced imperfect quality products σ i ¼ θi þα1i þ ::: þ αm i , Di : the demand rate of the i-th product, vji : the ratio of the rework rate of i-th item with j-th defective type to the i-th item production rate (vji Z1) Hm i : the maximum on-hand inventory of i-th item I i : the maximum on-hand inventory of i-th item, based on which the regular production process stops, H ji : the maximum on-hand inventory of the i-th item, based on which the rework process stops for j-th defective type, εi : the safety factor of the total allowable shortage for i-th item, W: the total available budget per period, Si : the machine setup time to produce i-th product, ci : the unit production cost of i-th product, r i : the unit rework cost of i-th product, hi : the unit holding cost of i-th product per unit time, di : the unit disposal cost of scrapped product i, π i : the unit backorder cost of i-th product per unit time, Ai : the setup production cost of i-th item, μi : the space required per unit of i-th item, δi : the ratio of the aisle space to the maximum level of on-hand inventory of i-th item, F i : the total required space of i-th item f i : the unit warehouse construction cost of i-th item per unit space.
8. The items with the percentage rework rate of α2i require less processing time than the ones with the percentage rework rate m1 of α1i and so on. In other words, vm Z ::: Z v1i Z1. i Z vi 9. All the parameters in the EPQ inventory model such as the production rate, the demand rate, and inventory cost are deterministic and known. 10. The total warehouse space of i-th item consists of aisles and storage spaces, where the aisle space for i-th item is considered a percentage, δi , of its required storage space. All production systems with the above conditions can benefit from the modeling and the solution procedure provided in the next sections. The proposed approach in this paper enables production managers to determine the optimal period length, the lot size, and the allowable shortage of each product so as the total cost, including setup, production, warehouse construction, holding, shortage, reworking, and disposal is minimized.
3. Mathematical modeling The aim of this research paper is to extend the two research works of Taleizadeh et al. (2010, 2013a) by taking into account reworkable items of various types that require different rework rates to become perfect quality items. Note that rework is not considered in Taleizadeh et al. (2010), while in Taleizadeh et al. (2013a) no scrap is assumed. In order to develop an inventory model that is even more applicable to real-world inventory problems, machine capacity, limited budget, and service level requirement are imposed. The parameters and the decision variables are defined in Section 3.1. Then, the inventory system under consideration is explained in Section 3.2. The total cost function is derived in Section 3.3. The constraints are presented in Section 3.4. At the end of this section, the mathematical formulation of the problem is proposed in Section 3.5.
3.1.2. Decision variables T: the cycle length (in year). Bi : the total shortage quantity of i-th item in a cycle.
3.1.3. Other notations 3.1. Parameters and decision variables CA: the annual setup cost, CP: the annual production cost, CR: the annual rework cost, CH: the annual holding cost, CB: the annual backorder cost, CD: the annual disposal cost, CC: the total warehouse construction cost, TC: the total cost.
The following parameters, decision variables, and notations are used throughout the paper for product i ; i ¼ 1; 2; :::; n with defective type of j ; j ¼ 1; 2; :::; m. 3.1.1. Parameters N: number of cycles per year, Q i : the production lot size of the i-th product in a cycle, P i : the production rate of the i-th product, θi : the proportion of produced scrapped products, αji : the proportion of produced i-th product with j-th defective type,
3.2. The inventory system Fig. 1 shows the on-hand inventory and shortages of product i þ4 per cycle. In this figure, t 1i and t m are the production uptimes, i
Inventory
Him
yim
Him-1
Hi2
-Di
yi 2
Hi1 yi 1
Ii ai
tim+3 ti1
ti2
ti3
tim+1
tim+2
Bi T
Fig. 1. The inventory of a product in a cycle.
tim+4
time
ai
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þ1 þ2 þ3 t 2i ; t 3i ; :::; t m are rework periods, and t m and t m are produci i i tion downtimes. Based on what was stated in Section 2, these periods in each cycle of product i are easily obtained using Eqs. (1)–(7) as follows:
t 3i
þ1 tm i
¼
ð2Þ Qi αm i m vi P i
ð3Þ
Hm i Di
ð4Þ
þ3 ¼ tm i
Bi Di
ð5Þ
ð15Þ
ð7Þ
e¼1
In addition, the inequality ai ¼ ð1 σ i ÞP i Di 4 0 must hold in þ4 the production uptimes t 1i and t m of a cycle. Besides, it is i assumed that the time at which rework period t 2i starts is the time at which the regular process periods t 1i ends, t 3i after t 2i , and so on. Moreover, scrapped items are disposed at the time the regular process stops. Then, based on Fig. 1 we have: I i ¼ ai Q i =P i Bi ð8Þ
ð16Þ
Nci Q i
Again, based on the joint policy we will have: CP ¼
n X
ci Q i =T
ð17Þ
i¼1
Then, using Eq. (12) we have: n X
ci Di =ð1 θi Þ
ð18Þ
i¼1
ð6Þ
t ei
n X i¼1
CP ¼
Bi ¼ ð1 σ i ÞP i Di
m þ4 X
T¼
Ai =T
3.3.2. Production cost The annual production cost of all items is easily obtained as: CP ¼
þ2 ¼ tm i
þ4 tm i
n X
ð1Þ
H 1i I i Q ¼ α1i 1 i v1i P i Di vi P i
Q ¼ α2i 2 i ; :::; vi P i
CA ¼
i¼1
Q Bi t 1i ¼ i P i ðð1 σ i ÞP i Di Þ t 2i ¼
where Ai is the setup production cost of i-th item. Moreover, based on a joint production policy N ¼ 1=T i ¼ 1=T and we have:
3.3.3. Rework cost For N reworks, each with a quantity of αji Q i and a cost of r i per unit of item i, the annual rework cost is derived as: CR ¼
n X m X i¼1j¼1
Nr i αji Q i ¼
n X m 1X r αj Q T i¼1j¼1 i i i
ð19Þ
m P Using Eq. (12) and noting that αji ¼ σ i θi , the annual rework j¼1 cost is obtained by:
CR ¼
n X
r i ðσ i θi Þ Di =ð1 θi Þ
ð20Þ
i¼1
and H 1i ¼ I i þ α1i y1i
Qi 1 1 ; y ¼ vi P i Di Z 0 v1i P i i
therefore 1 m Qi ¼ Hm þ αm i yi m i vi P i
Hm i
ð9Þ
0 1 m αji yji Q i@X A ¼ Ii þ P i j ¼ 1 vj
ð10Þ
i
Moreover, Appendix A shows that the cycle length is: T¼
ð1 θi ÞQ i Di
ð11Þ
Di T ð1 θi Þ
ð12Þ
Qi ¼
3.3.4. Holding cost Based on Fig. 1, the annual holding cost of the inventory system under consideration is obtained in Appendix B as: " n X hi I i 1 ðI i þ H 1i Þ 2 ðH 1i þ H 2i Þ 3 ðt i Þ þ ðt i Þ ðt Þ þ CH ¼ 2 2 T 2 i i¼1 # ðH m 1 þ H m Hm þ4 i Þ mþ1 ðt i þ ::: þ i Þ þ i ðt m Þ ð21Þ 2 2 i
3.3.5. Backorder cost Using Fig. 1, the annual backorder cost of the inventory system is shown in Eq. (22). CB ¼
i¼1
3.3. Total cost calculation The total costs of all items ðTC Þ is the sum of total setup cost ðCAÞ, total production cost ðCP Þ, total rework cost ðCRÞ, total holding cost ðCH Þ, total backorder cost ðCBÞ, total disposal cost ðCDÞ, and total warehouse construction cost ðCC Þ of all items. In other words TC ¼ CAþ CP þ CR þCH þ CB þCD þ CC
ð13Þ
In the following subsections, each cost in Eq. (13) is derived.
CA ¼
i¼1
NAi
ð14Þ
þ3 þ4 tm þ tm i i
ð22Þ
ð23Þ
3.3.6. Disposal cost For N disposals, each at a cost of di per unit of item i, the annual disposal cost of all items becomes: n X i¼1
3.3.1. Setup cost The annual setup cost of all items is:
2T
Then, from Eqs. (5) and (6) we have: n n X π i Bi Bi Bi 1 X π i P i ð1 σ i Þ 2 ¼ Bi þ CB ¼ 2T i ¼ 1 Di 2T Di ai i¼1
CD ¼
n X
n X π i Bi
Ndi θi Q i ¼
n 1X dθQ Ti¼1 i i i
Then, based on Eq. (12) it becomes:
n X Di di θ i CD ¼ ð1 θ i Þ i¼1
ð24Þ
ð25Þ
S.H.R. Pasandideh et al. / Int. J. Production Economics 169 (2015) 203–214
3.3.7. Warehouse construction cost As the space required per unit and the maximum on-hand inventory of i-th item are μi and H m i , respectively, the required m storage space of product i is μi H m i , of which an additional δi μi H i is needed for its aisle space. Thus, the total required space of i-th item becomes: F i ¼ μi H m i ð1 þ δ i Þ
ð26Þ
Then, based on Eqs. (9)–(12) the total warehouse construction cost is obtained by: CC ¼
n X
f iF i ¼
i¼1
n X
f i μi H m i ð1 þδi Þ
i¼1
2 0 1 3 j j n m X X α y f μ a D ð 1 þ δ Þ f μ D ð 1 þ δ Þ i i i i i i i i iA 4 i @ T f i μi ð1 þ δi ÞBi 5 Tþ i ¼ j P i ð1 θi Þ P i ð1 θ i Þ i¼1 j¼1 v i
ð27Þ Therefore, the total cost is calculated by: TC ¼ CA þCP þ CR þ CB þ CD þ CH þ CC
X
n n n X Ai X Di Di þ ¼ þ ci r i ðσ i θ i Þ T i ¼ 1 1 θi 1 θi i¼1 i¼1 !
n n 2 X X π i P i ð1 σ i Þ ðBi Þ Di di θi þ þ 2Di T 1 θi i¼1 i¼1 " n 1 1 2 X hi I i ðI þ H i Þ 2 ðH i þ H i Þ 3 ðt i Þ þ ðt i Þ ðt 1i Þ þ i þ 2 2 T 2 i¼1 # ðH m 1 þH m Hm þ2 i Þ mþ1 ðt i Þ þ i ðt m Þ þ ::: þ i 2 2 i 2 0 1 n m X αji yji f i μi ai Di ð1 þδi Þ f i μi Di ð1 þ δi Þ@ X 4 AT Tþ þ j P i ð1 θ i Þ P i ð1 θi Þ i¼1 j ¼ 1 vi f i μi ð1 þ δi ÞBi In Appendix C, we show that TC in Eq. (28) reduces to: ! n n n n X X X X ðBi Þ2 Δ1i Δ2i ðT Þ Δ3i ðBi Þ þ Δ4i þ TC ¼ Z ¼ T i¼1 i¼1 i¼1 i¼1 n X 1 þ Δ5i T i¼1
ð28Þ
The capacity of the single machine, the budget and the service level of each item are the four constraints of the model described in the following subsections. 3.4.1. Machine capacity constraint The summation of the total production, rework, and setup times for all products should be smaller than the cycle length. Therefore, t 1i þ t 2i þ t 3i þ ⋯ þ t i
i¼1
þ
n X
Si r T
ð30Þ
i¼1
In Appendix D, we show that Inequality (30) becomes: 0 1 n n P P ðBi Þ Si B C ðP i Di σ i P i Þ B C i¼1 i¼1 Min C ( ! ) T ZB ¼ T B C n m αj P P @ A Di i 1 j P i ð1 θ i Þ 1 þ i¼1
j¼1
vi
Item 1 2 3 4 5 6 7 8 9 10
Pi
Di
σi
α1i
α2i
α3i
θi
v1i
v2i
v3i
6000 6500 7000 7500 8000 8500 9000 9500 10,000 10,500
1000 1100 1200 1300 1400 1500 1600 1700 1800 1900
0.135 0.12 0.111 0.102 0.092 0.082 0.073 0.063 0.054 0.047
0.005 0.005 0.006 0.006 0.005 0.005 0.007 0.007 0.008 0.008
0.05 0.045 0.04 0.036 0.032 0.027 0.021 0.016 0.011 0.009
0.08 0.07 0.065 0.06 0.055 0.05 0.045 0.04 0.035 0.03
0.005 0.005 0.004 0.003 0.003 0.003 0.002 0.002 0.001 0.001
2 2 2 2 2 3 3 3 3 3
3 3 3 3 3 4 4 4 4 4
4 4 4 4 4 5 5 5 5 5
Table 2 General data (continued). Item
Ai ($)
hi
ci
ri
di
πi
μi (ft2)
δi
f i ($/ft2)
Si
εi
1 2 3 4 5 6 7 8 9 10
400 500 600 700 800 900 1000 1100 1200 1300
8 10 12 14 16 18 20 22 24 26
46 44 42 40 38 36 34 32 30 28
25 24 23 22 21 20 19 18 17 16
20 20 20 20 16 16 16 16 12 12
5 8 11 14 17 20 23 26 29 32
2 2 2 3 3 3 4 4 4 5
3 3 3 3 2 2 2 2 4 4
50 55 60 65 70 75 80 85 90 95
0.0004 0.0004 0.0005 0.0005 0.0006 0.0006 0.0007 0.0007 0.0008 0.0008
0.09 0.09 0.085 0.085 0.08 0.08 0.075 0.075 0.07 0.07
3.4.2. Budget constraint As there is a total available budget of W to produce Q i items for each product i with a production cost of ci , to rework αji Q i items for each product i with a rework cost of r i , to dispose θi Q i items for each product i with a disposal cost of di , and to construct the warehouse for all items, the budget constraint is obtained as: 0 1 f i μi ai Di ð1 þ δi Þ ci Q i þ r i α1i Q i þ ⋯ þ r i αm i Q i þ di θ i Q i þ P i ð1 θi Þ T C B 0 1 n B X C j j m B CrW ð32Þ B þ f i μi Di ð1 þ δi Þ@ X αi yi AT f μ ð1 þ δ ÞB C A i i i i P i ð1 θi Þ i¼1@ j v j¼1 i Then, according to Appendix E, Inequality (32) becomes:
3.4. The constraints
mþ1
Table 1 General data.
