- Email: [email protected]
Inventory system with expiration date: Pricing and replenishment decisions
Accepted Manuscript Inventory system withexpiration date: pricingand replenishment decisions Al-Amin Khan, Ali Akbar Shaikh, Gobinda Chandra Panda, Ioannis Konstantaras, Ata Allah Taleizadeh PII: DOI: Reference:
S0360-8352(19)30205-0 https://doi.org/10.1016/j.cie.2019.04.002 CAIE 5789
To appear in:
Computers & Industrial Engineering
Received Date: Revised Date: Accepted Date:
25 April 2018 27 March 2019 1 April 2019
Please cite this article as: Khan, A-A., Shaikh, A.A., Panda, G.C., Konstantaras, I., Taleizadeh, A.A., Inventory system withexpiration date: pricingand replenishment decisions, Computers & Industrial Engineering (2019), doi: https://doi.org/10.1016/j.cie.2019.04.002
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Inventory system with expiration date: pricing and replenishment decisions
Inventory system with expiration date: pricing and replenishment decisions Md. Al-Amin Khan Department of Mathematics, Jahangirnagar University, Savar, Dhaka-1342, Bangladesh, [email protected] Ali Akbar Shaikh Department of Mathematics, The University of Burdwan, Burdwan-713104, India, [email protected]
Gobinda Chandra Panda Department of Mathematics, Mahavir Institute of Engineering and Technology, BBSR, India752054, [email protected] Ioannis Konstantaras* Department of Business Administration, School of Business Administration, University of Macedonia, 156 Egnatia Str., Thessaloniki 54636, Greece, [email protected] Ata Allah Taleizadeh School of Industrial Engineering, College of Engineering, University of Tehran, Tehran, Iran, Email: [email protected]
Abstract Every product has a life cycle i.e., the product has a certain time period for useable conditions for mankind. After the certain time period, the product is not for useable conditions and which is known as expired product. However, price discount is a very much attractive strategy in the present business situations. Using these two concepts in together, we introduce two supply chain models by assuming the demand of the product to be dependent on price. Also, shortages are considered and these depend on the customer waiting time. In both models, we formulate the problem mathematically, considering price as a decision variable, and prove their optimality. Two proposed solution procedures have been ameliorated and then solved two numerical examples in order to validate our proposed models as well as to compare which model is more
1
economical. Two sensitivity analyses are performed to make a conclusion regarding of our paper.
Keyword: Inventory; expiration date; deterioration; discount; demand dependent on price; partial backlogging.
*Corresponding author: Tel. 00302310891695
Abstract Every product has a life cycle i.e., the product has a certain time period for useable conditions for mankind. After the certain time period, the product is not for useable conditions and which is known as expired product. However, price discount is a very much attractive strategy in the present business situations. Using these two concepts in together, we introduce two supply chain models by assuming the demand of the product to be dependent on price. Also, shortages are considered and these depend on the customer waiting time. In both models, we formulate the problem mathematically, considering price as a decision variable, and prove their optimality. Two proposed solution procedures have been ameliorated and then solved two numerical examples in order to validate our proposed models as well as to compare which model is more economical. Two sensitivity analyses are performed to make a conclusion regarding of our paper.
Keyword: Inventory; expiration date; deterioration; discount; demand dependent on price; partial backlogging. 1.
Introduction
Inventory control is an essential activity in business management. The function of inventory control starts from the very beginning i.e., from the processing of raw materials into complete products manufacturing, movement of products from warehouses to wholesalers, wholesalers to retailers and retailers to customers. In every stage, control or management of inventory is most
2
vital. Various inventory control models have been developed based on inventory control theories. Basically, it helps in bringing maximum profits to the companies or minimizes their costs. The proposed inventory model is developed considering some essential features related to the present marketing scenario. First, we have discussed about demand of the model as demand always plays a crucial role during the construction of any inventory model. Generally, from wholesaling to retailing business, the crucial factor affecting the demand is the price of a new or existing product, because we know that low-priced products have higher demand and high-priced products have lower demand. As a result, the decision of a customer during the purchase period is highly influenced by price of the goods. Consequently, demand depends on price is one of the most general cases in the inventory management system. That’s the reason why we have taken price-dependent demand as one of our basic assumptions. Again, a large part of the grocery products have limited lifetimes and when these products approach to their expiration dates, the deterioration rate gradually increases and finally reaches to 100% at their expiration dates. When the customers purchase these types of products, they always check the expiration date of the products and try to purchase which is further from the expiration date so that they can store the products for a long time. As a result, alongside the suitability of the price of a product, the expiration date of the product is also equally important to attract the customers during buying or purchasing. Products like fruits, vegetables, electronic gadgets, medicine, dried food, rice, milk, wheat and radioactive materials etc., which deteriorate or decline over time with an increasing rate and have also an expiry period after which they lose their usability and are not in a condition for sale. Accordingly, the deterioration rates for these types of products can be considered as an increasing function in terms of the expiration dates. So here we have considered the expiration date of the product as an assumption in our model. These two above factors such as price and the deterioration rate with the expiration date are most important features which simultaneously affect the purchase-decision of customers. As we have considered price-demand demand and have also paid attention to the expiration date or maximum lifespan of a product this may lead to higher demand and in consequence the probability of shortages. So another two faceted assumptions that we have considered in this model are: i) shortages may not occur and ii) shortages may occur with partially backorders. In most of the typical inventory models, one ordinary assumption is carrying cost per unit is constant during the entire cycle length. As the deterioration rate is gradually increasing, the 3
decision maker always wants to control this rate by providing better preserving facilities. Consequently, the holding cost per unit cannot be constant in the entire cycle length but, in practice, it is an increasing function of the storage time. In this work, the holding cost per unit has been considered as a linearly increasing function of the storage time. Another typical assumption in most of the inventory models is that the unit purchase cost is fixed whatever the order size. However, in practical scenario, it is a decreasing function according to the order size. So, finally, we assumed a discount environment where unit purchase cost is a decreasing step function based on increasing order size. In the recent past, many inventory models have been developed considering different assumptions. But to the best of our knowledge, combination of the factors such as pricedependent demand, time varying deterioration rate depending on the product’s maximum lifetime or expiration date and linearly increasing holding cost have not yet been considered under all-units quantity discount environment in a single model. 1.1. Literature Review There is a large volume of published studies describing the role of price-dependent demand in inventory models for deteriorating items. Mondal et al. (2003) first proposed a model for an ameliorating item where demand of the product is dependent on price. Then, Mukhopadhyay et al. (2004) introduced an ordering policy on a pricing inventory model for deteriorating items. Next onwards Mukhopadhyay et al. (2005) extended their previous problem introducing another feature that the deterioration rate follows two parameter Weibull distribution and demand of the product depends on price. Similarly, You (2006) formulated a model where demand depends on price with an advance sales system. Again, Roy and Chaudhuri (2007) designed a model for price-dependent demand for a deteriorating item with special sale of the product and Roy (2008) extended his previous problem by incorporating time dependent holding cost. Further, Maiti et al. (2009) built a price-dependent inventory problem with stochastic lead time in advance payment system. Next, Sridevi et al. (2010) has mentioned a price-dependent model under the Weibull distribution deterioration rate. Similarly, Sana (2011) designed a price sensitive model for perishable items while Maihami and Kamalabadi (2012) formulated a model where demand of the product is dependent on jointly time and price. Avinadav et al. (2013) have introduced a supply chain problem for a perishable item. Bhunia and Shaikh (2014) proposed an inventory problem with price effect on demand. After that, Ghorieshi et al. (2015) proposed a supply 4
problem with price affected demand in trade credit system. Then, Alfares and Ghaithan (2016) included quantity discounts offered to the customer with holding cost dependent on time. Recently, Jaggi et al. (2017) proposed an inventory problem for price-dependent demand with credit financing and non-instantaneous deterioration in two storage facilities. A considerable amount of literature has been published on inventory control models for deteriorating products. In this literature, Ghare and Schrader (1963) first introduced the concept of deterioration in the area of inventory control, while Philip (1974) has upgraded from constant deterioration to three parameters Weibull deterioration and proposed another model. Thereafter a lot of researches have been discussed by several authors with different types of deterioration. Much of the current literature on deterioration pays particular attention to inventory models on deteriorating items with expiration date. Hsu et al. (2006) first proposed an expiration rate dependent deteriorating inventory model. Hsu et al. (2007) further extended their previous problem considering the lead time is uncertain in nature. Later on, Jain and Singh (2011) modified Hsu et al. (2007) model in multi-echelon supply chain system by considering several factors like inflation, expiration date dependent deterioration, capital constraint and uncertain lead time. Next onwards, Wu et al. (2014) investigated a model for expiration rate dependent deterioration in two level trade credit systems. Then, Chen and Teng (2015) designed a timevarying deteriorating inventory model with upstream and downstream delay in payment system. Wu et al. (2016) introduced another concept of freshness of the product in stocking policy with expiration rate of the product in the inventory system. Teng et al. (2016) further designed lot size inventory policies where the deteriorating item having an expiration date and advance payment system is followed. Similarly, Feng et al. (2017) built an inventory model by considering same concept of Wu et al. (2016) except freshness of the product. Further, Wu et al. (2017) extended their previous model with expiration rate dependent deterioration and advance-cash-credit payment schemes. Recently, Tiwari et al. (2018) developed a profit maximizing inventory model under two level trade credit policy in a supply chain for a deteriorating product with pricedependent demand and time varying deterioration rate with maximum lifetime. Additionally, we have summarized the comparison between the present work and earlier published works in Table 1: 1.2. Research gap and our contribution
5
Alfares and Ghaithan (2016) introduced an inventory model for zero ending case and did not consider the effects of deterioration and shortages. In the present paper, we have generalized the Alfares and Ghaithan (2016) model by considering expiration rate dependent deterioration with partial backlogging shortages during shortage time of the product. Also, we proved the optimality for each model (zero ending inventory and partial backlogged shortages).
6
Table 1: Summarized between earlier published research and our present work. Deterioration
Discount
Holding cost
Situation
PBO FBO PBO FBO FBO PBO FBO No FBO PBO No No No No No No No PBO No No FBO No PBO PBO FBO PBO
Payment Mode AP AP No No AP DP No DP AP No DP DP DP DP No No AP AP DP No AP No AP No AP DP
No No No No Constant No Constant Expiration No Expiration Expiration Expiration Expiration Expiration No Expiration Time-varying Time-varying Expiration No Expiration Expiration No Constant No Expiration
Yes Yes No Yes No No Yes No No No No No Yes No No No No No Yes Yes No No No No No No
Constant Constant Constant Constant Constant Constant Constant Constant Constant Constant Constant Constant Constant Constant Constant Constant Constant Constant Constant Time-varying Constant Constant Constant Constant Constant Constant
Single Single Double Single Single Single Single No Single Single No No No No No Single No Single No No Single No Single Single Single Single
Solution methodology Soft computing Soft computing Soft computing Soft computing Soft computing Mathematically Mathematically Mathematically Mathematically Mathematically Mathematically Mathematically Mathematically Mathematically Mathematically Mathematically Mathematically Mathematically Mathematically Mathematically Mathematically Mathematically Mathematically Mathematically Mathematically Mathematically
No
No
No
No
Constant
Single
Mathematically
PBO FBO
No No
Time-varying No
No No
Time-varying Constant
Single Single
Mathematically Mathematically
No
No
No
No
Constant
Single
Mathematically
FBO
No
No
Constant
Single
Mathematically
PBO
No
Expiration
No Linked to Order
Time varying
Double
Mathematically
Authors
Demand
Shortage
Gupta et al. (2009) Maiti et al. (2009) Taleizadeh et al. (2010) Taleizadeh et al. (2010) Guira et al. (2013) Taleizadeh et al. (2013) Taleizadeh et al. (2013) Wu et al. (2014) Taleizadeh (2014) Tayal et al. (2014) Wang et al. (2014) Wu and Chan (2014) Chen and Teng (2015) Sarkar et al. (2015) Taleizadeh et al. (2015) Chen et al. (2016) Teng et al. (2016) Wu et al. (2016a) Wu et al. (2016b) Alfares and Ghaithan (2016) Wu et al. (2017) Feng et al.(2017) Taleizadeh(2017) Pal et al. (2017) Tavakoli and Taleizadeh (2017) Tiwari et al. (2018)
Das Roy et al. (2018)
Constant Price-dependent Constant Constant Variable Constant Constant Constant Constant Price and expiration rate Credit period dependent Constant Depends on credit period Constant Price-dependent Stock dependent Constant Constant Constant Price-dependent Constant Price-dependent Constant Price-dependent Constant Price-dependent Retail price and advertising expenditures Time-dependent Price-dependent Retail price and product quality Price-sensitive stochastic
This Paper
Price dependent
Taleizadeh (2018) Pervin et al. (2018) Roy et al. (2018) Modak et al. (2018)
PBO: Partial Back-ordering, FBO: Full Back-ordering, AP: Advanced Payment, DP: Delayed Payment
7
We also proposed two solution algorithms and solved each model. In the present work, we have modified the model of Alfares and Ghaithan (2016) under the given considerations: (i)
Expiration date dependent deterioration has been introduced.
(ii)
Two separate cases have been considered (zero ending case and shortage case).
(iii)
Compared which model is more economical.
The remainder of this paper is organized as follows. The basic notation of the problem is presented in Section 2 while problem’s definition is briefly discussed along with the assumptions in Section 3. The mathematical formulation of the proposed models is given in Section 4 and the proofs of the optimality are described in section 5. An algorithm of the solution procedure is presented in Section 6 while Section 7 provides numerical results. Section 8represents the sensitivity analysis of our work and the main conclusions, comparisons and further research are described in Section 9. 2. Problem description To describe the problem, we define first the notations which have been used in our proposed work. Notations K
Units $/unit Constant Constant $/unit
Description replenishment cost per order constant part of demand rate (a 0) coefficient of the price in the demand rate (b 0) purchasing cost
$/unit $/unit
selling price shortage cost
$/unit
opportunity cost
(u )
Constant
rate of deterioration at any time u (0 (u) 1)
g
$/unit $/unit Week
constant part of holding cost coefficient of linearly time dependent holding cost the maximum lifetime or expired date
S
Constant
backlogging parameter
Units
the initial number of inventories (for the model with shortages)
R
Units
Q
Units
maximum number of partially backordered quantity (for the model with shortages) number of order placed per cycle
t1 t2
Week
time period when stock reaches to zero for shortages case
Week
time period of shortages
T
Week
cycle length (for shortages case T t1 t 2 )
a b cj p
cs cl
h
8
TP1 ( p, T )
$/week
the total profit of the problem for non shortages case
TP2 ( p, t1 , t 2 )
$/week
the total profit of the problem for partially backlogged shortages case
$/unit Week
Decision variable selling price time period when stock reaches to zero for shortages case
Week
time period of shortages
Week
cycle length for the model without shortages
p
t1 t2 T
3. Problem definition We assume that a supplier in order to stimulate demand, increase cash flow, and run smooth business offers a temporary price discounts on unit purchase costs of a deteriorating product based on order sizes as follows: purchase cost of a unit is discontinuous and it is a decreasing step function according to the order quantity. Since this discount directly affects the order size of the retailers, sometimes it leads to stock-out problems due to heavy demand and unavailability of adequate stock in the system. Therefore, we need to discuss both cases: with and without shortages. The model without any shortages will be discussed first and then the model with shortages. Our proposed works are developed under the following considerations: 1. The models are developed for a single deteriorating product.
2. Replenishment rate is infinite and instant. 3. It’s well known that if the price of an item is low then its demand appears highly in the market. From this view point, we assume the same as in Alfares and Ghaithan (2016), i.e., the demand is dependent linearly on price and can be expressed as: D( p) a bp
(1)
where a is the amount of demanded products when selling price p is zero, b is the steepness of the demand and a bp 0 . 4. The deterioration rate (u ) is a time varying increasing function. This rate depends on the product’s maximum lifetime or expired date . In addition, we have considered this rate similar to Wang et al. (2014) and Chen and Teng (2014) which is expressed bellow:
(u )
1 1 u
, 0 u T .
(2)
Remarkable point is that the replenishment cycle length T cannot be exceeding the maximum lifetimeτ because of after the maximum lifetime product cannot be sold. 9
5. Any replacement or repair cannot be taken into account for deteriorated products during the cycle length. We can say in other way, once a product has sold it cannot returnable or repairable. 6. Partially backlogged shortages are considered and it depends on the waiting time of the customers up to the arrival of next lot of the products. The partial backlogging rate L( y ) is a decreasing function of the customer waiting time and it can be express as: L( y )
1 , 1 y
(3)
where y indicates the customers’ waiting times up to arrival of the product and 0 . Notice that if 0 , the partial backlogging rate convert in fully backlogged shortages. 7. Holding cost of the products depends on the duration of the stocking time of each unit and also proportional to the purchase cost c j . Similar to Alfares and Ghaithan (2016), the holding cost consists of two parts: a constant part g and a linearly increasing with the storage time part h . Therefore, the holding cost function can be expressed as follows:
H (u) ci ( g hu) .