ð29Þ
where Δoi ; o ¼ 1; 2; 3; 4; 5 are specified for all items in Appendix C.
n X
207
ð31Þ
Tr
W 0 1 0 0 11 9 ¼ T Budget 8 m m > > X X αji yji > j f μ D ð 1 þ δ Þ D > > i i i i i @ci þ di θi þ r i AA > > > αi A þ Pi ð1 θi Þ @ai þ @ = n < ð1 θi Þ j P j¼1 j ¼ 1 vi > > > > i¼1> > Δ3 > > ; : f i μi ð1 þδi Þ2Δi 1 i
ð33Þ
3.4.3. Service level constraint Based on the total shortage quantity and the safety factor of i-th item, Bi and εi , respectively, the service level constraint is derived as: N Bi rεi Di
ð34Þ
Moreover, employing the joint production policy, we have: Bi ð35Þ ¼ T SL TZ i εi D i Then, Appendix F shows that: ! Δ3 ξi ¼ εi 1i Z 0 ; 8 i ¼ 1; 2; :::; n 2Δi Di
ð36Þ
208
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Table 3 Values of ai ; ξi ; ΔOi ; τfor all products. Item
ai
ξi
Δ1i
Δ2i
Δ3i
Δ4i
Δ5i
τ
1 2 3 4 5 6 7 8 9 10
4190 4620 5023 5435 5864 6303 6743 7201.5 7660 8106.5
0.0742838960 0.0801689265 0.0778144128 0.0765810375 0.0747697771 0.0755600600 0.0698936361 0.0704983150 0.0633521412 0.0625025788
12.9799546539379 20.8056277056277 28.5282778386091 36.2720571792513 44.1099356850516 52.0274278914803 59.9730455194276 68.0782921654394 76.2137888018567 84.2737089145488
333798.179619482 403082.388459499 478290.350973602 838341.902929786 729072.528886392 836971.271815920 1265413.29921775 1427114.52388657 2657747.01523145 3696845.43126392
407.988632979713 449.991637031310 491.981826176133 793.970027362086 645.973419043241 692.995959075757 979.981405205473 1041.98788407927 1823.97060567568 2400.97484116908
49597.9899497487 51805.0251256281 53663.8554216867 55074.6238716148 56051.9558676028 56611.8355065196 56723.0460921844 56433.8677354709 55699.0990990991 54675.8758758759
400 500 600 700 800 900 1000 1100 1200 1300
12613894.704
objective function Z are taken with respect to T and Bi in Appendix H. Setting the equations obtained by taking the derivatives, the optimal cycle length and backorder level are given by:
Table 4 Values of Bi . B1 ¼ 0:407971433593460 B2 ¼ 0:280723316550802 B3 ¼ 0:223835193381573 B4 ¼ 0:284110174629736 B5 ¼ 0:190078540956709 B6 ¼ 0:172883367260570 B7 ¼ 0:212088244897210 B8 ¼ 0:198659291969368
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u P 5 u Δi u i u ! T ¼u 2 n u P 2 P ðΔ3i Þ t Δi ¼τ 4Δ1 i¼1
i
B9 ¼ 0:310626962203211 B10 ¼ 0:369785917528195
Bi ¼
Δ3i T 2Δ1i
;
ð39Þ
i
i ¼ 1; 2; :::; n
ð40Þ
3.5. The final EPQ model
In short, the proposed 8-step solution procedure flows.
Using the objective function in Eq. (29) and the constraints in Inequalities (31), (33), and (35), the final EPQ model of the problem under investigation is presented as follows: ! X X X X 1 X B2i Z¼ Δ1i Δ2i ðTÞ Δ3i ðBi Þ þ Δ4i þ Δ5i þ T T i i i i i
Step 1. Initial feasibility If τ 40 ; ξi Z 0 , and ai ¼ ð1 σ i ÞP i Di 4 0 for all products, then go to Step 2. Otherwise, the EPQ inventory model is infeasible, and then go to Step 8. Step 2. Value of the cycle length Calculate T using Eq. (39), Calculate Bi using Eq. (40). Step 3. Secondary feasibility ( !) n n n m αj P P P P ðBi Þ Di i Si and 1 1 þ If are j ðP i Di σ i P i Þ P i ð1 θi Þ
s:t: : T Z T Min T r T Budget T Z T SL i Bi Z 0 T 40
; ;
i ¼ 1; 2; :::; n
i¼1
i ¼ 1; 2; :::; n ð37Þ
4. Solution procedure The objective function of the nonlinear mathematical model presented in (37) is convex. To show this, in the Appendix G we prove that the Hessian matrix of the objective function is positive, i.e.: 2 3 2 3 T 6 P 7 6 7 6 B 7 P 5 A 2Δi 6 76 1 7 6 i i Δ41 Δ42 6 7 Δ4n 7 7 6 B2 7 ¼ i ð38Þ Z0 ; ; :::; H¼6 2 ; 6 7 6 7 2 T T T T76 6 T 7 6 74 ⋮ 5 4 5 Bn Besides, the constraints in Inequalities (31), (33) and (35) all are given in linear forms and hence all are convex as well. As a result, the mathematical model in (37) is a convex nonlinear programming and the local minimum is the global solution. To find the optimal common cycle length and the optimal backorder level of each product, the partial derivatives of the
i¼1
i¼1
j¼1
vi
simultaneously either positive or negative, then go to Step 4. Otherwise, the EPQ inventory model is infeasible. Go to Step 8. Step 4. Checking the shortage level Calculate T SL n o i by Eq. (35), SL SL SL SL Calculate T SL max using T Max ¼ Max T 1 ; T 2 ; :::; T n Step 5. Constraints boundaries Calculate T Min using Eq. (31), calculate T Budget by Eq. (33), n o calculate T nMin using T nMin ¼ Max T Min ; T SL Max calculate T nMax using T nMax ¼ T Budget Step 6. Checking the constraints This step involves four conditions to determine the optimal values of the decision variables as follows. Condition 1. If T nMax o T nMin , then the EPQ inventory model is infeasible and then go to Step 8; else, Condition 2. If T nMax Z T Z T nMin , then T n ¼ T and go to Step 7; else, Condition 3. If T Z T nMax , then T n ¼ T nMax and go to Step 7; else, Condition 4. If T r T nMin , then T n ¼ T nMin and go to Step 7. Step 7. Finding the optimal solution Based on the values of T n and Bni , obtain Q ni using Eq. (12) and Z n by Eq. (28). Step 8. Terminating the proposed solution procedure
S.H.R. Pasandideh et al. / Int. J. Production Economics 169 (2015) 203–214
209
Table 5 Values of T SL i . T SL 1
T SL 2
T SL 3
T SL 4
T SL 5
T SL 6
T SL 7
T SL 8
T SL 9
T SL 10
0.0045330
0.00283558
0.00219446
0.00257113
0.00169712
0.00144069
0.00176740
0.00155811
0.00246529
0.0027803
5. A numerical example
Table 6 Constraints' boundaries.
Consider an imperfect single-machine production system with 10 products, the total available budget of $4,000,000,000 per period. As well, let the general data on each product be the one listed in Tables 1 and 2. Then, the optimal solution is obtained based on the proposed solution procedure as follows. Step 1. Initial feasibility In order to check whether the problem has a feasible solution space, all ai ; ξi ; Δoi and τ are first calculated and they are shown in Table 3. As the conditions for initial feasibility hold, the problem has a feasible solution space and we go to Step 2. Step 2. Value of the cycle length Using Eq. (39), T is obtained as: vP ffiffiffiffiffiffiffiffiffiffiffi u rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u Δ5i t i 8500 ¼ T¼ ¼ 0:0259588149281040 12613894:7041 τ Moreover based on Eq. (40), Bi 's are calculated and they are shown in Table 4. Step 3. Secondary feasibility n n n n m αj o P P P P ðBi Þ Di i Si o 0, jÞ ðP i Di σ i P i Þ o 0 and 1 P i ð1 θi Þð1 þ i¼1
i¼1
i¼1
j¼1
Value
Step 5. Constraints boundaries The constraints' boundaries are calculated and they are given in Table 6 Step 6. Checking the constraints As the third condition holds, i.e. T nMax ZT Z T nMin , we have T n ¼ T ¼ 0:0259588149281040. Step 7. Finding the optimal solution Based on T n ¼ 0:0259588149281040; Bni and Q ni are computed and they are shown in Table 7. Moreover, the minimum annual inventory cost is Z n ¼$ 1,201,220.69088020. Step 8. Terminate the procedure solution The inventory graph of the first item is illustrated in Fig. 2. Similar graphs can be provided for the rest of the items based on T¼ 0.026.
6. Sensitivity analysis This subsection shows the results of some sensitivity analyses performed on important parameters of the problem, including the production rate, the demand rate, the available budget, and the safety factor of each product. In these analyses, the value of each parameter is changed 7 10% and 730% of its original values while
T SL Max
T Budget
T nMin
T nMax
0.000314
0.004533
0.0341848
0.004533
0.042731
maintaining the original values of the other parameters. Table 8 contains the effects of these changes on T Min , T SL , T Budget , T n , and Z n . Besides, the following points are observed based on Table 8.
T Min is insensitive to both changes in the values of the available
vi
then the problem is feasible and go to Step 4 T Min ¼ 0:000314130978925651. Step 4. Checking the shortage level Using Eq. (35), T SL i s are obtained and they are presented in Table 5, based on which n o SL SL SL ¼ T SL T SL max ¼ Max T 1 ; T 2 ; :::; T n 1 ¼ 0:004533:
T Min
budget and the safety factor. It is moderately and highly sensitive to the changes in the values of the annual demand rate and the annual production rate, respectively. T SL is insensitive to the changes in the value of the available budget. It is moderately sensitive to the changes in the value of the annual demand rate and it is highly sensitive to both changes in the values of the annual production rate and the safety factor. T Budget is insensitive to the changes in the value of the safety factor. It is highly sensitive to the changes of all the values of the annual demand rate, the annual production rate, and the safety factor. T n and Z n are slightly sensitive to the changes in the value of the annual production rate. They are highly sensitive to the changes in the value of the annual demand rate. Moreover, they are insensitive to the changes in the values of the available budget and the safety factor.
From a managerial point of view, as the total cost (Z n ) including setup, production, rework, disposal, holding and construction, has the highest sensitivity to the demand rates (Di s), planners or managers should pay more attention in determining the quantities of customers' demand. In other words, wrong demand prediction would impose additional costs to the production system. Moreover, the cycle length is very sensitive to the changes in the production capacity but is not as sensitive as the total cost to the demand rate. In general, the results in Table 8 indicate that the demand rate and the production rate should be close to each other in order to have a desirable production system. Moreover, parameters such as budget, warehouse capacity, and safety factor have little impact on the total cost, but their improper values may lead into infeasible productions.
7. Conclusion and future research In this paper, an imperfect multiproduct single-machine EPQ problem was mathematically formulated. This imperfect production system included two different types of items, scrap and rework, where there were m types of failures that required rework. As the nonlinear programming problem was shown to be convex, an exact solution process involving eight steps was proposed to find the solution that would minimize total inventory cost, including setup, production, warehouse construction, holding, shortage, reworking, and disposal. In addition, the available total budget was scarce and that there was a lower bound on the service
210
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Table 7 Values of Bni ; Q ni .
Bni Q ni
1
2
3
4
5
6
7
8
9
10
0.407971 26.08926
0.280723 28.69818
0.223835 31.27568
0.284111 33.84800
0.190079 36.45169
0.172883 39.05538
0.212088 41.61733
0.198659 44.21842
0.310627 46.77264
0.369786 49.37112
Inventory
21.11
23000
19.11 17.88 17.77
-1000
17000 11000
0.000505
4190 0.00424
0.00007 0.00001
0.407
time
0.02111 0.000087
4190
T=0.026 Fig. 2. The inventory of the first product in a cycle.
Table 8 The effects of parameter changes. Parameters
Pi
Di
W
εi
% changes
þ 30 þ 10 10 30 þ 30 þ 10 10 30 þ 30 þ 10 10 30 þ 30 þ 10 10 30
% changes in T Min
T SL
T Budget
Tn
Zn
65.2309 20.2513 þ 21.7963 þ 80.8799 10.6784 5.4108 þ 9.3815 þ 70.9078 0 0 0 0 0 0 0 0
24.9811 9.9975 þ 12.5130 þ 50.2101 9.2565 3.5899 þ 4.2720 þ 15.7742 0 0 0 0 23.0766 9.09058 þ 11.1115 þ 42.8576
26.8168 10.8848 þ 13.9143 þ 57.8377 17.6024 7.0318 þ 8.7039 þ 33.9575 þ 30.0002 þ 10.0002 9.9998 29.9999 0 0 0 0
2.4887 1.0030 þ 1.2682 þ 5.17611 9.2507 3.5881 þ 4.2694 þ 15.7664 0 0 0 7.8178 0 0 0 0
þ 1.3914 þ 0.5523 0.6827 2.6830 þ 19.2019 þ 6.5771 6.7804 21.0694 0 0 0 þ 0.1629 0 0 0 0
level of each product. Moreover, the warehouse space of i-th item was determined based on its maximum on-hand inventory and its required aisles space. A numerical example was solved in order to demonstrate the applicability of the proposed solution procedure, as well as to perform some sensitivity analyses on the effects of some parameter changes on the solutions obtained. For future research, there are some recommendations as follows:
Some parameters can be considered stochastic, fuzzy, or a
combination of stochastic and fuzzy. The proposed inventory model can be extended to include discrete product quantities to be transported with limited capacity vehicles. Shortage can be considered as a combination of backorder and lost sale. Produced items can be considered perishable items.
Acknowledgments This research was supported by the Tecnológico de Monterrey Research Group in Industrial Engineering and Numerical Methods 0822B01006.
Appendix A. Determining the cycle length From Eq. (7) we have: þ1 þ2 þ3 þ4 T ¼ t 1i þ t 2i þ t 3i þ ::: þ t m þ tm þt m þt m i i i i
T¼
α1 Q i α2 Q i αm Q i H m B Qi Bi Bi þ i1 þ i2 þ ::: þ im þ i þ i þ P i P i Di σ i P i vi P i vi P i vi P i Di Di P i Di σ i P i
S.H.R. Pasandideh et al. / Int. J. Production Economics 169 (2015) 203–214
1 Q Qi 1 i αm ðð1 σ i ÞP i Di ÞQP ii Bi αi vi P i Di v1i P i αm i ðvi P i Di Þvm Pi Q i α1i Q i α2i Q i B i i Qi T ¼ þ 1 þ 2 þ ::: þ m þ þ þ ::: þ þ i P i vi P i vi P i vi P i Di Di Di Di αm Q i ðP Di σ i P i ÞQ i Bi α1i v1i P i Q i α1i Q i α m vm P i Q i α m Qi B Q i α1i Q i α2i Q i þ 1 þ 2 þ ::: þ im þ i þ 1 1 þ :::: þ i mi im þ i P i vi P i vi P i vi P i P i Di Di vi P i Di vi P i Di vi P i Di vi P i Q i α1 Q 1 DP ii θi þ α1i þ α2i þ::: þ αm αm Q i i Qi i T¼ þ þ i þ ::: þ i Pi Di Di Di m m X X Qi αji Q i αji Q i Q i Q i θi Q i Q θi Q i ð1 θi ÞQ i j¼1 j¼1 T¼ þ þ ¼ i ¼ P i Di P i Di Di Di Di Di T¼
Appendix B. Calculating the holding cost Based on Eqs. (8)–(10) we have: I i ¼ ai
Qi Bi Pi
-
ðI i Þ2 ¼
ðai Þ2 Q 2 ðP i Þ
2
þ ðBi Þ2 2
ai Q i Bi Pi
and
H 1i
2
¼ ðI i Þ2 þ
2 2 1
α1i yi v1i
Qi Pi
8 2 9 2 2> > α1i y1i 2 2 ai ai Qi > > > > Q ð Q Þ þ ð B Þ 2 B þ i i i i = < Pi Pi Pi v1i 2α1i y1i þ v1 P Q i I i ¼ 2 1 1 1 1 i > > i α y Qi Qi > > > > þ2ai αi 1yi 2 vi 1 i ; : Pi P i Bi v i
H 2i
2
¼
i
8 2 9 2 2 2> 1 1 > α1i y1i > > Qi Qi > > ai ðQ i Þ2 þ ðBi Þ2 2 ai Q i Bi þ αi 1yi þ 2a > > i > > Pi Pi Pi Pi vi v1i > > > > > > !2 > > = < 2 2 2 2 2 1 1 2 2 2 2 αi yi Q i αi yi α i yi αi yi 1 Qi Qi Qi Hi þ 2 ¼ 2 B þ þ 2a i i Pi Pi Pi v1i v2i v2i > > Pi vi > > > > > > > > 2 2 2 1 1 2 2 > > αi yi αi yi αi yi > > Qi Qi > > > > 1 2 P i Bi þ2 P ; : 2 v2 v v i i
i
i
therefore 8 !2 !2 !2 9 2 X m > > m m 2 X > > αji yji αji yji αji yji Qi Q i Bi X > > ai 2 2 2 ai Qi > > ðQ i Þ þ ðBi Þ 2 P i Q i Bi þ P i þ 2ai 2 > > > > Pi j j j > > P P i i = vi vi vi j¼1 j¼1 j¼1 m 2 < Hi ¼ ! ! 2 j j j 1 > > m k k > > 2X α y X α y > > i i i i > > þ 2 Qi > > > > P j k i > > v ; : v j¼2
i k¼1
i
then " # n 1 X hi I i 1 ðI i þ H 1i Þ 2 ðH 1i þH 2i Þ 3 ðH m þ Hm Hm mþ2 i i Þ mþ1 i CH ¼ ðt i Þ þ ðt i Þ þ ::: þ ðt i ðt Þ þ ðt Þþ Þ 2 2 2 T 2 i 2 i i¼1 Based on Eqs. (1)–(4), we have: 2 3 α1i Q i α2i Q i 1 Qi Bi 1 1 Qi 2 2 Qi I þ α y þ α y þ 2H þ ::::: þ 2I 1 1 2 2 i i i i i v Pi i i v Pi Pi ai vi P i vi P i 7 1X 6 i i 7 CH ¼ h6 m 5 m 2 2T i i 4 m 1 α Q H ð Þ i i m Qi i þ þ αm y þ 2H i i i vm P i vm P i Di i
i
0 0 0 9 8 !1 ! 11 j j 2 m m 2 2 X 2 X > > α α > > j hi a i Di D h D > > i i i i i AðT Þ þ @ @ > > yi AAðT Þ ðT Þ þ @hi ai P i ð1 θi Þ > > 2 P i ð1 θi Þ 2 P i ð1 θi Þ j j > > > > v v > > j ¼ 1 j ¼ 1 i i > > > > > > 0 1 0 1 ! ! ! > > > > j j j j1 m m > > 2 X 2 k X X > > α α y α > > Di hi Di Di ai hi Di i k i i i > > @ A @ A ð T Þ þ y ð T Þ þ þ h ð T Þ > > 2 i i P i ð1 θi Þ 2 P i ð1 θi Þ > > j j k ðP i ð1 θ i ÞÞ > > v vi > > i j ¼ 1 vi k ¼ 1 j¼1 > > > > > > 0 1 0 1 n < = ! ! ! X 2 2 j j j j j 1 m m 2 X k k X X CH ¼ α y α y α y h D h D i i i i i i 1 i i i i @ A @ A > > þ 2 P ð1 θ Þ ðT Þ þ ðT Þ > i¼1> > > i i ðP i ð1 θi ÞÞ2 > > vji vji k ¼ 1 vki > > j¼1 j¼1 > > > > > > 2 2 > > > > hi ðBi Þ hi ðBi Þ > > þ þ > > > > 2Di T 2ai T > > > > 0 1 0 1 > > ! ! > > 2 > > j j j m m > > X X αi yi αi > > h a h h D h D > > i i i i i i i @ A @ Að B i Þ > > B ð B Þ ð Þ ð Þ B i i i > > P i ð1 θi Þ P i ð1 θi Þ P i ð1 θi Þ P i ð1 θi Þ j j > > ; : vi j¼1 j ¼ 1 vi
211
212
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Appendix C. Coefficients of the objective function in Eq. (29)
P i
B2
Δ1i ð Ti Þ ¼
P h hi i
i
π i P i ð1 σ i Þ hi 2Di 2Di þ 2ai þ
ðBi Þ2 T
;
0
Δ1i 4 0 !1 0 A þ @hi
!2 11 3 6 yji AA 7 2 j 6 7 vji 6 7 j ¼ 1 vi j¼1 6 0 7 1 1 0 6 7 ! ! ! ! 2 jX j j j1 j 1 m m 6 7 2 X k k k X X α α y α α y 6 @ 7 Di hi D i i k i i i i i @ A A þ y 6 þ hi P i ð1 θi Þ 7 2 i j ðP i ð1 θi ÞÞ 7 vki vji k ¼ 1 vki X6 X j ¼ 1 vi k ¼ 1 j¼1 6 7 2 6 7ðT Þ; 0 1 0 1 Δi ðTÞ ¼ ! ! 6 7 2 j j j j m m X 7 2X α y 2 α y i i 6 Di ai hi Di i i i i A 1 6 þ @hi Di 7 A þ hi Di þ @ðP ð1 2 6 7 P i ð1 θi Þ 2 2 P i ð1 θi Þ j j θi ÞÞ i 6 7 v v j¼1 j¼1 i i 6 7 0 1 6 7 6 7 j j m X αi yi A 6 f i μi ai Di ð1 þ δi Þ f i μi Di ð1 þ δi Þ@ 7 4 þ P ð1 θ Þ þ P ð1 θ Þ 5 j i i i i v j¼1 i 2
2
hi ai Di 2 P i ð1 θi Þ
þ @hi ai
m 2 X αji
Di P i ð1 θi Þ
m 2 X
Di P i ð1 θi Þ
0 @
αji
8 X <
9 0 ! ! 1 j j j 2 m m = X X α y α h i i i i @ai þDi þ A þf i μi ð1 þ δi Þ ðBi Þ; þ Di Δ3i ðBi Þ ¼ j j : P i ð1 θi Þ ; vi i i j¼1 j ¼ 1 vi X X Di ; Δ4i 4 0 Δ4i ¼ ðci þ r i ðσ i θi Þ þdi θi Þ ð 1 θi Þ i i X 1 X 1 ¼ ; Δ5i 40 Δ5i ðAi Þ T T i i X
Δ2i 4 0
Δ3i 4 0
Appendix D. The maximum capacity of the single machine
n X
n X þ1 t 1i þ t 2i þ t 3i þ ⋯ þ t m Si r T þ i
i¼1
i¼1
Then, based on Eqs. (1)–(4), we have: ! n X αm Q i Q i Bi α1i Q i α2i Q i þ 1 þ 2 þ ⋯ þ im þ Si r T P i ai vi P i vi P i vi P i i¼1
0 0 1 1 n m X X αji @Q i @1 þ A Bi þ Si A rT j Pi ai i¼1 j ¼ 1v i
Finally, using Eq. (12), we have: 0 1 1 j m X α D B i i i @ @1 þ AT þSi A r T j P i ð1 θi Þ ai i¼1 j ¼ 1v n X
0
-
i¼1
i
0
n X
0 0 11 j n m X X α Bi D i i @1 þ AA Si rT @1 j ai P ð1 θ i Þ i¼1 i j ¼ 1v i
1
n n X X Bi B C Si B C ai B C i ¼ 1 i ¼ 1 8 0 19 ¼ T Min C T ZB B C j = n < m B C X X αi Di @1 A @1 þ A j : P i ð1 θ i Þ ; i¼1 j ¼ 1v i
Appendix E. Determining the budget constraint
0 1 1 m X αji yji @ci Q i þ r i α1 Q i þ ⋯ þ r i αm Q i þ di θi Q i þ f i μi ai Di ð1 þδi ÞT þ f i μi Di ð1 þ δi Þ@ AT f i μi ð1 þ δi ÞBi A r W i i j P i ð1 θ i Þ P i ð1 θi Þ i¼1 j ¼ 1 vi n X
n X
0
0
0
m X
1
0
1
1
m αji yji f μ a D ð1 þ δ i Þ f μ D ð1 þ δ i Þ@ X @Q i @ci þ di θi þ r i AT f i μi ð1 þ δi ÞBi A r W Tþ i i i αji A þ i i i i j P i ð1 θi Þ P i ð1 θ i Þ i¼1 j¼1 j ¼ 1 vi
S.H.R. Pasandideh et al. / Int. J. Production Economics 169 (2015) 203–214
Using Eq. (12) and Appendix H, we have: 1 0 1 0 1 j j m m 3 X X α y Δ D T f μ a D ð 1 þ δ Þ f μ D ð 1 þ δ Þ i i i i i i j i iA @ @c þ di θi þ r i @ T f i μi ð1 þ δi Þ i 1 T A r W Tþ i i αi A þ i i j P i ð1 θi Þ P i ð1 θ i Þ ð1 θ i Þ i 2Δi i¼1 j¼1 j¼1 v n X
0
i
Tr
8
W
9 ¼ T Budget 0 0 11 j j m 3 = X α y Δ Di @ j A f i μi Di ð1 þ δi Þ@ i i AA f i μi ð1 þ δi Þ i 1 c þ di θ i þ r i ai þ @ αi þ j :ð1 θi Þ i P i ð1 θ i Þ 2Δi ; i¼1 j¼1 j¼1 v 0
n < X
m X
1
i
Appendix F. Finding the initial feasibility From Eq. (33) and Appendix H, we have: ! Δ3i T εi Di 1 Z0 2Δi
8 n > < T Z0
Δ3i > : ξi ¼ εi 2Δ1i Di Z 0
-
8 i ¼ 1; 2; :::; n
Appendix G. Hessian matrix to prove the convexity of the objective function (Z)
Z¼
X
Δ1i
i
! X X X 1 X ðBi Þ2 Δ2i ðT Þ Δ3i ðBi Þ þ Δ4i þ Δ5i þ T T i i i i 2
6 6 6 6 6 6 H ¼ ½T; B1 ; B2 ; :::; Bn 6 6 6 6 6 4 P ∂Z ¼ ∂T
i
Δ1i ðBi Þ2
P i
∂2 Z ∂2 T
∂2 Z ∂T∂B1
∂2 Z ∂T∂B2
∂2 Z ∂B1 ∂T
∂2 Z ∂2 B1
∂2 Z ∂B1 ∂B2
∂2 Z ∂B2 ∂T
∂2 Z ∂B2 ∂B1
∂2 Z ∂ 2 B2
∂2 Z ∂T∂Bn
⋯
∂2 Z ∂B1 ∂Bn
⋯
∂2 Z ∂B2 ∂Bn
⋮
⋮
⋮
⋮