(4)
8. The unit purchase cost ( c j ) in the discount environment is a discontinuous decreasing step function according to the order size Q as follows:
qi , i 1, 2,3,........, n 1 (q1 q2 ........ qn qn1 ) are quantities which determine the n price breaks with the unit purchase cost ci , i 1, 2,3,........, n (c1 c2 ........ cn ) . 4. Description of mathematical modelling Here, we describe the modelling of the two problems. 4.1. Inventory model without shortages The inventory model starts with Q units of a deteriorating item and the stock level gradually decays because of customers’ demand and deterioration throughout the cycle length T and finally drops at zero at u T . After that, a new order will be placed then and there and the whole inventory system will be repeated (see Fig. 1). So, at any time u the level of inventory I (u ) can be expressed in the following governing differential equation:
10
Inventory
Q
Q
Time
T
Fig. 1. Pictorial presentation of proposed problem of without shortages case dI (u ) 1 I (u ) (a bp), 0 u T du 1 u
(5)
with conditions I (0) Q and I (T ) 0 .Using I (T ) 0 , we have: 1 u I (u ) (a bp)(1 u ) ln . 1 T
(6)
From Eq. (6) with the help of the condition I (0) Q , the order size is: 1 Q (a bp)(1 ) ln . 1 T So, the entire cycle length, T , can be calculated as follows: Q ( a bp )(1 ) . T (1 ) 1 e
(7)
(8)
The profit of the entire inventory system through the cycle length involves the following components which are stated below: 1) Ordering cost of (OC): K
1 2) Acquisition cost (AC): c j Q c j (a bp)(1 ) ln 1 T
11
T
3) Holding cost (HC): c j ( g hu ) I (u )du 0
T
T
1 u 1 u c j g (a bp) (1 u ) ln du c j h(a bp) u (1 u ) ln du 1 T 1 T 0 0 (1 ) 2 1 T 2 (1 )T c j g (a bp) ln 2 1 T 4 2 1 3 1 T 3 (1 )T 2 1 2 T c j h(a bp) ln 12 6 6 1 T 9 T
4) Sale Revenue (SR): p (a bp)du p(a bp)T 0
So, the profit function of the system can be expressed as: 1 TP1 ( p, T ) SR OC AC HC T (1 )2 1 T 2 (1 )T 1 2 ln ap bp T K c j (a bp)(1 ) ln c j g (a bp) 2 1 T 1 T 4 2 1 3 2 1 1 T 3 (1 )T 2 1 T T c j h(a bp) 6 ln 1 T 9 12 6
(9)
(10)
We need to determine the retailer’s optimum cycle length, T * , as well as selling price of the product, p * , in such a way that maximize retailer’s profit TP1 ( p, T ) . Now we investigate the model with partial backlogged shortages due to the waiting time of the customers. 4.2. Model with partial backlogged shortages In this case, an order of Q (S R) units of a deteriorating product which has fixed life time is placed at the beginning of a replenishment cycle. Shortly after, R units are utilized to fulfill the total accumulated backlogged demand; consequently, the remaining on hand inventory level becomes S . Then the inventory level gradually decreases not only for the customer’s demand but also the deterioration over the interval [0, t1 ] and finally drops at zero at time u t1 . After that, shortages are appeared which are accumulated depend upon waiting time of the customers in time interval [t1 , t1 t 2 ] . A new order is placed again at this instant and the entire inventory system is reiterated. The whole inventory situation over the entire cycle length can be presented in Fig. 2. The inventory level I (u ) satisfies the following two governing differential equations: 12
Inventory
S
t2 Time Backorders
t1 Lost Sales
Fig. 2. Graphical presentation of the proposed problem with partial backlogged shortages dI1 (u ) 1 I1 (u ) (a bp) , 0 u t1 du 1 u
(11)
dI 2 (u ) (a bp) , t1 u t1 t 2 du 1 (t1 t 2 u )
(12)
with the conditions I1 (0) S , I1 (t1 ) 0 and I 2 (t1 t 2 ) R . From Eqs. (11) and (12) and using the conditions I1 (t1 ) 0 , I 2 (t1 t 2 ) R , we have:
1 u and I 1 (u ) (a bp)(1 u ) ln 1 t1
I 2 (u )
(a bp)
ln1 (t1 t 2 u ) R .
(13)
(14)
Using the initial condition I1 (0) S , from Eq. (13) one can get:
1 . S (a bp)(1 ) ln 1 t1
(15)
The total backordered quantity, with the help of continuity of I (u ) at u t1 , is:
R
(a bp) ln 1 δ t2 . δ
(16)
13
Therefore, the total number of order quantities for each replenishment cycle is:
1 1 ln 1 t 2 . Q (a bp) (1 ) ln 1 t1
(17)
The cycle length of the shortage period is: 1 Q 1 ( a bp) (1 ) ln 1 t1 t2 e 1 .
(18)
The profit of the entire inventory system involves the following components: 1) Ordering cost (OC): K
1 1 ln 1 t 2 2) Acquisition cost (AC): c j Q c j (a bp) (1 ) ln 1 t1 t1
3) Holding cost (HC): c j ( g hu ) I 1 (u )du 0
1 1 u 1 u du c j h(a bp) u (1 u ) ln du c j g (a bp) (1 u ) ln 1 t1 1 t1 0 0 (1 ) 2 1 t1 2 (1 )t1 c j g (a bp) ln 2 1 t 4 2 1 (1 ) 3 1 t13 (1 )t1 2 (1 ) 2 t1 c j h(a bp) ln 6 1 t 9 12 6 1 t1 t 2 c (a bp) ln(1 t 2 ) 4) Shortage cost (SC): c s I 2 (u )du s t2 t1
t1
t
t1 t 2
5) Lost sale cost (LSC): cl
t1
ln(1 t 2 ) 1 1 (a bp)du cl (a bp) t 2 1 (t1 t 2 u )
t1
1 6) Sale Revenue (SR): p (a bp)du pR (ap bp 2 ) t1 ln(1 t 2 ) 0 So, the profit function of the system can be expressed as: TP2 ( p, t1 , t 2 )
1 SR OC AC HC SC LSC t1 t 2
(19)
14
1 2 (ap bp ) t1 ln(1 t 2 ) K 1 1 c j (a bp ) (1 ) ln ln(1 t 2 ) 1 t 1 (1 ) 2 1 t1 2 (1 )t1 1 ln c j g (a bp ) t1 t 2 2 1 t1 4 2 3 2 2 3 c h(a bp ) (1 ) ln 1 t1 (1 )t1 (1 ) t1 1 t 9 j 6 12 6 1 cs ln(1 t 2 ) cl (a bp ) t 2
(20)
Our objective is to find the optimum selling price per unit p * , t1* and t 2 * in order to maximize retailer’s profit per unit time TP2 ( p, t1 , t 2 ) . 5. Theoretical results Here, we present the concavity of the objective function for the problem with no shortage case first and then for the problem with partial backlogged shortages. To investigate the concavity for both models, some results have been used from Cambini and Martein (2009). Based on the Theorems 3.2.9 and 3.2.10 in Cambini and Martein (2009), the function of the form
( x)
f ( x) , x Rn g ( x)
(21)
is (strictly) pseudo-concave, if f (x) is a non-negative, differentiable and (strictly) concave function but g (x) is a positive, differentiable and convex function. 5.1 Inventory model with no shortage case Here we describe the optimality of the first problem by using the results of the Theorems 3.2.9 and 3.2.10 in Cambini and Martein (2009). The problem’s optimal policy can be obtained in two folds. Firstly, the optimal replenishment T * is found which optimizes the retailer’s total profit for any certain selling price p 0 . Secondly, the optimal p * is determined which maximizes the retailer’s total profit per unit time using the optimal replenishment T * . Theorem 1. For a certain p 0 , TP1 ( p, T ) is a pseudo-concave function of T and so there exists a unique T such that TP1 ( p, T ) is maximized. Proof. See Appendix A. 15
Lemma 1. For all u 0 ,
1 3 4u (1 ) is always positive. 1 u 2
Proof. See Appendix B. According to the necessary condition (maximization or minimization) to find the optimum cycle length T * , calculate first order derivative with respect to T of the objective function TP1 ( p, T ) for any given p , set the result to zero and after rearranging terms, we get
1 K c j (1 ) ln a bp 1 T
c j gT 2 4
T (1 ) (1 ) 2 1 g h 2 6 1 T
2T 3 (1 )T 2 c jh 0 12 9
(22)
Theorem 2. For any given positive values of cycle length T 0 , TP1 ( p, T ) is a concave function in p and so there exists a unique p such that TP1 ( p, T ) is maximized. Proof. See Appendix C. The necessary condition to find the optimal price p * can be found for any T by setting the first order partial derivative of the objective function with respect to p i.e., Eq. (C1) equal to zero, we get: T 2 (1 )T 1 (1 ) (1 ) 2 1 g h c gb j 2 6 2 1 T 4
a 2bp T c j b(1 ) ln
T 3 (1 )T 2 1 2 T 0 c j hb 12 6 9
(23)
5.2. Inventory model with partial backlogged shortages Now we investigate the concavity of TP2 ( p, t1 , t 2 ) with respect to p , t1 and t 2 for the model with partial backlogged shortages in two folds. Firstly, the joint concavity of TP2 ( p, t1 , t 2 ) in (t1 , t 2 ) is observed for any certain p 0 and then optimal (t1 , t 2 ) is determined. Secondly, the concavity of *
*
TP2 ( p, t1 , t 2 ) is investigated in p for fixed (t1 , t2 ) . For any certain p 0 , the first order partial derivatives of TP2 ( p, t1 , t 2 ) with respect to t1 and
t 2 are:
16
TP2 (t1 , t2 p) 1 TP2 ( p, t1 , t2 ) t1 t1 t2 (1 τ ) 2 t (1 τ ) 1 τ p c c g 1 j j 1 τ t1 2 a bp 2(1 τ t1 ) 2 t1 t2 (1 τ )3 t12 (1 τ )t1 (1 τ ) 2 c j h 6(1 τ t ) 3 6 6 1
TP2 (t1 , t 2 p) t 2
c c 1 a bp 1 TP2 ( p, t1 , t 2 ) p c j cl s cl s t1 t 2 t1 t 2 (1 t 2 )
(24)
(25)
The necessary condition to find the optimal t1* and t 2 * are:
TP2 (t1 , t 2 p) t1
0 and
TP2 (t1 , t 2 p) t 2
0 i.e.