⋮
∂2 Z ∂Bn ∂T
∂2 Z ∂Bn ∂B1
∂2 Z ∂Bn ∂B2
⋯
Δ5i þ
T2
∂Z 2Δ1i Bi Δ3i ¼ ∂Bi T
⋯
X
Δ2i
i
∂2 Z 2Δ1i ¼ T ∂ 2 Bi
∂2 Z ¼ ∂2 T
∂2 Z ∂2 Bn
3 2
7 7 7 7 7 7 7 7 7 7 7 5
T
3
6B 7 6 1 7 6 7 6 B2 7 6 7 6 ⋮ 7 4 5 Bn
P P 2 Δ1i ðBi Þ2 þ 2 Δ5i i
i
T3
2Δ1i Bi ∂2 Z ∂2 Z ∂2 Z ¼ ¼ ¼ ∂Z∂Bi ∂Z∂T ∂Bi ∂T T2
therefore 2 P 2
6 6 6 6 6 6 H ¼ ½T; B1 ; B2 ; :::; Bn 6 6 6 6 6 6 4
Δ5i þ 2
P
i
T
T2 2Δ12 B2 T2
⋮ 2Δ1n Bn T2
T
2Δ11 B1
2Δ12 B2
2Δ11 T
0
T2
2Δ11 B1
2
3 2 6 7 6 P 7 6 76 62 Δ5i 76 6 i 76 6 76 ; 0; 0; :::; 0 ¼6 76 2 T 6 76 6 74 6 7 4 5
Δ1i ðBi Þ2
i 3
0 ⋮ 0
T2
2Δ12 T
2Δ1n Bn T2
⋯ ⋯
0
⋯
0
⋮
⋮
0
⋯
⋮ 2Δ1n T
3 7 7 7 7 7 7 7 7 7 7 5
2
3 T 6B 7 6 1 7 6 7 6 B2 7 6 7 6 ⋮ 7 4 5 Bn
3
P 5 B1 7 2Δi 7 7 i B2 7 Z0 7¼ T ⋮ 7 5 Bn
Hessian Matrix is P:S:D
213
214
S.H.R. Pasandideh et al. / Int. J. Production Economics 169 (2015) 203–214
Appendix H. Finding the optimal value of the decision variables
∂Z ∂T ∂Z ∂Bi
P
¼ ¼
Δ1i ðBi Þ2
i
P
Δ5i
i
T2
þ
P i
2Δ1i Bi 3 T Δi
P Δ2i ¼ 0
-
T2 ¼
¼0
-
Bi ¼
Δ5i þ
i
P
Pi
Δ1i ðBi Þ2
Δ2i
Δ3i 2Δ1i
;
i
T
thus P T2 ¼
i
Δ5i þ
P i
P i
Δ1i
Δ3i T 2Δ1i
2
P ¼
Δ2i
i
P ðΔ3i T Þ2 Δ5i þ 1 4Δi i P 2 Δi
-
i
T2
X i
Δ2i T 2
3 2 n X Δi i¼1
4Δ1i
¼
X
Δ5i
i
then P i
2
T ¼
P i
Δ2i
Δ5i 2 n P ðΔ3i Þ
i¼1
!
4Δ1i
finally vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u P 5 u Δi u i ! T ¼u u 2 n u P 2 P ðΔ3i Þ t Δi 1 4Δ i
i¼1
Bi ¼
Δ3i T 2Δ1i
i ¼ 1; 2; :::; n
i
References Abboud, N.E., Sfairy, R.G., 1997. Time-limited free back-orders EOQ model. Appl. Math. Model. 21 (1), 21–25. Bakker, M., Riezebos, J., Teunter, R.H., 2012. Review of inventory systems with deterioration since 2001. Eur. J. Oper. Res. 221 (2), 275–284. Barzoki, M.R., Jahanbazi, M., Bijari, M., 2011. Effects of imperfect products on lot sizing with work in process inventory. Appl. Math. Comput. 217 (21), 8328–8336. Cárdenas-Barrón, L.E., 2009. Economic production quantity with rework process at a single-stage manufacturing system with planned backorders. Comput. Ind. Eng. 57 (3), 1105–1113. Chiu, S.W., Wang, S.L., Chiu, Y.S.P., 2007. Determining the optimal run time for EPQ model with scrap, rework, and stochastic breakdowns. Eur. J. Oper. Res. 180 (2), 664–676. Covert, R.P., Philip, G.C., 1973. An EOQ model for items with Weibull distribution deterioration. AIIE Trans. 5 (4), 323–326. Dey, O., Giri, B.C., 2014. Optimal vendor investment for reducing defect rate in a vendor–buyer integrated system with imperfect production process. Int. J. Prod. Econ. 155, 222–228. Eroglu, A., Ozdemir, G., 2007. An economic order quantity model with defective items and shortages. Int. J. Prod. Econ. 106 (2), 544–549. Ghare, P.M., Schrader, G.P., 1963. A model for an exponentially inventory. J. Ind. Eng. 14 (5), 238–243. Goyal, S.K., Giri, B.C., 2001. Recent trends in modeling of deteriorating inventory. Eur. J. Oper. Res. 134 (1), 1–16. Hadley, G., Whitin, T.M., 1963. Analysis of Inventory Systems. Prentice-Hall, Englewood Cliffs, NJ. Harris, F.W., 1913. How many parts to make at once. Fact.: Mag. Manag. 10 (2) 135–136 and 152. Hayek, P.K., Salameh, M.K., 2001. Production lot sizing with the reworking of imperfect quality items produced. Prod. Plan. Control 12 (6), 584–590. Li, J., Wang, S., Cheng, T.C., 2008. Analysis of postponement strategy by EPQ-based models with planned backorders. Omega 36 (5), 777–788. Rogers, J., 1958. A computational approach to the economic lot scheduling problem. Manag. Sci. 4 (3), 264–291. Sarkar, B., Cárdenas-Barrón, L.E., Sarkar, M., Singghi, M.L., 2014. An EPQ inventory model with random defective rate, rework process and backorders for a single stage production system. J. Manuf. Syst. 33 (3), 423–435. Sarkar, B., Moon, I., 2011. An EPQ model with inflation in an imperfect production system. Appl. Math. Comput. 217 (13), 6159–6167. Skouri, K., Konstantaras, I., Lagodimos, A.G., Papachristos, S., 2014. An EOQ model with backorders and rejection of defective supply batches. Int. J. Prod. Econ. 155, 148–154. Taft, E.W., 1918. The most economical production lot. Iron Age 101 (18), 1410–1412. Tai, A.H., 2013. Economic production quantity models for deteriorating/imperfect products and service with rework. Comput. Ind. Eng. 66 (4), 879–888. Taleizadeh, A.A., Kalantari, S.S., Cárdenas-Barrón, L.E., 2015. Determining optimal price, replenishment lot size and number of shipments for an EPQ model with rework and multiple shipments. J. Ind. Manag. Optim. 11 (4), 1059–1071. Taleizadeh, A.A., Cárdenas-Barrón, L.E., Mohammadi, B., 2014. A deterministic multi product single machine EPQ model with backordering, scrapped products, rework and interruption in manufacturing process. Int. J. Prod. Econ. 150, 9–27. Taleizadeh, A.A., Jalali-Naini, S.G., Wee, H.M., Kuo, T.C., 2013a. An imperfect multi-product production system with rework. Sci. Iran. 20 (3), 811–823. Taleizadeh, A.A., Niaki, S.T.A., Najafi, A.A., 2010. Multi-product single-machine production system with stochastic scrapped production rate, partial backordering and service level constraint. J. Comput. Appl. Math. 233 (8), 1834–1849. Taleizadeh, A.A., Wee, H.M., Jalali-Naini, S.G., 2013b. Economic production quantity model with repair failure and limited capacity. Appl. Math. Model. 37 (5), 2765–2774. Treviño-Garza, G., Castillo-Villar, K.K., Cárdenas-Barrón, L.E., 2015. Joint determination of the lot size and number of shipments for a family of integrated vendor-buyer systems considering defective products. Int. J. Syst. Sci. 46 (9), 1705–1716. Uthayakumar, R., Valliathal, M., 2011. Fuzzy economic production quantity model for weibull deteriorating items with ramp type of demand. Int. J. Strateg. Decis. Sci. 2 (3), 55–90. Wee, H.M., Wang, W.T., Kuo, T.C., Cheng, Y.L., Huang, Y.D., 2014. An economic production quantity model with non-synchronized screening and rework. Appl. Math. Comput. 233, 127–138. Wee, H.M., Wang, W.T., Cárdenas-Barrón, L.E., 2013. An alternative analysis and solution procedure for the EPQ model with rework process at a single-stage manufacturing system with planned backorders. Comput. Ind. Eng. J. 64 (2), 748–755. Weiss, H.J., 1982. Economic order quantity models with nonlinear holding costs. Eur. J. Oper. Res. 9 (1), 56–60. Whitin, T.M., 1953. The Theory of Inventory Management. Princeton University Press, Princeton, NJ, USA. Widyadana, G.A., Wee, H.M., 2012. An economic production quantity model for deteriorating items with multiple production setups and rework. Int. J. Prod. Econ. 138 (1), 62–67.
Contents lists available at ScienceDirect
Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe
A multiproduct single machine economic production quantity model for an imperfect production system under warehouse construction cost Seyed Hamid Reza Pasandideh a,1, Seyed Taghi Akhavan Niaki b,2, Amir Hossein Nobil c,3, Leopoldo Eduardo Cárdenas-Barrón d,n a
Department of Industrial Engineering, Faculty of Engineering, Kharazmi University, Tehran, Iran Department of Industrial Engineering, Sharif University of Technology, Tehran, Iran c Faculty of Mechanical and Industrial Engineering, Islamic Azad University, Qazvin Branch, Qazvin, Iran d School of Engineering and Sciences Tecnológico de Monterrey, E. Garza Sada 2501 Sur, C.P. 64849 Monterrey, Nuevo León, Mexico b
art ic l e i nf o
a b s t r a c t
Article history: Received 10 March 2015 Accepted 6 August 2015 Available online 15 August 2015
This paper develops an economic production quantity (EPQ) inventory model for a multiproduct singlemachine lot-sizing problem with nonconforming items including scrap and rework, where reworks are classified into several groups based on failure severity. In this inventory model, shortage is allowed and is backordered. The aim is to determine the optimal period length, the lot size, and the allowable shortage of each product so as the total cost, including setup, production, warehouse construction, holding, shortage, reworking, and disposal is minimized. Besides, the available total budget is scarce and that there is a lower bound on the service level of each product. While the convexity is proved, an exact method is proposed to solve the nonlinear programming problem. A numerical example is provided at the end to demonstrate the applicability of the proposed solution procedure as well as to perform a sensitivity analysis. & 2015 Elsevier B.V. All rights reserved.