(26)
2 2t 3 (1 )t1 2 c ln(1 t 2 ) t K c j g 1 c j h 1 p c j cl s (a bp) 4 12 9
1 t t g (1 ) (1 ) 2 1 2 1 c j (1 )ln 2 6 1 t 1 1 u 1
(27)
t1 2 (1 )t1 (1 ) 2 cs t1 (1 ) t 2 p cl c j g c j h 0 2 6 6 2 3
and cs p cl
t12 (1 )t1 cs ln(1 t2 ) t1 t2 K t p c c c g j l j 1 (1 t2 ) (a bp) 2 4
t13 (1 )t12 (1 ) 2 t1 1 g (1 ) (1 ) 2 c j (1 ) ln cjh 1 0. 2 6 12 6 1 t1 9
(28)
* * On solving Eqs. (27) and (28), we can obtain the optimal values of t1 and t 2 , say t1 and t 2 .
Theorem 3. For certain p 0 , TP2 ( p, t1 , t 2 ) is jointly pseudo-concave function in (t1 , t 2 ) and so there exists a unique pair of values (t1 , t 2 ) such that TP2 ( p, t1 , t 2 ) is maximized. Proof. See Appendix D. Theorem 4. For any given t1 0, t2 0 , TP2 ( p, t1 , t 2 ) is a concave function of p and so there exists a unique p such that TP2 ( p, t1 , t 2 ) is maximized. Proof. See Appendix E. The necessary condition to find the optimal selling price per unit p * is:
17
TP2 ( p, t1 , t2 ) 0 i.e., p t 2 (1 τ )t1 c ln(1 δt2 ) c (a 2bp)t1 a 2bp c j b cl s b b cl s t 2 c j gb 1 δ δ δ 2 4 1 τ t13 (1 τ )t12 (1 τ ) 2 t1 (1 τ ) (1 τ ) 2 c j b(1 τ ) ln h c j hb 1 g 0 2 6 12 6 9 1 τ t1
(29)
6. Solution procedure In the discount environment, suppose that there are n price breaks ( n quantity discounts) which is offered from supplier to retailer. Here, we have considered the purchasing cost ( c j ) is a decreasing step function measured by these n price breaks. Suppose that qi , i 1, 2,3,........, n 1
(q1 q2 ........ qn qn1 ) are quantities which determine the n price breaks with the corresponding unit purchase cost ci , i 1, 2,3,........, n (c1 c2 ........ cn ) . There are two possibilities of the optimal order quantity: 1) the optimal order size (Q) may lie between any two price breaks, qn Q qn1 , or 2) the optimal order size (Q) may occur on any of the price breaks
Q q n . For the first case, the optimal order size can be found from Eq. (7) by using the necessary conditions (22) and (23) for the model without shortages and from Eq. (17) by using the necessary conditions Eqs. (27), (28) and (29) for the model with shortages. For the second case, if the optimal order size is any price breaks, then Q is fixed. So from Eq. (8) (without shortages model) we see that T depends on p and also from Eq. (18) (with shortages model) we see that t 2 depends on p and t1 . Hence, in this situation, the derived necessary conditions for both models are not suitable and thus we need to calculate new necessary conditions for both cases. 6.1 Necessary conditions when optimal order size (Q) occurs on any of the price breaks Let us derive the necessary condition for the model without shortages when the optimal order size (Q) occurs on any of the price breaks (q n ) first, and then the necessary conditions for second model with shortages case. 6.1.1 The model without shortages If we substitute each T with the help of Eq. (8), the total profit function TP1 contains only one independent variable namely p . Differentiating Eq. (7) with respect to p , one can get
18
dT b(1 T ) 1 ln dp (a bp) 1 T
.
(30)
The necessary condition to find the optimal selling price per unit p * can be found by calculating the first order derivative
dTP1 with the help of Eq. (30). Equating the result to zero and after dp
simplifying the terms, the necessary condition can be written as: 2 1 1 T (1 ) c j (1 ) ln 1 g 2 h 6 1 T 1 T 1 2 a 2 bp T b (1 T ) ln 2 3 2 1 T 2T T (1 )T K c jh c j g 4 12 (a bp) 9
1 c j bT (1 ) ln 1 T
1 (1 ) h 1 g 2 6
2
(31)
T 3 (1 )T 2 1 2 T T 2 (1 )T h 0 g 2 12 6 9 4
6.1.2 The model with shortages If we substitute each t 2 with the help of Eq. (18), the total profit function TP2 contains only two independent variables, namely, p and t1 . Consequently, in the shortages case, only two decision variables exist under which the total profit function TP2 ( p, t1 ) has to be maximized. Differentiating Eq. (17)with respect to p, anyone can find
1 1 t 2 b(1 t 2 ) ln(1 t 2 ) . (1 ) ln p (a bp) 1 t1
(32)
Again, differentiating Eq. (17) with respect to t1 , we find:
t2 1 τ (1 δ t2 ) . t1 1 τ t1
(33)
To find the optimal price of the product p * , the necessary condition can be found for any given
t1 by calculating the first order partial derivative
TP2 with the help of Eq. (32). Setting this p
result with equal to zero and after rearranging terms, we have:
19
1 τ 1 b(1 δt2 ) (1 τ ) ln ln(1 δt2 ) 1 τ t1 δ 2 p c cs t K c (1 τ ) ln 1 τ 1 g (1 τ ) h (1 τ ) l j 1 δ (a bp ) 2 6 1 τ t1 cs 1 t1 t2 p c j cl ln(1 δt2 ) δ δ (1 δt2 ) 2 2 2 3 c g t1 (1 τ )t1 c h t1 (1 τ )t1 1 τ t1 j j 4 2 9 12 6 (a 2bp)t a 2bp c b b c cs ln(1 δt2 ) b c cs t 1 j l l 2 δ δ δ 1 τ (1 τ ) (1 τ ) 2 (t1 t2 ) c j b(1 τ ) ln h 0. 1 g 2 6 1 τ t1 2 2 2 3 c g t1 (1 τ )t1 c h t1 (1 τ )t1 1 τ t1 j j 4 2 9 12 6
(34)
* Again, the necessary condition for finding the optimal t1 can be found for any given p by
calculating the first order partial derivative
TP2 with the help of Eq. (33). Setting this result t1
equal to zero and after rearranging terms, the necessary condition is: 1 (t1 t 2 ) p c j 1 t1
t1 2 (1 )t1 (1 ) 2 (1 ) (1 ) 2 t1 (1 ) 1 g h c g c h j j 2 6 2 6 6 2 3
cs cs ln(1 t2 ) t1 t2 K p cl t1 p c j cl 1 t2 (a bp) 1 (1 ) (1 ) 2 1 (1 t2 ) c j (1 ) ln h 0. 1 g 1 t1 2 6 1 t1 2 3 2 2 c j g t1 (1 )t1 c j h t1 (1 )t1 (1 ) t1 2 12 6 4 9
(35)
We develop two algorithms for the aforementioned described models in order to obtain the optimal solutions under this discount environment. Additionally, we assume that the value of Q should be considered to the nearest integer value, while the values of p, t1 , t 2 , T and TPi should be
20
considered up to3-decimals. Now we present the flowchart and algorithm for both models which are given below. 6.1.3 Flowchart for the two models Before going to the discussion of solution algorithms, we have supplied two flowcharts, in figures 3 and 4 of our solution technique for each model in order to elucidate the solution algorithms. Inside the flowchart, we have used the equations numbers as well as mathematical expression. The first flowchart is for inventory model for without shortages case and the second one is for the inventory model with partially backlogged shortages case. 6.2 Algorithm for the model without shortages Step 1:
Initialize TPmax ( p, T ) 0 and i n .