Keywords: Inventory control EPQ NLP model Nonconforming items Convex programming Exact solution
1. Introduction Harris (1913) proposed the classical economic order quantity (EOQ) inventory model. It is well-known that this inventory model has assumptions that are not practical in many real-world problems. For instance, it considers that there is no shortage of inventory, the parameters are deterministic, all items are perfect quality. For that reasons, several academicians and researchers have extended it; making it more applicable in real-world inventory problems. Hadley and Whitin (1963) not only provided a summary of the basic EOQ inventory models, but also extended the Harris (1913) inventory model to include shortage. Another extension is due to Abboud and Sfairy (1997) who developed an EOQ inventory model under the effect of time limited free back-orders. They assumed in their research that during a stock-out period the customers would be willing to wait for a limited time to take delivery of their order at no additional charge. In
n
Corresponding author. Tel.: þ 52 81 83284235; fax: þ 52 81 83284153. E-mail addresses: [email protected] (S.H.R. Pasandideh), [email protected] (S.T.A. Niaki), [email protected] (A.H. Nobil), [email protected] (L.E. Cárdenas-Barrón). 1 Tel.: þ98 21 88830891; fax: þ98 21 88329213. 2 Tel.: þ98 21 66165740; fax: þ98 21 66022702. 3 Tel.: þ98 9123390684; fax: þ98 2166022702. http://dx.doi.org/10.1016/j.ijpe.2015.08.004 0925-5273/& 2015 Elsevier B.V. All rights reserved.
this new variant of the EOQ inventory model, a fraction of the customers are typically lost and hence the lost opportunity costs are considered. Another unrealistic assumption of the classical EOQ inventory model is that all items are assumed to have a perfect quality. However, deteriorating items are of great importance in the inventory system of modern organizations. Deterioration is regularly defined as defective, scrapped, obsolescence, change, and perishable. Whitin (1953) was the first researcher who introduced deteriorating goods in terms of becoming old-fashioned after a specified period. A decade later, Ghare and Schrader (1963) presented an EOQ inventory model for exponentially decaying inventories with deteriorating items. Covert and Philip (1973) extended the EOQ inventory model for deteriorating items with survival time with a Weibull distribution. Later, Weiss (1982) analyzed an inventory system with deteriorating items. While all parameters are assumed to be deterministic in his work, the maintenance cost is assumed to be a nonlinear function of time. Eroglu and Ozdemir (2007) proposed an EOQ inventory model with backordered shortages and defective items including scrap items with the rate of θ and imperfect items with the rate of 1 θ which are sold at a discounted selling price as a single lot. Interested readers are referred to Goyal and Giri (2001) who presented a review of deteriorating inventory control literature since 1990s, and Bakker et al. (2012) who provided another review on deteriorating inventory
204
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literature from 2001 to 2012. Recently, Skouri et al. (2014) developed a single-echelon inventory installation under the EOQ strategy with backorders and rejection of defective supply batches to study the effects of supply quality on system cost. When the items are manufactured in a production system (instead of purchasing) then the so-called economic production quantity (EPQ) inventory model can be used to determine the optimal production quantity. The classical EPQ inventory model was first introduced in 1918 by Taft (1918), in which all the parameters are assumed deterministic and that shortage is not allowed. At the end of the fifties, Rogers (1958) extended the EPQ inventory model to include multiple items. Since then, many extensions of this inventory model have been proposed to relax its constraining assumptions. For instance, Hayek and Salameh (2001) investigated the effect of imperfect quality items on the EPQ inventory model. Their model is derived based on reworking of imperfect quality items produced. Afterwards, Chiu et al. (2007) developed an inventory model that determines the optimal run time in an EPQ model with rework, scrap, and random machine breakdowns. Li et al. (2008) evaluated the effect of a postponement strategy on a manufacturer in a two-echelon supply chain that utilizes EPQ-based inventory models with planned backorders. After, Cárdenas-Barrón (2009) developed an EPQ inventory model with planned backorders to determine the optimal production quantity in a single-stage manufacturing system. Later, Taleizadeh et al. (2010) developed an EPQ model for a multiproduct single-machine production system with shortage, scrapped production rate, and service level constraint. In their model, shortage is permitted to occur in a mixture of lost sales and backorder. The objective of their research is to determine the permissible shortage level, the period length, and the lot sizes for each product such that the total inventory cost is minimized. Barzoki et al. (2011) presented an EPQ inventory model that considers work in process inventory and imperfect quality products. Sarkar and Moon (2011) introduced an EPQ model with random demand and imperfect quality items under the effect of inflation. Their inventory model assumes that the life cycle of an imperfect item follows a Weibull distribution. Moreover, Uthayakumar and Valliathal (2011) proposed an EPQ inventory model for Weibull deteriorating items in a fuzzy environment. They used triangular fuzzy numbers for costs' coefficients of holding, production, setup, shortage and lost opportunity. Afterward, Widyadana and Wee (2012) developed an EPQ model for deteriorating items with rework process and multiple production setups. They considered the (m,1) policy for one cycle in which the facility can produce items in m production setups and one rework setup. Tai (2013) proposed two single-period EPQ inventory models for deteriorating/imperfect items with rework. He assumes that imperfect quality items need to be reworked and that the perfect quality items might be deteriorating. The first inventory model is a single production-rework system and the second model is a system that considers several production plants and one rework plant. Besides, Taleizadeh et al. (2013a) developed a single-machine multiproduct EPQ inventory model with imperfect quality items that are reworked. The goal of their research is to minimize the total cost subject to budget and service level constraints. In addition, Taleizadeh et al. (2013b) presented an EPQ inventory model with random defective items, production capacity limitation and failure in repair. Their inventory model obtains the optimal period lengths, order quantities and backordered quantities. Recently, Dey and Giri (2014) developed a single vendor-single buyer EOQ inventory control model with imperfect production process. They assumed the scrapped rate to be an additional control parameter together with the number of shipments from the vendor to the buyer. Wee et al. (2014) proposed an EPQ model with the rework process and non-synchronized screening. They developed a solution procedure to determine the optimal solution. Moreover, Taleizadeh et al. (2014) derived an EPQ model with scrapped items, rework process, interrupted manufacturing process,
and backordering with the purpose of minimizing the expected total cost. Other relevant and recent works that address defective items and their rework are Wee et al. (2013), Sarkar et al. (2014), Taleizadeh et al. (2015) and Treviño-Garza et al. (2015). In an attempt to propose another variant of the EPQ inventory model that is more applicable in real-world inventory problems, a multiproduct single-machine EPQ inventory model for an imperfect production system is proposed in this paper. In this inventory model, each product is assumed in a combination of imperfect and perfect quality items, where imperfect items include scrap and rework. Also, a construction cost is considered to provide a warehouse space for each product. Moreover, there is a limited budget, and that the shortage is allowed in backorder form. The main objective is to determine the optimal period length, the lot size and the allowable shortage for each product so as the total cost, including setup, production, warehouse construction, holding, shortage, reworking, and disposal is minimized. While the convexity is proved, an exact method based on derivatives is proposed to solve the nonlinear optimization problem. The remainder of this paper is organized as follows. Section 2 defines the problem and the assumptions are explicitly stated too. Section 3 develops the mathematical optimization model. Section 4 presents the solution procedure. Finally, Section 5 solves a numerical example. Finally, Section 6 gives a conclusion and discuses some future research directions.
2. Problem definition and assumptions The single-machine imperfect production problem of this research assumes that perfect as well as imperfect quality products are produced at certain percentages on a single machine. Furthermore, all imperfect products are classified as reworked and scrapped. In this inventory problem, the annual constant production rate of item i in a regular production time, (P i ), is assumed to be greater than to the annual constant demand rate of product i (Di ), where the annual constant imperfect production rate is σ i P i : Mathematically speaking, ð1 σ i Þ P i 4 Di or ai ¼ ð1 σ i Þ P i Di 4 0. In addition, the σ i parameter considers two types of parameters, the proportion of re-workable products (αji ) and the proportion of scrap items (θi ). After termination of the regular production, scrapped items are disposed and the rework process starts with the v1i P i ; v2i P i ; :::; and vm i P i rates, where it is assumed no scrapped item is produced during the rework process. As the rework process of a product usually does not require more time compared to its corresponding regular production time, the rework rate is greater than or equal to the regular production rate for all products, i.e. vji Z1. As a result, the rework production rate vji P i of the product i is greater than or equal to the demand rate (Di ). In other words, vji P i 4 Di or yji ¼ vji P i Di 4 0. Additionally, the following conditions are assumed in to model the multiproduct single machine EPQ inventory problem. 1. All products are manufactured on a single machine. Thus, the production cycle length of all items is equal. In other words, T 1 ¼ T 2 ¼ ::: ¼ T n ¼ T: 2. Shortages are allowed and take the backorder form. 3. The production process includes perfect as well as imperfect quality items. 4. The imperfect quality items include reworked and scrapped items. 5. For each item, there are m types of failures that require rework. 6. The number of re-workable items with percentage rework rate of α1i is less than the quantity of re-workable items with the percentage rework rate of α2i and so on. In other words, m1 αm Z::: Z α1i . i Z αi 7. The rework rates (vji ) are proportions of the regular production rate.
S.H.R. Pasandideh et al. / Int. J. Production Economics 169 (2015) 203–214
205
σ i : the proportion of produced imperfect quality products σ i ¼ θi þα1i þ ::: þ αm i , Di : the demand rate of the i-th product, vji : the ratio of the rework rate of i-th item with j-th defective type to the i-th item production rate (vji Z1) Hm i : the maximum on-hand inventory of i-th item I i : the maximum on-hand inventory of i-th item, based on which the regular production process stops, H ji : the maximum on-hand inventory of the i-th item, based on which the rework process stops for j-th defective type, εi : the safety factor of the total allowable shortage for i-th item, W: the total available budget per period, Si : the machine setup time to produce i-th product, ci : the unit production cost of i-th product, r i : the unit rework cost of i-th product, hi : the unit holding cost of i-th product per unit time, di : the unit disposal cost of scrapped product i, π i : the unit backorder cost of i-th product per unit time, Ai : the setup production cost of i-th item, μi : the space required per unit of i-th item, δi : the ratio of the aisle space to the maximum level of on-hand inventory of i-th item, F i : the total required space of i-th item f i : the unit warehouse construction cost of i-th item per unit space.
8. The items with the percentage rework rate of α2i require less processing time than the ones with the percentage rework rate m1 of α1i and so on. In other words, vm Z ::: Z v1i Z1. i Z vi 9. All the parameters in the EPQ inventory model such as the production rate, the demand rate, and inventory cost are deterministic and known. 10. The total warehouse space of i-th item consists of aisles and storage spaces, where the aisle space for i-th item is considered a percentage, δi , of its required storage space. All production systems with the above conditions can benefit from the modeling and the solution procedure provided in the next sections. The proposed approach in this paper enables production managers to determine the optimal period length, the lot size, and the allowable shortage of each product so as the total cost, including setup, production, warehouse construction, holding, shortage, reworking, and disposal is minimized.
3. Mathematical modeling The aim of this research paper is to extend the two research works of Taleizadeh et al. (2010, 2013a) by taking into account reworkable items of various types that require different rework rates to become perfect quality items. Note that rework is not considered in Taleizadeh et al. (2010), while in Taleizadeh et al. (2013a) no scrap is assumed. In order to develop an inventory model that is even more applicable to real-world inventory problems, machine capacity, limited budget, and service level requirement are imposed. The parameters and the decision variables are defined in Section 3.1. Then, the inventory system under consideration is explained in Section 3.2. The total cost function is derived in Section 3.3. The constraints are presented in Section 3.4. At the end of this section, the mathematical formulation of the problem is proposed in Section 3.5.
3.1.2. Decision variables T: the cycle length (in year). Bi : the total shortage quantity of i-th item in a cycle.
3.1.3. Other notations 3.1. Parameters and decision variables CA: the annual setup cost, CP: the annual production cost, CR: the annual rework cost, CH: the annual holding cost, CB: the annual backorder cost, CD: the annual disposal cost, CC: the total warehouse construction cost, TC: the total cost.
The following parameters, decision variables, and notations are used throughout the paper for product i ; i ¼ 1; 2; :::; n with defective type of j ; j ¼ 1; 2; :::; m. 3.1.1. Parameters N: number of cycles per year, Q i : the production lot size of the i-th product in a cycle, P i : the production rate of the i-th product, θi : the proportion of produced scrapped products, αji : the proportion of produced i-th product with j-th defective type,
3.2. The inventory system Fig. 1 shows the on-hand inventory and shortages of product i þ4 per cycle. In this figure, t 1i and t m are the production uptimes, i
Inventory
Him
yim
Him-1
Hi2
-Di
yi 2
Hi1 yi 1
Ii ai
tim+3 ti1
ti2
ti3
tim+1
tim+2
Bi T
Fig. 1. The inventory of a product in a cycle.