Step 2:
Set j 1and initialize p ( j ) p1 , where p1 is the solution of a 2bp 0 .
Step 3:
Solve Eq. (22) to obtain T ( j ) by using the values of K , a, b, g , h, m and ci . Substitute the ( j) ( j) value of T ( j ) in Eq. (23) and solve for p1 . Set p ( j 1) p1 .
Step 4:
If p ( j 1) p ( j ) 10 4 , then set ( p, T ) p ( j 1) , T ( j ) and go to Step 5. Otherwise go to step 3 with j j 1 .
Step 5:
Substitute the values of p and T in Eq. (7) in order to calculate the corresponding order size Q . i) If Q belongs to the right unit purchase cost range (qn Q qn1 ) , then this solution is feasible. Calculate TPi ( p, T ) from Eq. (10) with the help of the optimal values of p and T . Now if TPi ( p, T ) TPmax ( p, T ) , set TPmax ( p, T ) TPi ( p, T ) . Go to Step 8. ii) If Q does not belong to the right unit purchase cost range (qn Q qn1 ) , then obtained solution is infeasible and go to Step 6.
21
Begin
Initialize
TPmax ( p, T ) 0 and i n
Input all the values of given parameters
Set j 1 and p ( j ) p1 , where p1 is the solution of a 2bp 0
Solve Eq. (22) for T
( j)
and using this T
( j)
( j 1)
( j)
Set p
p1
solve Eq. (23) for p1
( j)
.
Set j j 1
No
p ( j 1) p ( j ) 10 4
Set Q qn
Yes
Set ( p, T ) p
( j 1)
,T ( j)
Q T (1 ) 1 e ( a bp )(1 )
Calculate the corresponding Q from Eq. (7)
Plug T into Eq. (31) and solve for p No
q n Q q n 1
Calculate T from Eq. (8) with derived p
Yes Calculate
TPi ( p, T ) from Eq. (10)
TPi ( p, T ) TPmax ( p, T )
Yes
TPmax ( p, T ) TPi ( p, T )
No Print maximum profit TPmax ( p, T ) with optimal
End
p * and T *
No
i2
Yes
Set i i 1
Fig.3. Flowchart of the solution technique for without shortages case 22
Begin
Initialize
TPmax ( p, t1 , t 2 ) 0 and i n
Input all the values of given parameters
Set j 1 and p ( j ) p1 , where p1 is the solution of a 2bp 0
Solve Eqs. (27) and (28) for t1
( j)
solve Eq. (29) for p1
, t2
( j)
( j)
and using this t
1
. Set p
( j 1)
p1
( j)
, t2
( j)
( j)
No
p ( j 1) p ( j ) 10 4
Set Q qn
Yes
Set ( p, t1 , t 2 ) p ( j 1) , t1
( j)
, t2
( j)
1 Q 1 ( a bp) (1 ) ln 1 t1 t2 e 1
Calculate the corresponding Q from Eq. (17)
No
q n Q q n 1 Yes Calculate
Set j j 1
Plug t 2 into Eqs. (34)& (35) and solve for p & t 1
Calculate t 2 from Eq. (18) with derived p & t 1
TPi ( p, t1 , t 2 ) from Eq. (20)
TPi ( p, t1 , t 2 ) TPmax ( p, t1 , t 2 )
Yes
TPmax ( p, t1 , t 2 ) TPi ( p, t1 , t 2 )
No Print maximum profit TPmax ( p, t1 , t1 ) with optimal
No
i2
Yes
Set i i 1
*
p * ,t1 and t 2 * Fig.4. Flowchart of the solution technique for with shortages case End
23
Step 6:
Set Q qn in Eq. (8) and calculate the corresponding cycle length T in terms of p . Plug the cycle length into Eq. (31) and solve for p .Calculate the corresponding optimal cycle length from Eq. (8) and finally calculate the corresponding TPi ( p, T ) from Eq. (10) with the help of the optimal values of p and T . If TPi ( p, T ) TPmax ( p, T ) , set
TPmax ( p, T ) TPi ( p, T ) . Go to Step 7. Step 7:
If i 2 , go to Step 2 with i i 1 ; otherwise go to Step 8.
Step 8:
Find the total profit per unit time is TPmax ( p, T ) by using the optimal value of p and T .
6.3Algorithm for the model with partially backlogged shortages Step 1:
Initialize TPmax ( p, t1 , t 2 ) 0 and i n .
Step 2:
Set j 1and initialize p ( j ) p1 , where p1 is the solution of a 2bp 0 .
Step 3:
Solve Eqs. (27) and (28) to obtain t1 , t 2
( j)
( j)
by
substituting all the input values
( K , a, b, g , h, m, cl , cs ) and ci . Substitute the values of t1( j ) , t 2 ( j ) in Eq. (E3) and solve for p1 . Set p ( j 1) p1 . ( j)
Step 4:
( j)
If p ( j 1) p ( j ) 10 4 , then set ( p, t1 , t 2 ) p ( j 1) , t1 , t 2 ( j)
( j)
and go to Step 5. Otherwise
go to step 3 with j j 1 . Step 5:
Substitute the values of p,t1 and t 2 in Eq. (17) in order to calculate the corresponding order size Q . i) If Q belongs to the right unit purchase cost range (qn Q qn1 ) , then this solution is feasible. Calculate TPi ( p, t1 , t 2 ) from Eq. (20) with the help of the optimal values of
p,t1 and t 2 . Now if TPi ( p, t1 , t 2 ) TPmax ( p, t1 , t 2 ) , set TPmax ( p, t1 , t 2 ) TPi ( p, t1 , t 2 ) . Go to Step 8. ii) If Q does not belong to the right unit purchase cost range (qn Q qn1 ) , then this solution is infeasible and go to Step 6. Step 6:
Set Q qn in Eq. (18) and calculate the corresponding t 2 in terms of p and t1 .Plug the expression of t 2 into Eqs. (34)and (35) and solve for p and t1 . Calculate the
24
corresponding t 2 from Eq. (18) and finally calculate the corresponding TPi ( p, t1 , t 2 ) from Eq.
(20)
with the
help
of the
optimal
values
of
p,t1 and t 2 . If
TPi ( p, t1 , t 2 ) TPmax ( p, t1 , t 2 ) , set TPmax ( p, t1 , t 2 ) TPi ( p, t1 , t 2 ) .Go to Step 7. Step 7:
If i 2 , go to Step 2 with i i 1 ; otherwise go to Step 8.
Step 8:
Find the total profit per unit time is TPmax ( p, t1 , t 2 ) with the optimal values of p,t1 and t 2 .
7. Numerical illustration and comparisons In this section, we have illustrated two numerical examples for both cases with the help of our developed algorithms in order to study the applicability of our theoretical results and also to achieve managerial insights of our proposed problems. The data of the numerical examples are taken from Alfares and Ghaithan (2016) with some additional data to adapt our models. Example 1: Model without shortages Let us consider the following example Quantity
0 q1 Q q2 100
100 q2 Q q3 200
200 q3 Q q4
Purchase cost
c1 $5 /unit
c2 $4.75 /unit
c3 $4.5 /unit
Along with
K $520 /order, a 100, b 1.5, g $0.2 /unit/week , h $0.05 /unit/week, 4
weeks. Step 1: Initialize TPmax ( p, T ) 0 and i 3 . Iteration 1: i 3 Step 2: Set j 1and initialize p (1) p1
a 33.33333 . 2b
Step 3: c3 $4.5 (200 Q ) The solutions of Eqs. (22) and (23) by executing the Step 3 to Step 4 in the computational algorithm 6.2, we can obtain p $37.204 , T 2.398 weeks. Step 5: With the help of the values of p and T , the corresponding lot size from Eq. (7) is Q 144 units and the retailer’s total profit from Eq. (10) is TP3 ( p, T ) $1085.265 . Since Q does not belong to the following range (200 Q ) , this is not a feasible solution and hence goes to step 6. Step 6: c3 $4.5 , Q 200 units Setting Q 200 units in Eq. (8) and then calculate the corresponding cycle length T in terms of p .Plug this derived cycle length into Eq. (31) and solve for p gives: p $36.606 . Then the corresponding cycle length from Eq. (8) is T 2.941 weeks. Finally, with the help of the optimal values of p and T from Eq. (10), we get 25
TP3 ( p, T ) $1067.524 .