tim+4
time
ai
206
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þ1 þ2 þ3 t 2i ; t 3i ; :::; t m are rework periods, and t m and t m are produci i i tion downtimes. Based on what was stated in Section 2, these periods in each cycle of product i are easily obtained using Eqs. (1)–(7) as follows:
t 3i
þ1 tm i
¼
ð2Þ Qi αm i m vi P i
ð3Þ
Hm i Di
ð4Þ
þ3 ¼ tm i
Bi Di
ð5Þ
ð15Þ
ð7Þ
e¼1
In addition, the inequality ai ¼ ð1 σ i ÞP i Di 4 0 must hold in þ4 the production uptimes t 1i and t m of a cycle. Besides, it is i assumed that the time at which rework period t 2i starts is the time at which the regular process periods t 1i ends, t 3i after t 2i , and so on. Moreover, scrapped items are disposed at the time the regular process stops. Then, based on Fig. 1 we have: I i ¼ ai Q i =P i Bi ð8Þ
ð16Þ
Nci Q i
Again, based on the joint policy we will have: CP ¼
n X
ci Q i =T
ð17Þ
i¼1
Then, using Eq. (12) we have: n X
ci Di =ð1 θi Þ
ð18Þ
i¼1
ð6Þ
t ei
n X i¼1
CP ¼
Bi ¼ ð1 σ i ÞP i Di
m þ4 X
T¼
Ai =T
3.3.2. Production cost The annual production cost of all items is easily obtained as: CP ¼
þ2 ¼ tm i
þ4 tm i
n X
ð1Þ
H 1i I i Q ¼ α1i 1 i v1i P i Di vi P i
Q ¼ α2i 2 i ; :::; vi P i
CA ¼
i¼1
Q Bi t 1i ¼ i P i ðð1 σ i ÞP i Di Þ t 2i ¼
where Ai is the setup production cost of i-th item. Moreover, based on a joint production policy N ¼ 1=T i ¼ 1=T and we have:
3.3.3. Rework cost For N reworks, each with a quantity of αji Q i and a cost of r i per unit of item i, the annual rework cost is derived as: CR ¼
n X m X i¼1j¼1
Nr i αji Q i ¼
n X m 1X r αj Q T i¼1j¼1 i i i
ð19Þ
m P Using Eq. (12) and noting that αji ¼ σ i θi , the annual rework j¼1 cost is obtained by:
CR ¼
n X
r i ðσ i θi Þ Di =ð1 θi Þ
ð20Þ
i¼1
and H 1i ¼ I i þ α1i y1i
Qi 1 1 ; y ¼ vi P i Di Z 0 v1i P i i
therefore 1 m Qi ¼ Hm þ αm i yi m i vi P i
Hm i
ð9Þ
0 1 m αji yji Q i@X A ¼ Ii þ P i j ¼ 1 vj
ð10Þ
i
Moreover, Appendix A shows that the cycle length is: T¼
ð1 θi ÞQ i Di
ð11Þ
Di T ð1 θi Þ
ð12Þ
Qi ¼
3.3.4. Holding cost Based on Fig. 1, the annual holding cost of the inventory system under consideration is obtained in Appendix B as: " n X hi I i 1 ðI i þ H 1i Þ 2 ðH 1i þ H 2i Þ 3 ðt i Þ þ ðt i Þ ðt Þ þ CH ¼ 2 2 T 2 i i¼1 # ðH m 1 þ H m Hm þ4 i Þ mþ1 ðt i þ ::: þ i Þ þ i ðt m Þ ð21Þ 2 2 i
3.3.5. Backorder cost Using Fig. 1, the annual backorder cost of the inventory system is shown in Eq. (22). CB ¼
i¼1
3.3. Total cost calculation The total costs of all items ðTC Þ is the sum of total setup cost ðCAÞ, total production cost ðCP Þ, total rework cost ðCRÞ, total holding cost ðCH Þ, total backorder cost ðCBÞ, total disposal cost ðCDÞ, and total warehouse construction cost ðCC Þ of all items. In other words TC ¼ CAþ CP þ CR þCH þ CB þCD þ CC
ð13Þ
In the following subsections, each cost in Eq. (13) is derived.
CA ¼
i¼1
NAi
ð14Þ
þ3 þ4 tm þ tm i i
ð22Þ
ð23Þ
3.3.6. Disposal cost For N disposals, each at a cost of di per unit of item i, the annual disposal cost of all items becomes: n X i¼1
3.3.1. Setup cost The annual setup cost of all items is:
2T
Then, from Eqs. (5) and (6) we have: n n X π i Bi Bi Bi 1 X π i P i ð1 σ i Þ 2 ¼ Bi þ CB ¼ 2T i ¼ 1 Di 2T Di ai i¼1
CD ¼
n X
n X π i Bi
Ndi θi Q i ¼
n 1X dθQ Ti¼1 i i i
Then, based on Eq. (12) it becomes:
n X Di di θ i CD ¼ ð1 θ i Þ i¼1
ð24Þ
ð25Þ
S.H.R. Pasandideh et al. / Int. J. Production Economics 169 (2015) 203–214
3.3.7. Warehouse construction cost As the space required per unit and the maximum on-hand inventory of i-th item are μi and H m i , respectively, the required m storage space of product i is μi H m i , of which an additional δi μi H i is needed for its aisle space. Thus, the total required space of i-th item becomes: F i ¼ μi H m i ð1 þ δ i Þ
ð26Þ
Then, based on Eqs. (9)–(12) the total warehouse construction cost is obtained by: CC ¼
n X
f iF i ¼
i¼1
n X
f i μi H m i ð1 þδi Þ
i¼1
2 0 1 3 j j n m X X α y f μ a D ð 1 þ δ Þ f μ D ð 1 þ δ Þ i i i i i i i i iA 4 i @ T f i μi ð1 þ δi ÞBi 5 Tþ i ¼ j P i ð1 θi Þ P i ð1 θ i Þ i¼1 j¼1 v i
ð27Þ Therefore, the total cost is calculated by: TC ¼ CA þCP þ CR þ CB þ CD þ CH þ CC
X
n n n X Ai X Di Di þ ¼ þ ci r i ðσ i θ i Þ T i ¼ 1 1 θi 1 θi i¼1 i¼1 !
n n 2 X X π i P i ð1 σ i Þ ðBi Þ Di di θi þ þ 2Di T 1 θi i¼1 i¼1 " n 1 1 2 X hi I i ðI þ H i Þ 2 ðH i þ H i Þ 3 ðt i Þ þ ðt i Þ ðt 1i Þ þ i þ 2 2 T 2 i¼1 # ðH m 1 þH m Hm þ2 i Þ mþ1 ðt i Þ þ i ðt m Þ þ ::: þ i 2 2 i 2 0 1 n m X αji yji f i μi ai Di ð1 þδi Þ f i μi Di ð1 þ δi Þ@ X 4 AT Tþ þ j P i ð1 θ i Þ P i ð1 θi Þ i¼1 j ¼ 1 vi f i μi ð1 þ δi ÞBi In Appendix C, we show that TC in Eq. (28) reduces to: ! n n n n X X X X ðBi Þ2 Δ1i Δ2i ðT Þ Δ3i ðBi Þ þ Δ4i þ TC ¼ Z ¼ T i¼1 i¼1 i¼1 i¼1 n X 1 þ Δ5i T i¼1
ð28Þ
The capacity of the single machine, the budget and the service level of each item are the four constraints of the model described in the following subsections. 3.4.1. Machine capacity constraint The summation of the total production, rework, and setup times for all products should be smaller than the cycle length. Therefore, t 1i þ t 2i þ t 3i þ ⋯ þ t i
i¼1
þ
n X
Si r T
ð30Þ
i¼1
In Appendix D, we show that Inequality (30) becomes: 0 1 n n P P ðBi Þ Si B C ðP i Di σ i P i Þ B C i¼1 i¼1 Min C ( ! ) T ZB ¼ T B C n m αj P P @ A Di i 1 j P i ð1 θ i Þ 1 þ i¼1
j¼1
vi
Item 1 2 3 4 5 6 7 8 9 10
Pi
Di
σi
α1i
α2i
α3i
θi
v1i
v2i
v3i
6000 6500 7000 7500 8000 8500 9000 9500 10,000 10,500
1000 1100 1200 1300 1400 1500 1600 1700 1800 1900
0.135 0.12 0.111 0.102 0.092 0.082 0.073 0.063 0.054 0.047
0.005 0.005 0.006 0.006 0.005 0.005 0.007 0.007 0.008 0.008
0.05 0.045 0.04 0.036 0.032 0.027 0.021 0.016 0.011 0.009
0.08 0.07 0.065 0.06 0.055 0.05 0.045 0.04 0.035 0.03
0.005 0.005 0.004 0.003 0.003 0.003 0.002 0.002 0.001 0.001
2 2 2 2 2 3 3 3 3 3
3 3 3 3 3 4 4 4 4 4
4 4 4 4 4 5 5 5 5 5
Table 2 General data (continued). Item
Ai ($)
hi
ci
ri
di
πi
μi (ft2)
δi
f i ($/ft2)
Si
εi
1 2 3 4 5 6 7 8 9 10
400 500 600 700 800 900 1000 1100 1200 1300
8 10 12 14 16 18 20 22 24 26
46 44 42 40 38 36 34 32 30 28
25 24 23 22 21 20 19 18 17 16
20 20 20 20 16 16 16 16 12 12
5 8 11 14 17 20 23 26 29 32
2 2 2 3 3 3 4 4 4 5
3 3 3 3 2 2 2 2 4 4
50 55 60 65 70 75 80 85 90 95
0.0004 0.0004 0.0005 0.0005 0.0006 0.0006 0.0007 0.0007 0.0008 0.0008
0.09 0.09 0.085 0.085 0.08 0.08 0.075 0.075 0.07 0.07
3.4.2. Budget constraint As there is a total available budget of W to produce Q i items for each product i with a production cost of ci , to rework αji Q i items for each product i with a rework cost of r i , to dispose θi Q i items for each product i with a disposal cost of di , and to construct the warehouse for all items, the budget constraint is obtained as: 0 1 f i μi ai Di ð1 þ δi Þ ci Q i þ r i α1i Q i þ ⋯ þ r i αm i Q i þ di θ i Q i þ P i ð1 θi Þ T C B 0 1 n B X C j j m B CrW ð32Þ B þ f i μi Di ð1 þ δi Þ@ X αi yi AT f μ ð1 þ δ ÞB C A i i i i P i ð1 θi Þ i¼1@ j v j¼1 i Then, according to Appendix E, Inequality (32) becomes:
3.4. The constraints
mþ1
Table 1 General data.
ð29Þ
where Δoi ; o ¼ 1; 2; 3; 4; 5 are specified for all items in Appendix C.
n X
207
ð31Þ
Tr
W 0 1 0 0 11 9 ¼ T Budget 8 m m > > X X αji yji > j f μ D ð 1 þ δ Þ D > > i i i i i @ci þ di θi þ r i AA > > > αi A þ Pi ð1 θi Þ @ai þ @ = n < ð1 θi Þ j P j¼1 j ¼ 1 vi > > > > i¼1> > Δ3 > > ; : f i μi ð1 þδi Þ2Δi 1 i
ð33Þ
3.4.3. Service level constraint Based on the total shortage quantity and the safety factor of i-th item, Bi and εi , respectively, the service level constraint is derived as: N Bi rεi Di
ð34Þ
Moreover, employing the joint production policy, we have: Bi ð35Þ ¼ T SL TZ i εi D i Then, Appendix F shows that: ! Δ3 ξi ¼ εi 1i Z 0 ; 8 i ¼ 1; 2; :::; n 2Δi Di
ð36Þ
208
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Table 3 Values of ai ; ξi ; ΔOi ; τfor all products. Item
ai
ξi
Δ1i
Δ2i
Δ3i
Δ4i
Δ5i
τ
1 2 3 4 5 6 7 8 9 10
4190 4620 5023 5435 5864 6303 6743 7201.5 7660 8106.5
0.0742838960 0.0801689265 0.0778144128 0.0765810375 0.0747697771 0.0755600600 0.0698936361 0.0704983150 0.0633521412 0.0625025788
12.9799546539379 20.8056277056277 28.5282778386091 36.2720571792513 44.1099356850516 52.0274278914803 59.9730455194276 68.0782921654394 76.2137888018567 84.2737089145488
333798.179619482 403082.388459499 478290.350973602 838341.902929786 729072.528886392 836971.271815920 1265413.29921775 1427114.52388657 2657747.01523145 3696845.43126392
407.988632979713 449.991637031310 491.981826176133 793.970027362086 645.973419043241 692.995959075757 979.981405205473 1041.98788407927 1823.97060567568 2400.97484116908
49597.9899497487 51805.0251256281 53663.8554216867 55074.6238716148 56051.9558676028 56611.8355065196 56723.0460921844 56433.8677354709 55699.0990990991 54675.8758758759
400 500 600 700 800 900 1000 1100 1200 1300
12613894.704
objective function Z are taken with respect to T and Bi in Appendix H. Setting the equations obtained by taking the derivatives, the optimal cycle length and backorder level are given by:
Table 4 Values of Bi . B1 ¼ 0:407971433593460 B2 ¼ 0:280723316550802 B3 ¼ 0:223835193381573 B4 ¼ 0:284110174629736 B5 ¼ 0:190078540956709 B6 ¼ 0:172883367260570 B7 ¼ 0:212088244897210 B8 ¼ 0:198659291969368
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u P 5 u Δi u i u ! T ¼u 2 n u P 2 P ðΔ3i Þ t Δi ¼τ 4Δ1 i¼1
i
B9 ¼ 0:310626962203211 B10 ¼ 0:369785917528195
Bi ¼
Δ3i T 2Δ1i
;
ð39Þ
i
i ¼ 1; 2; :::; n
ð40Þ
3.5. The final EPQ model
In short, the proposed 8-step solution procedure flows.