Since TP3 ( p, T ) $1067.524 0 TPmax ( p, T ) ,
set
TPmax ( p, T ) $1067.524 .Go to step 7. Step 7: Set i 2 and go to step 2. Iteration 2: i 2 a Step 2: Set j 1and initialize p (1) p1 33.33333 . 2b Step 3: c3 $4.75 (100 Q 200) The solutions of Eqs. (22) and (23) by executing the Step 3 to Step 4 in the computational algorithm 6.2, we can obtain p $37.379 , T 2.361 weeks. Step 5: Using these values of p and T , the corresponding lot size from Eq. (7) is Q 140 units and the retailer’s total profit from Eq. (10) is TP2 ( p, T ) $1066.409 .Since Q belongs to the following range (100 Q 200) , this solution is feasible and TP2 ( p, T ) $1066.409 TPmax ( p, T ) $1067.524 . Therefore, TPmax ( p, T ) $1067.524 which has found in step 6 of iteration 1 and hence go to step 8. * Step 8: The optimal solution is p * $36.606 , T 2.941weeks, TPmax ( p, T ) $1067.524 .
T * 2.941weeks, Q * 200 units, Therefore the optimal solution is p * $36.606 , TPmax ( p, T ) $1067.524 .The concavity can be investigated from the Fig. 5 of the total profit of the problem against p . 1070 1060 1050
Total Profit
1040 1030 1020 1010 1000 990 980 32
34
36
38
40
42
44
p
Fig. 5. TP1 ( p, T ) versus p .
Example 2: Model with partially backlogged shortages The data are same those are used in Example 1. Additionally, we consider that
0.4, cs 3 and cl 5 for the backlogged shortages. Then the optimal solution of the problem can be founded using algorithm 5.2 as follows: 26
Step 1: Initialize TPmax ( p, t1 , t 2 ) 0 and i 3 . Iteration 1: i 3 a Step 2: Set j 1and initialize p (1) p1 33.33333 . 2b Step 3: c3 $4.5 (200 Q ) The values of p,t1 and t 2 from the Eqs. (27), (28) and (29) can be found by executing the Step 3 to Step 4 in the computational algorithm 6.3, we can obtain p $36.928 , t1 2.27 weeks, t 2 0.491week. Step 5: With the help of the values of p,t1 and t 2 , the corresponding lot size from Eq. (17) is Q 155 units and the retailer’s total profit from Eq. (20) is TP3 ( p, t1 , t 2 ) $1117.701 . Since Q does not belong to the following range (200 Q ) , this is not a feasible solution and hence goes to step 6. Step 6: c3 4.5 , Q 200 Setting Q 200 in Eq. (18) and then calculate the shortages period t 2 in terms of p and t1 . Plugging the shortages period into Eqs. (34) and (35) and solving we get p $36.462 , t1 2.699 weeks and the corresponding shortages period from Eq. (18), t 2 0.594 week. With the help of these derived values and all given values, from Eq. (20), TP3 ( p, t1 , t 2 ) $1107.94 . Since TP3 ( p, t1 , t 2 ) $1107.94 0 TPmax ( p, t1 , t 2 ) , set
TPmax ( p, t1 ,2) $1107.94 .Go to step 7. Step 7: Set i 2 and go to step 2. Iteration 2: i 2 Step 2: Set j 1and initialize p (1) p1 Step 3: c3 $4.75 (100 Q 200)
a 33.33333 2b
The values of p,t1 and t 2 from the Eqs. (27), (28) and (29)can be found by executing the Step 3 to Step 4 in the computational algorithm 6.3, we can obtain p $37.083 , t1 2.227 weeks, t 2 0.504 week. Step 5: Using these values of p,t1 and t 2 , the corresponding lot size from Eq. (17) is Q 151 units and the retailer’s total profit from Eq. (20) is TP2 ( p, t1 , t 2 ) $1100.798 . Since Q belongs to the following range (100 Q 200) , this solution is feasible and TP2 ( p, t1 , t 2 ) $1100.798 TPmax ( p, t1 , t 2 ) $1107.94 .Therefore,
TPmax ( p, t1 , t 2 ) $1107.94 which has found in step 6 of iteration 1 and hence go to step 8. Step 8: The optimal solution is TPmax ( p, t1 , t 2 ) $1107.94 .
p * $36.462 , t1* 2.699 weeks, t 2 0.594 week, *
27
Therefore
the
optimal
is p * $36.462 ,
solution
t1 2.699 weeks, *
t 2 0.594 week, *
T * t1 t 2 3.293 weeks, S * 176 units, R * 24 units, Q * 200 units and TPmax ( p, t1 , t 2 ) $1107.94 . *
*
The concavity can be investigated from the Fig. 6 of the total profit against p and t1 . 1100
1200
1050 1100 1000 1000
Total Profit
950 900
900
800
850 800
700 750 600 3
700 2.5 t1
2
30
32
36
34
40
38
42
p
Fig. 6.Concavity of TP2 ( p, t1 , t 2 ) against p and t1 .
Moreover, we can observe the optimal solution easily from the two line diagrams, namely
p and TP2 ( p, t1 , t 2 ) versus t1 presenting in Fig. 7(a) and Fig. 7(b)
TP2 ( p, t1 , t 2 ) versus respectively.
1120
1110 1100
1100
1090 1080
Total Profit
Total Profit
1080 1070 1060 1050
1060
1040
1020
1040 1000
1030 1020 32
33
34
35
36 p
37
38
Fig. 7(a). TP2 ( p, t1 , t 2 ) versus p
39
40
980 2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3
t1
Fig. 7(b). TP2 ( p, t1 , t 2 ) versus t1
28
Closing this section, we compare our numerical results with those in the Alfares and Ghaithan (2016) paper. These can be found in the following table 2: Table 2: Comparisons of our results with Alfares and Ghaithan (2016) Proposed model Alfares and Ghaithan (2016) Inventory model with partial backlogged The Alfares and Ghaithan (2016) results are shortage gives better result than Alfares and given below: Ghaithan (2016). If we consider the expiration date is large number (say, TP=1207.20, T=4.423,Q=200,p=36.52. 9999 weeks)and considered partial backlogged shortage with the length of the waiting time of the customer then we get better results. In addition, we are giving the both result TP= 1217.78,t1= 4.092, T= 4.445, Q=200, R=14.935, p= 36.51897.
From the above table, we can conclude that the proposed model is more profitable in the perspective of profit maximization. 8.
Sensitivity analysis
Here, we have described two sensitivity analyses of the optimal solutions with respect to different parameters for both models which are presented in Table 3 and Table 4. Changing the value of one parameter at a time and keeping same values of the rest parameters, we observed the effects of different parameters on the objective function and different variables i.e., on the price of the product (p), zero ending time (t1 ) , shortages period (t 2 ) , cycle length of the model (T), maximum stock level (S), highest shortage level (R), order quantities (Q) and total profit of the whole inventory system (TP) based on the used examples. We changed the value ofone parameter in a relative steps of 10% (-20%, -10%, 10%, +20%)but once at a time. Table 3: Sensitivity analysis of different parameters for example 1 Parameter
Original Value
K
520
a
100
b
1.5
*
Changes Value
p*
T*
Q*
TP1
416 468 572 624 80 90 110 120 1.2
37.200 37.292 36.726 36.845 30.948 34.151 39.985 43.353 44.949
2.186 2.278 2.948 2.955 2.572 2.458 2.753 2.585 2.902
127 134 200 200 121 131 200 200 200
1112.120 1088.827 1049.863 1032.246 549.514 790.710 1384.560 1734.529 1480.968
29
g
0.2
h
0.05
4
c1 , c2 , c3
5, 4.75, 4.5
1.35 1.65 1.80 0.16 0.18 0.22 0.24 0.04 0.045 0.055 0.06 3.2 3.6 4.4 4.80 4, 3.8, 3.6 4.5, 4.275, 4.05 5.5, 5.225, 4.95 6, 5.7, 5.4
40.315 33.570 31.847 36.595 36.600 37.419 37.459 36.627 36.616 36.595 37.386 37.447 37.410 36.628 36.643 36.200 36.403 37.709 38.036
2.921 2.960 2.383 2.940 2.940 2.334 2.308 2.942 2.941 2.940 2.343 2.165 2.268 3.026 3.103 2.916 2.928 2.296 2.239
200 200 138 200 200 138 136 200 200 200 139 133 137 200 200 200 200 134 127
1251.100 917.613 793.570 1083.805 1075.664 1060.310 1054.310 1071.286 1069.405 1065.644 1064.118 1040.495 1054.595 1079.896 1090.328 1149.024 1108.212 1031.367 997.265
From Table 3, we can make the following observations: (i) The total profit for the first model TP1* decreases with respect to the augmenting values of
K , b, g , h, c1 , c2 and c3. It indicates that the retailer will earn fewer amounts if the values of the said parameters increase. Also, it increases for the rest of the parameters a and . So, retailer can earn more money if these two parameters’ values increase. Additionally, the demand parameter a has the greatest positive impact on increasing the retailer’s total profit because the higher value of a helps the retailer to earn the revenue to a great extent by increasing the demand. (ii) Optimal order size (Q*) rises when the values of the parameters K , a and are changed from -20% to -10% but remains constant for the changing from +10% to +20%. It is also observed that the order size declines with respect to the parameters b, h, c1 , c2 and c3 , if we increase the values of these parameters up to+20% and remains constant for -20% to -10% whereas we observed that order size remains constant if we change the value of g from -20% to -10% and decreases if we change the value from +10% to +20%. It indicates that if the value of g i.e., constant part of the unit holding cost increases, the decision maker (retailer) wants to reduce the total holding cost by making a small order size.