Using the objective function in Eq. (29) and the constraints in Inequalities (31), (33), and (35), the final EPQ model of the problem under investigation is presented as follows: ! X X X X 1 X B2i Z¼ Δ1i Δ2i ðTÞ Δ3i ðBi Þ þ Δ4i þ Δ5i þ T T i i i i i
Step 1. Initial feasibility If τ 40 ; ξi Z 0 , and ai ¼ ð1 σ i ÞP i Di 4 0 for all products, then go to Step 2. Otherwise, the EPQ inventory model is infeasible, and then go to Step 8. Step 2. Value of the cycle length Calculate T using Eq. (39), Calculate Bi using Eq. (40). Step 3. Secondary feasibility ( !) n n n m αj P P P P ðBi Þ Di i Si and 1 1 þ If are j ðP i Di σ i P i Þ P i ð1 θi Þ
s:t: : T Z T Min T r T Budget T Z T SL i Bi Z 0 T 40
; ;
i ¼ 1; 2; :::; n
i¼1
i ¼ 1; 2; :::; n ð37Þ
4. Solution procedure The objective function of the nonlinear mathematical model presented in (37) is convex. To show this, in the Appendix G we prove that the Hessian matrix of the objective function is positive, i.e.: 2 3 2 3 T 6 P 7 6 7 6 B 7 P 5 A 2Δi 6 76 1 7 6 i i Δ41 Δ42 6 7 Δ4n 7 7 6 B2 7 ¼ i ð38Þ Z0 ; ; :::; H¼6 2 ; 6 7 6 7 2 T T T T76 6 T 7 6 74 ⋮ 5 4 5 Bn Besides, the constraints in Inequalities (31), (33) and (35) all are given in linear forms and hence all are convex as well. As a result, the mathematical model in (37) is a convex nonlinear programming and the local minimum is the global solution. To find the optimal common cycle length and the optimal backorder level of each product, the partial derivatives of the
i¼1
i¼1
j¼1
vi
simultaneously either positive or negative, then go to Step 4. Otherwise, the EPQ inventory model is infeasible. Go to Step 8. Step 4. Checking the shortage level Calculate T SL n o i by Eq. (35), SL SL SL SL Calculate T SL max using T Max ¼ Max T 1 ; T 2 ; :::; T n Step 5. Constraints boundaries Calculate T Min using Eq. (31), calculate T Budget by Eq. (33), n o calculate T nMin using T nMin ¼ Max T Min ; T SL Max calculate T nMax using T nMax ¼ T Budget Step 6. Checking the constraints This step involves four conditions to determine the optimal values of the decision variables as follows. Condition 1. If T nMax o T nMin , then the EPQ inventory model is infeasible and then go to Step 8; else, Condition 2. If T nMax Z T Z T nMin , then T n ¼ T and go to Step 7; else, Condition 3. If T Z T nMax , then T n ¼ T nMax and go to Step 7; else, Condition 4. If T r T nMin , then T n ¼ T nMin and go to Step 7. Step 7. Finding the optimal solution Based on the values of T n and Bni , obtain Q ni using Eq. (12) and Z n by Eq. (28). Step 8. Terminating the proposed solution procedure
S.H.R. Pasandideh et al. / Int. J. Production Economics 169 (2015) 203–214
209
Table 5 Values of T SL i . T SL 1
T SL 2
T SL 3
T SL 4
T SL 5
T SL 6
T SL 7
T SL 8
T SL 9
T SL 10
0.0045330
0.00283558
0.00219446
0.00257113
0.00169712
0.00144069
0.00176740
0.00155811
0.00246529
0.0027803
5. A numerical example
Table 6 Constraints' boundaries.
Consider an imperfect single-machine production system with 10 products, the total available budget of $4,000,000,000 per period. As well, let the general data on each product be the one listed in Tables 1 and 2. Then, the optimal solution is obtained based on the proposed solution procedure as follows. Step 1. Initial feasibility In order to check whether the problem has a feasible solution space, all ai ; ξi ; Δoi and τ are first calculated and they are shown in Table 3. As the conditions for initial feasibility hold, the problem has a feasible solution space and we go to Step 2. Step 2. Value of the cycle length Using Eq. (39), T is obtained as: vP ffiffiffiffiffiffiffiffiffiffiffi u rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u Δ5i t i 8500 ¼ T¼ ¼ 0:0259588149281040 12613894:7041 τ Moreover based on Eq. (40), Bi 's are calculated and they are shown in Table 4. Step 3. Secondary feasibility n n n n m αj o P P P P ðBi Þ Di i Si o 0, jÞ ðP i Di σ i P i Þ o 0 and 1 P i ð1 θi Þð1 þ i¼1
i¼1
i¼1
j¼1
Value
Step 5. Constraints boundaries The constraints' boundaries are calculated and they are given in Table 6 Step 6. Checking the constraints As the third condition holds, i.e. T nMax ZT Z T nMin , we have T n ¼ T ¼ 0:0259588149281040. Step 7. Finding the optimal solution Based on T n ¼ 0:0259588149281040; Bni and Q ni are computed and they are shown in Table 7. Moreover, the minimum annual inventory cost is Z n ¼$ 1,201,220.69088020. Step 8. Terminate the procedure solution The inventory graph of the first item is illustrated in Fig. 2. Similar graphs can be provided for the rest of the items based on T¼ 0.026.
6. Sensitivity analysis This subsection shows the results of some sensitivity analyses performed on important parameters of the problem, including the production rate, the demand rate, the available budget, and the safety factor of each product. In these analyses, the value of each parameter is changed 7 10% and 730% of its original values while
T SL Max
T Budget
T nMin
T nMax
0.000314
0.004533
0.0341848
0.004533
0.042731
maintaining the original values of the other parameters. Table 8 contains the effects of these changes on T Min , T SL , T Budget , T n , and Z n . Besides, the following points are observed based on Table 8.
T Min is insensitive to both changes in the values of the available
vi
then the problem is feasible and go to Step 4 T Min ¼ 0:000314130978925651. Step 4. Checking the shortage level Using Eq. (35), T SL i s are obtained and they are presented in Table 5, based on which n o SL SL SL ¼ T SL T SL max ¼ Max T 1 ; T 2 ; :::; T n 1 ¼ 0:004533:
T Min
budget and the safety factor. It is moderately and highly sensitive to the changes in the values of the annual demand rate and the annual production rate, respectively. T SL is insensitive to the changes in the value of the available budget. It is moderately sensitive to the changes in the value of the annual demand rate and it is highly sensitive to both changes in the values of the annual production rate and the safety factor. T Budget is insensitive to the changes in the value of the safety factor. It is highly sensitive to the changes of all the values of the annual demand rate, the annual production rate, and the safety factor. T n and Z n are slightly sensitive to the changes in the value of the annual production rate. They are highly sensitive to the changes in the value of the annual demand rate. Moreover, they are insensitive to the changes in the values of the available budget and the safety factor.
From a managerial point of view, as the total cost (Z n ) including setup, production, rework, disposal, holding and construction, has the highest sensitivity to the demand rates (Di s), planners or managers should pay more attention in determining the quantities of customers' demand. In other words, wrong demand prediction would impose additional costs to the production system. Moreover, the cycle length is very sensitive to the changes in the production capacity but is not as sensitive as the total cost to the demand rate. In general, the results in Table 8 indicate that the demand rate and the production rate should be close to each other in order to have a desirable production system. Moreover, parameters such as budget, warehouse capacity, and safety factor have little impact on the total cost, but their improper values may lead into infeasible productions.
7. Conclusion and future research In this paper, an imperfect multiproduct single-machine EPQ problem was mathematically formulated. This imperfect production system included two different types of items, scrap and rework, where there were m types of failures that required rework. As the nonlinear programming problem was shown to be convex, an exact solution process involving eight steps was proposed to find the solution that would minimize total inventory cost, including setup, production, warehouse construction, holding, shortage, reworking, and disposal. In addition, the available total budget was scarce and that there was a lower bound on the service
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Table 7 Values of Bni ; Q ni .
Bni Q ni
1
2
3
4
5
6
7
8
9
10
0.407971 26.08926
0.280723 28.69818
0.223835 31.27568
0.284111 33.84800
0.190079 36.45169
0.172883 39.05538
0.212088 41.61733
0.198659 44.21842
0.310627 46.77264
0.369786 49.37112
Inventory
21.11
23000
19.11 17.88 17.77
-1000
17000 11000
0.000505
4190 0.00424
0.00007 0.00001
0.407
time
0.02111 0.000087
4190
T=0.026 Fig. 2. The inventory of the first product in a cycle.
Table 8 The effects of parameter changes. Parameters
Pi
Di
W
εi
% changes
þ 30 þ 10 10 30 þ 30 þ 10 10 30 þ 30 þ 10 10 30 þ 30 þ 10 10 30
% changes in T Min
T SL
T Budget
Tn
Zn
65.2309 20.2513 þ 21.7963 þ 80.8799 10.6784 5.4108 þ 9.3815 þ 70.9078 0 0 0 0 0 0 0 0
24.9811 9.9975 þ 12.5130 þ 50.2101 9.2565 3.5899 þ 4.2720 þ 15.7742 0 0 0 0 23.0766 9.09058 þ 11.1115 þ 42.8576
26.8168 10.8848 þ 13.9143 þ 57.8377 17.6024 7.0318 þ 8.7039 þ 33.9575 þ 30.0002 þ 10.0002 9.9998 29.9999 0 0 0 0
2.4887 1.0030 þ 1.2682 þ 5.17611 9.2507 3.5881 þ 4.2694 þ 15.7664 0 0 0 7.8178 0 0 0 0
þ 1.3914 þ 0.5523 0.6827 2.6830 þ 19.2019 þ 6.5771 6.7804 21.0694 0 0 0 þ 0.1629 0 0 0 0
level of each product. Moreover, the warehouse space of i-th item was determined based on its maximum on-hand inventory and its required aisles space. A numerical example was solved in order to demonstrate the applicability of the proposed solution procedure, as well as to perform some sensitivity analyses on the effects of some parameter changes on the solutions obtained. For future research, there are some recommendations as follows:
Some parameters can be considered stochastic, fuzzy, or a
combination of stochastic and fuzzy. The proposed inventory model can be extended to include discrete product quantities to be transported with limited capacity vehicles. Shortage can be considered as a combination of backorder and lost sale. Produced items can be considered perishable items.
Acknowledgments This research was supported by the Tecnológico de Monterrey Research Group in Industrial Engineering and Numerical Methods 0822B01006.