30
(iii) The optimal cycle length for the first model (T * ) rises with respect to the augmenting values of the parameters K and whereas it declines if we increase the value of the parameter a. This reveals that if the ordering cost increases, retailer wants to decrease it by increasing order size and also if maximum lifetime of the product increases, retailer makes a large order; consequently, the optimal cycle length increases. On the other hand, if the demand parameter a increases, retailer can sell his/her products so quickly because of customers’ high demand. It is to be noted that for rest of the parameters the cycle length is increasing first and then decreases. (iv) Optimal price of the product ( p* ) increases if we change the values of the parameters
a, g , c1 , c2 and c3 whereas it decreases if we change the values of b and . So, this reveals that if the unit purchase cost and unit carrying cost increase, retailer wants to cover his/her costs by increasing unit selling price of the product and also if the value of b increases, as a result, product’s demand decreases and then retailer makes the decision to decrease the selling price in order to boost customers demand. Additionally, it is to be noted that when we change the parameter value of K it proliferates first then declines whereas if we change the parameter value of h it decreases initially and then increases i.e., exactly reverse of the parameter K. Table 4: Sensitivity analysis of different parameters for example 2 Parameter
Original Value
K
520
a
100
b
1.5
g
0.2
h
0.05
*
New Values
p*
t1
416 468 572 624 80 90 110 120 1.2 1.35 1.65 1.80 0.16 0.18 0.22 0.24 0.04 0.045
36.945 36.347 36.576 36.691 30.574 33.062 39.849 43.228 44.807 40.170 33.428 30.900 36.472 36.467 36.456 36.450 36.483 36.472
2.064 2.698 2.700 2.702 2.397 2.887 2.533 2.385 2.700 2.699 2.701 2.703 2.713 2.706 2.692 2.686 2.704 2.702
t2
*
0.434 0.577 0.613 0.63 0.722 0.733 0.493 0.416 0.485 0.54 0.647 0.7 0.559 0.577 0.612 0.629 0.585 0.589
*
S*
R*
Q*
TP2
119 176 175 175 111 174 177 179 179 178 174 173 177 176 175 175 176 176
18 24 25 25 22 26 23 21 21 22 26 27 23 24 25 25 24 24
137 200 200 200 133 200 200 200 200 200 200 200 200 200 200 200 200 200
1140.559 1123.774 1092.196 1076.543 588.405 828.368 1421.220 1768.092 1515.206 1288.512 960.879 838.926 1119.877 1113.881 1102.053 1096.220 1110.470 1109.203
31
4
0.4
cl
5
cs
3
c1 , c2 , c3
5, 4.75, 4.5
0.055 0.06 3.2 3.6 4.4 4.8 0.32 0.36 0.44 0.48 4 4.5 5.5 6 2.4 2.7 3.3 3.6 4, 3.8, 3.6 4.5, 4.275, 4.05 5.5, 5.225, 4.95 6, 5.7, 5.4
36.451 36.440 36.383 36.427 36.489 36.510 36.456 36.459 36.465 36.469 36.452 36.457 36.466 36.470 36.448 36.455 36.468 36.474 36.094 36.278 36.643 36.824
2.697 2.695 2.510 2.610 2.779 2.850 2.658 2.680 2.716 2.730 2.693 2.696 2.702 2.705 2.690 2.695 2.704 2.708 2.714 2.707 2.692 2.684
0.599 0.603 0.651 0.62 0.572 0.554 0.688 0.638 0.556 0.523 0.609 0.602 0.587 0.58 0.616 0.605 0.583 0.574 0.491 0.541 0.648 0.703
176 176 174 175 177 177 172 174 177 179 175 176 176 176 175 175 176 177 179 178 174 172
24 24 26 25 23 23 28 26 23 21 25 24 24 24 25 25 24 23 21 22 26 28
200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200
1106.681 1105.426 1085.561 1097.718 1116.637 1124.116 1114.191 1110.839 1105.407 1103.175 1108.798 1108.364 1107.524 1107.116 1109.241 1108.580 1107.319 1106.717 1178.092 1142.728 1073.723 1040.073
From Table 4, we can make the following observations: (i) The
total
profit
for
the
second
model TP2* declines
with
respect
to
K , b, g , h, , cl , cs , c1, c2 and c3. Among these mentioned parameters the demand parameter b has the greatest negative effect on the retailer’s total profit since if b increases, customers’ demand decreases and as a result retailer’s total revenue also abates. On the other hand, total profit increases with respect to the parameters a and . Moreover, the demand parameter a has the greatest positive effect on increasing the retailer’s total profit because the higher value of a helps the retailer to earn the revenue to a great extent by increasing the demand. (ii) Optimal order size (Q*) increases if we increase the values from -20% to -10% of K and a but it is fixed for all other changes of the values of K and a as well as for all changes of the rest parameters. Hence, if the ordering cost increases, retailer wants to decrease it by increasing order size and if the demand parameter a increases, customers’ demand also increases and consequently, decision maker makes a large order size. (iii) Maximum shortage level (R*)increases with respect to the changes of the parameters
K , b, g , c1 , c2 and c3 whereas it decreases with respect to the changes of the parameters
32
, , cl and cs .It is also remarkable that the time varying holding cost parameter h has no impact on the optimal value of maximum shortages. (iv) Highest stock level (S*) increases when we change the values of , a, E, , cl and cs whereas it decreases with respect to the changes of the parameters b, g , c1 , c2 and c3 . Among these mentioned parameters the demand parameter a has the greatest impact on stock level in a positive way as the higher value of a aggrandizes the demand and hence retailer wants to stock in a large extent. It is also remarkable that the optimal value of the highest stock level has no change if we change the value of h. (v) The shortages period of the inventory system (t 2 * ) increases if we change the values of the parameters K , b, g , h, , c1 , c2 and c3. Again, it decreases if we change the values of
, cl and cs . In addition, both the demand parameters a and b have the greatest effect on the optimal shortages period. It reflects the fact that for a higher value of a or a lower value of b , the demand augments significantly and then products are consumed faster. As a result, the duration of the shortages period waxes. (vi) Inventory
ending
time
(t1* ) increases
when
we
change
the
parameters
K , , , cl , cs , c1 , c2 and c3 whereas decreases if we change the values of the parameters h and g. In addition, it is to be noted that if we change the values of the parameters a and b ,
initially, the value of t1 increases and then decreases. (vii) Optimal price of the product ( p* ) increases if we change the values of the parameters
a, g , , cl , cs , c1 , c2 and c3 whereas it decreases if we change the value of the parameters
b and . Additionally, it is to be noted that when we change the value of K , selling price rises first then declines whereas if we change the value of h it declines initially and then rises. Among the all parameters the demand parameter b has the foremost important impact on the selling price of the product. For the lower values of b the optimal selling price increases most. Though the parameter b has the greatest impact on increasing p * but the parameter b has more effect on increasing the total profit. It is important to mention that a higher selling price does not always give a higher total profit. 9. Conclusion and future research directions 33