Appendix A. Determining the cycle length From Eq. (7) we have: þ1 þ2 þ3 þ4 T ¼ t 1i þ t 2i þ t 3i þ ::: þ t m þ tm þt m þt m i i i i
T¼
α1 Q i α2 Q i αm Q i H m B Qi Bi Bi þ i1 þ i2 þ ::: þ im þ i þ i þ P i P i Di σ i P i vi P i vi P i vi P i Di Di P i Di σ i P i
S.H.R. Pasandideh et al. / Int. J. Production Economics 169 (2015) 203–214
1 Q Qi 1 i αm ðð1 σ i ÞP i Di ÞQP ii Bi αi vi P i Di v1i P i αm i ðvi P i Di Þvm Pi Q i α1i Q i α2i Q i B i i Qi T ¼ þ 1 þ 2 þ ::: þ m þ þ þ ::: þ þ i P i vi P i vi P i vi P i Di Di Di Di αm Q i ðP Di σ i P i ÞQ i Bi α1i v1i P i Q i α1i Q i α m vm P i Q i α m Qi B Q i α1i Q i α2i Q i þ 1 þ 2 þ ::: þ im þ i þ 1 1 þ :::: þ i mi im þ i P i vi P i vi P i vi P i P i Di Di vi P i Di vi P i Di vi P i Di vi P i Q i α1 Q 1 DP ii θi þ α1i þ α2i þ::: þ αm αm Q i i Qi i T¼ þ þ i þ ::: þ i Pi Di Di Di m m X X Qi αji Q i αji Q i Q i Q i θi Q i Q θi Q i ð1 θi ÞQ i j¼1 j¼1 T¼ þ þ ¼ i ¼ P i Di P i Di Di Di Di Di T¼
Appendix B. Calculating the holding cost Based on Eqs. (8)–(10) we have: I i ¼ ai
Qi Bi Pi
-
ðI i Þ2 ¼
ðai Þ2 Q 2 ðP i Þ
2
þ ðBi Þ2 2
ai Q i Bi Pi
and
H 1i
2
¼ ðI i Þ2 þ
2 2 1
α1i yi v1i
Qi Pi
8 2 9 2 2> > α1i y1i 2 2 ai ai Qi > > > > Q ð Q Þ þ ð B Þ 2 B þ i i i i = < Pi Pi Pi v1i 2α1i y1i þ v1 P Q i I i ¼ 2 1 1 1 1 i > > i α y Qi Qi > > > > þ2ai αi 1yi 2 vi 1 i ; : Pi P i Bi v i
H 2i
2
¼
i
8 2 9 2 2 2> 1 1 > α1i y1i > > Qi Qi > > ai ðQ i Þ2 þ ðBi Þ2 2 ai Q i Bi þ αi 1yi þ 2a > > i > > Pi Pi Pi Pi vi v1i > > > > > > !2 > > = < 2 2 2 2 2 1 1 2 2 2 2 αi yi Q i αi yi α i yi αi yi 1 Qi Qi Qi Hi þ 2 ¼ 2 B þ þ 2a i i Pi Pi Pi v1i v2i v2i > > Pi vi > > > > > > > > 2 2 2 1 1 2 2 > > αi yi αi yi αi yi > > Qi Qi > > > > 1 2 P i Bi þ2 P ; : 2 v2 v v i i
i
i
therefore 8 !2 !2 !2 9 2 X m > > m m 2 X > > αji yji αji yji αji yji Qi Q i Bi X > > ai 2 2 2 ai Qi > > ðQ i Þ þ ðBi Þ 2 P i Q i Bi þ P i þ 2ai 2 > > > > Pi j j j > > P P i i = vi vi vi j¼1 j¼1 j¼1 m 2 < Hi ¼ ! ! 2 j j j 1 > > m k k > > 2X α y X α y > > i i i i > > þ 2 Qi > > > > P j k i > > v ; : v j¼2
i k¼1
i
then " # n 1 X hi I i 1 ðI i þ H 1i Þ 2 ðH 1i þH 2i Þ 3 ðH m þ Hm Hm mþ2 i i Þ mþ1 i CH ¼ ðt i Þ þ ðt i Þ þ ::: þ ðt i ðt Þ þ ðt Þþ Þ 2 2 2 T 2 i 2 i i¼1 Based on Eqs. (1)–(4), we have: 2 3 α1i Q i α2i Q i 1 Qi Bi 1 1 Qi 2 2 Qi I þ α y þ α y þ 2H þ ::::: þ 2I 1 1 2 2 i i i i i v Pi i i v Pi Pi ai vi P i vi P i 7 1X 6 i i 7 CH ¼ h6 m 5 m 2 2T i i 4 m 1 α Q H ð Þ i i m Qi i þ þ αm y þ 2H i i i vm P i vm P i Di i
i
0 0 0 9 8 !1 ! 11 j j 2 m m 2 2 X 2 X > > α α > > j hi a i Di D h D > > i i i i i AðT Þ þ @ @ > > yi AAðT Þ ðT Þ þ @hi ai P i ð1 θi Þ > > 2 P i ð1 θi Þ 2 P i ð1 θi Þ j j > > > > v v > > j ¼ 1 j ¼ 1 i i > > > > > > 0 1 0 1 ! ! ! > > > > j j j j1 m m > > 2 X 2 k X X > > α α y α > > Di hi Di Di ai hi Di i k i i i > > @ A @ A ð T Þ þ y ð T Þ þ þ h ð T Þ > > 2 i i P i ð1 θi Þ 2 P i ð1 θi Þ > > j j k ðP i ð1 θ i ÞÞ > > v vi > > i j ¼ 1 vi k ¼ 1 j¼1 > > > > > > 0 1 0 1 n < = ! ! ! X 2 2 j j j j j 1 m m 2 X k k X X CH ¼ α y α y α y h D h D i i i i i i 1 i i i i @ A @ A > > þ 2 P ð1 θ Þ ðT Þ þ ðT Þ > i¼1> > > i i ðP i ð1 θi ÞÞ2 > > vji vji k ¼ 1 vki > > j¼1 j¼1 > > > > > > 2 2 > > > > hi ðBi Þ hi ðBi Þ > > þ þ > > > > 2Di T 2ai T > > > > 0 1 0 1 > > ! ! > > 2 > > j j j m m > > X X αi yi αi > > h a h h D h D > > i i i i i i i @ A @ Að B i Þ > > B ð B Þ ð Þ ð Þ B i i i > > P i ð1 θi Þ P i ð1 θi Þ P i ð1 θi Þ P i ð1 θi Þ j j > > ; : vi j¼1 j ¼ 1 vi
211
212
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Appendix C. Coefficients of the objective function in Eq. (29)
P i
B2
Δ1i ð Ti Þ ¼
P h hi i
i
π i P i ð1 σ i Þ hi 2Di 2Di þ 2ai þ
ðBi Þ2 T
;
0
Δ1i 4 0 !1 0 A þ @hi
!2 11 3 6 yji AA 7 2 j 6 7 vji 6 7 j ¼ 1 vi j¼1 6 0 7 1 1 0 6 7 ! ! ! ! 2 jX j j j1 j 1 m m 6 7 2 X k k k X X α α y α α y 6 @ 7 Di hi D i i k i i i i i @ A A þ y 6 þ hi P i ð1 θi Þ 7 2 i j ðP i ð1 θi ÞÞ 7 vki vji k ¼ 1 vki X6 X j ¼ 1 vi k ¼ 1 j¼1 6 7 2 6 7ðT Þ; 0 1 0 1 Δi ðTÞ ¼ ! ! 6 7 2 j j j j m m X 7 2X α y 2 α y i i 6 Di ai hi Di i i i i A 1 6 þ @hi Di 7 A þ hi Di þ @ðP ð1 2 6 7 P i ð1 θi Þ 2 2 P i ð1 θi Þ j j θi ÞÞ i 6 7 v v j¼1 j¼1 i i 6 7 0 1 6 7 6 7 j j m X αi yi A 6 f i μi ai Di ð1 þ δi Þ f i μi Di ð1 þ δi Þ@ 7 4 þ P ð1 θ Þ þ P ð1 θ Þ 5 j i i i i v j¼1 i 2
2
hi ai Di 2 P i ð1 θi Þ
þ @hi ai
m 2 X αji
Di P i ð1 θi Þ
m 2 X
Di P i ð1 θi Þ
0 @
αji
8 X <
9 0 ! ! 1 j j j 2 m m = X X α y α h i i i i @ai þDi þ A þf i μi ð1 þ δi Þ ðBi Þ; þ Di Δ3i ðBi Þ ¼ j j : P i ð1 θi Þ ; vi i i j¼1 j ¼ 1 vi X X Di ; Δ4i 4 0 Δ4i ¼ ðci þ r i ðσ i θi Þ þdi θi Þ ð 1 θi Þ i i X 1 X 1 ¼ ; Δ5i 40 Δ5i ðAi Þ T T i i X
Δ2i 4 0
Δ3i 4 0
Appendix D. The maximum capacity of the single machine
n X
n X þ1 t 1i þ t 2i þ t 3i þ ⋯ þ t m Si r T þ i
i¼1
i¼1
Then, based on Eqs. (1)–(4), we have: ! n X αm Q i Q i Bi α1i Q i α2i Q i þ 1 þ 2 þ ⋯ þ im þ Si r T P i ai vi P i vi P i vi P i i¼1
0 0 1 1 n m X X αji @Q i @1 þ A Bi þ Si A rT j Pi ai i¼1 j ¼ 1v i
Finally, using Eq. (12), we have: 0 1 1 j m X α D B i i i @ @1 þ AT þSi A r T j P i ð1 θi Þ ai i¼1 j ¼ 1v n X
0
-
i¼1
i
0
n X
0 0 11 j n m X X α Bi D i i @1 þ AA Si rT @1 j ai P ð1 θ i Þ i¼1 i j ¼ 1v i
1
n n X X Bi B C Si B C ai B C i ¼ 1 i ¼ 1 8 0 19 ¼ T Min C T ZB B C j = n < m B C X X αi Di @1 A @1 þ A j : P i ð1 θ i Þ ; i¼1 j ¼ 1v i
Appendix E. Determining the budget constraint
0 1 1 m X αji yji @ci Q i þ r i α1 Q i þ ⋯ þ r i αm Q i þ di θi Q i þ f i μi ai Di ð1 þδi ÞT þ f i μi Di ð1 þ δi Þ@ AT f i μi ð1 þ δi ÞBi A r W i i j P i ð1 θ i Þ P i ð1 θi Þ i¼1 j ¼ 1 vi n X
n X
0
0
0
m X
1
0
1
1
m αji yji f μ a D ð1 þ δ i Þ f μ D ð1 þ δ i Þ@ X @Q i @ci þ di θi þ r i AT f i μi ð1 þ δi ÞBi A r W Tþ i i i αji A þ i i i i j P i ð1 θi Þ P i ð1 θ i Þ i¼1 j¼1 j ¼ 1 vi
S.H.R. Pasandideh et al. / Int. J. Production Economics 169 (2015) 203–214
Using Eq. (12) and Appendix H, we have: 1 0 1 0 1 j j m m 3 X X α y Δ D T f μ a D ð 1 þ δ Þ f μ D ð 1 þ δ Þ i i i i i i j i iA @ @c þ di θi þ r i @ T f i μi ð1 þ δi Þ i 1 T A r W Tþ i i αi A þ i i j P i ð1 θi Þ P i ð1 θ i Þ ð1 θ i Þ i 2Δi i¼1 j¼1 j¼1 v n X
0
i
Tr
8
W
9 ¼ T Budget 0 0 11 j j m 3 = X α y Δ Di @ j A f i μi Di ð1 þ δi Þ@ i i AA f i μi ð1 þ δi Þ i 1 c þ di θ i þ r i ai þ @ αi þ j :ð1 θi Þ i P i ð1 θ i Þ 2Δi ; i¼1 j¼1 j¼1 v 0
n < X
m X
1
i
Appendix F. Finding the initial feasibility From Eq. (33) and Appendix H, we have: ! Δ3i T εi Di 1 Z0 2Δi
8 n > < T Z0
Δ3i > : ξi ¼ εi 2Δ1i Di Z 0
-
8 i ¼ 1; 2; :::; n
Appendix G. Hessian matrix to prove the convexity of the objective function (Z)
Z¼
X
Δ1i
i
! X X X 1 X ðBi Þ2 Δ2i ðT Þ Δ3i ðBi Þ þ Δ4i þ Δ5i þ T T i i i i 2
6 6 6 6 6 6 H ¼ ½T; B1 ; B2 ; :::; Bn 6 6 6 6 6 4 P ∂Z ¼ ∂T
i
Δ1i ðBi Þ2
P i
∂2 Z ∂2 T
∂2 Z ∂T∂B1
∂2 Z ∂T∂B2
∂2 Z ∂B1 ∂T
∂2 Z ∂2 B1
∂2 Z ∂B1 ∂B2
∂2 Z ∂B2 ∂T
∂2 Z ∂B2 ∂B1
∂2 Z ∂ 2 B2
∂2 Z ∂T∂Bn
⋯
∂2 Z ∂B1 ∂Bn
⋯
∂2 Z ∂B2 ∂Bn
⋮
⋮
⋮
⋮
⋮
∂2 Z ∂Bn ∂T
∂2 Z ∂Bn ∂B1
∂2 Z ∂Bn ∂B2
⋯
Δ5i þ
T2
∂Z 2Δ1i Bi Δ3i ¼ ∂Bi T
⋯
X
Δ2i
i
∂2 Z 2Δ1i ¼ T ∂ 2 Bi
∂2 Z ¼ ∂2 T
∂2 Z ∂2 Bn
3 2
7 7 7 7 7 7 7 7 7 7 7 5
T
3
6B 7 6 1 7 6 7 6 B2 7 6 7 6 ⋮ 7 4 5 Bn
P P 2 Δ1i ðBi Þ2 þ 2 Δ5i i
i
T3
2Δ1i Bi ∂2 Z ∂2 Z ∂2 Z ¼ ¼ ¼ ∂Z∂Bi ∂Z∂T ∂Bi ∂T T2
therefore 2 P 2
6 6 6 6 6 6 H ¼ ½T; B1 ; B2 ; :::; Bn 6 6 6 6 6 6 4
Δ5i þ 2
P
i
T
T2 2Δ12 B2 T2
⋮ 2Δ1n Bn T2
T
2Δ11 B1
2Δ12 B2
2Δ11 T
0
T2
2Δ11 B1
2
3 2 6 7 6 P 7 6 76 62 Δ5i 76 6 i 76 6 76 ; 0; 0; :::; 0 ¼6 76 2 T 6 76 6 74 6 7 4 5
Δ1i ðBi Þ2
i 3
0 ⋮ 0
T2
2Δ12 T
2Δ1n Bn T2
⋯ ⋯
0
⋯
0
⋮
⋮
0
⋯
⋮ 2Δ1n T
3 7 7 7 7 7 7 7 7 7 7 5
2
3 T 6B 7 6 1 7 6 7 6 B2 7 6 7 6 ⋮ 7 4 5 Bn
3
P 5 B1 7 2Δi 7 7 i B2 7 Z0 7¼ T ⋮ 7 5 Bn
Hessian Matrix is P:S:D
213
214
S.H.R. Pasandideh et al. / Int. J. Production Economics 169 (2015) 203–214
Appendix H. Finding the optimal value of the decision variables
∂Z ∂T ∂Z ∂Bi
P
¼ ¼
Δ1i ðBi Þ2
i
P
Δ5i
i
T2
þ
P i
2Δ1i Bi 3 T Δi
P Δ2i ¼ 0
-
T2 ¼
¼0
-
Bi ¼
Δ5i þ
i
P
Pi
Δ1i ðBi Þ2
Δ2i
Δ3i 2Δ1i
;
i
T
thus P T2 ¼
i
Δ5i þ
P i
P i
Δ1i
Δ3i T 2Δ1i
2
P ¼
Δ2i
i
P ðΔ3i T Þ2 Δ5i þ 1 4Δi i P 2 Δi
-
i
T2
X i
Δ2i T 2
3 2 n X Δi i¼1
4Δ1i
¼
X
Δ5i
i
then P i
2
T ¼
P i
Δ2i
Δ5i 2 n P ðΔ3i Þ
i¼1
!
4Δ1i
finally vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u P 5 u Δi u i ! T ¼u u 2 n u P 2 P ðΔ3i Þ t Δi 1 4Δ i
i¼1
Bi ¼
Δ3i T 2Δ1i
i ¼ 1; 2; :::; n
i
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