Every product has a fixed lifetime i.e., the products will be damaged or destroyed or deteriorated totally after certain time periods. After this time period, this type product cannot be perfect in usable condition for mankind. So, manufacturers/suppliers want to sell this type of products within their lifetime. If they cannot able to sell these products within maximum lifetime period, then they totally lose the invested amount. Consequently, to shun this situation and sell their products within the expiration date times they offer price discounts on unit purchase cost based on the order size. In this connection, if anyone wants to buy something in a shopping mall or other wholesale markets, he/she can observe price discount on the purchased amount or purchased quantity. So, it has a huge impact in the retail or wholesale markets. Under this discount environment, in this work, we have introduced two inventory models, namely, (i) zero ending inventory model and (ii) partial backlogging shortages inventory model for an expiration rate dependent deteriorating goods with variable demand depends on price. Shortages are accumulated with a decreasing rate of the customers waiting time. Two solution algorithms along with the flowcharts in order to solve our proposed problems have been constructed. The optimality for each model has been investigated mathematically. To observe the applicability of the algorithms, we solved two numerical examples by using the proposed algorithms, validated the obtained results graphically by using MATLAB software and conducted two sensitivity analyses. Our basic observations are that the demand parameters a and b for both models have the greatest effect on increasing the retailer’s total profit which reflects the fact that the decision maker (retailer) can increase the profit by following proper marketing policies for boosting customers’ demand. Another way retailer can increase his/her total profit by giving meticulous concentration to negotiate the unit purchase cost with the manufacturer/supplier. Alfares and Ghaithan (2016) studied an inventory model for zero ending case and didn’t consider the effects of deterioration and shortages. Comparing our results with those obtained in Alfares and Ghaithan (2016),we observe that the retailer’s optimal cycle length, T * , in our models are lower than the corresponding one in the model of Alfares and Ghaithan (2016). In the model of Alfares and Ghaithan (2016), the products are non-deteriorating and do not have any limited lifespan or expiration date. In contrast to the Alfares and Ghaithan (2016) model, the products in our models do not have only maximum lifetime but also a gradually increasing deterioration rate which reaches to 100% at their expiration dates. In our examples, the maximum lifetime of the products 34
has been considered as 4 weeks. That’s why the retailer wants to vend his/her products within this period and consequently, T * is lower than the optimal cycle length of Alfares and Ghaithan (2016). The optimal selling price ( p * ) in our model without shortages is higher than the corresponding one in the model of Alfares and Ghaithan (2016). But p * for the model with shortages is lower than the optimal selling price in Alfares and Ghaithan (2016). As our models are applicable for deteriorating products, the retailer tries to cover his/her loss by increasing the selling price per unit from the very beginning of the cycle. That’s why p * is higher than that of Alfares and Ghaithan (2016). If the retailer lessens the selling price in order to boost the customers’ demand, then the products have been consumed so faster and hence shortages have been occurred. This situation has been reflected in the proposed model with shortages. It is interesting to mention that, if we consider the products with a higher maximum lifetime (say,
9999 weeks), then the model without shortages gives the same results of Alfares and Ghaithan (2016) and the shortage case gives better result than Alfares and Ghaithan (2016) in the perspective of profit maximization. This model is more applicable for products like fruit, vegetables, electronic gadgets, gasoline, alcohol, medicine, dried food, rice, wheat, volatile substances and radioactive materials, etc. which deteriorate or decline over time and have also an expiry period after which they lose their usability and are not in a condition for sale. In the perspective of the retail marketing, suppliers buy these products directly from the manufacturer/producer and they transport the products to their local area and sale to their retailers or any shopping mall. Retailers sell the products directly to the customers. The proposed models are directly applicable between the manufacturer/producer to suppliers and suppliers to retailers. One of the interesting extensions of the current work would be possible by incorporating multiple equal sized advance payments scheme or multiple equal sized delay payments scheme. Relaxing zero ending inventory by non-ending inventory a profit maximization problem could be another possible extension. Also, anyone can extend this model by taking an imprecise value of inventory parameters (for instance, fuzzy-valued inventory parameters and defuzzyfy by any defuzzification technique, fuzzy-valued inventory cost and then apply alpha-cut and convert into interval or directly interval-valued inventory cost).
35
Acknowledgments The authors thank the three anonymous referees for their constructive comments and suggestions that improved the paper. Appendix A: Let us assume the following auxiliary functions from the equation (10), (1 )2 1 T 2 (1 )T 1 f1 (T ) ap bp 2 T K c j (a bp )(1 ) ln ln c j g (a bp ) 2 1 T 1 E T 4 2
(A1)
1 3 1 T 3 (1 )T 2 1 2 T c j h(a bp) ln 12 6 6 1 T 9
g1 (T ) T .
(A2)
Hence, we can write the total profit function in the following way TP1 ( p, T )
f1 (T ) . Now taking g1 (T )
the derivatives of f1 (T ) with respect to T for any given p , we have c j g (a bp ) (1 ) 2 1 2 f1 (T ) ap bp c j (a bp) T (1 ) 1 T 2 1 T
(A3)
c j h(a bp ) 1 3 2 2T 2 (1 )T 1 6 1 T
and f1 (T ) c j (a bp)
To
show
1 3 1 T
2
1 3 1 T
2
that
1
1 T the
2
c j h(a bp) 1 3 c j g (a bp) (1 )2 . 1 4 T (1 ) 2 6 1 T 2 1 T 2
function f1 (T ) is
4T (1 ) is
always
strictly
non-negative.
concave, Using
we
need
Lemma
1,
to we
(A4)
insure
that
see
that
4T (1 ) is always positive. As a result, f1 (T ) 0 for all T 0 and hence
f1 (T ) is differentiable, non-negative function and concave. Moreover, the function g1 (T ) T is positive, differentiable and convex function. Accordingly, the total profit function TP1 ( p, T ) is a strictly pseudo-concave function of T for any certain p , and hence there exists a unique optimal solution T * .
□
Appendix B: 36
Let us consider that 3 1 (u ) 4u (1 ). 1 u 2
(B1)
Therefore, at the point u 0 the value of is 3 1 (0) (1 ) 0 1 2
2 1
(B2)
3
Also, (u )
1 u 3
4 0 . This indicates that (u ) is an increasing function of u .
Consequently, (u ) is always positive for all u 0 .
□
Appendix C: For a settled cycle length T , differentiate the objective function TP1 ( p, T ) twice partially with respect to the decision variable p , we get (1 ) 2 1 T 2 (1 )T 1 ln a 2bp T c j b(1 ) ln c j gb 2 1 T 1 T 4 2 TP 1 1 3 1 T 3 (1 )T 2 1 2 T p T c j hb 6 ln 1 T 9 12 6
(C1)
2TP 2b 0 . (C2) p 2 Therefore, TP1 ( p, T ) is the concave function of p for any given value of T 0 and with this value there exists a unique p such that TP1 ( p, T ) is maximized. This completes the proof.
□
Appendix D: Let us assume the following auxiliary functions from the equation (20), 1 1 1 f 2 (t1 , t2 ) (ap bp 2 ) t1 ln(1 t2 ) K c j (a bp) (1 ) ln ln(1 t2 ) 1 t1
(1 ) 2 1 t12 (1 )t1 c j g (a bp) ln 2 1 t1 4 2 (1 )3 1 t13 (1 )t12 (1 )2 t1 c j h(a bp) ln 12 6 1 t1 9 6
(D1)
c ln(1 t2 ) cl s (a bp) t2
37
and
g 2 (t1 , t 2 ) t1 t 2 . Hence, from Eq. (20) we can write TP2 ( p, t1 , t 2 )
(D2)
f 2 (t1 , t 2 ) .To use results of Cambini and g 2 (t1 , t 2 )
Martein (2009), we need to show that f 2 (t1 , t 2 ) is a non-negative, differentiable and (strictly) joint concave function with respect to t1 and t 2 . The second order partial derivatives with respect to t1 and t 2 are:
1 h (1 ) 3 g (1 ) 2 c ( a bp ) 1 4 t ( 1 ) (D3) j 1 2 1 t1 2 2 (1 t1 ) 2 6 (1 t1 ) 2 t1 2 f 2 (t1 , t 2 ) 2 f 2 (t1 , t 2 ) (D4) 0 t 2 t1 t1t 2
2 f 2 (t1 , t 2 )
c p c j cl s t 2 2 f 2 (t1 , t 2 ) 2 t Therefore, the Hessian matrix for f 2 (t1 , t 2 ) is: H ii 2 1 f 2 (t1 , t 2 ) t t 2 1 and the first principal minor is: 2 f 2 (t1 , t 2 ) 2
H11
(a bp) 2 (1 t 2 )
(D5)
2 f 2 (t1 , t 2 ) t1t 2 , 2 f 2 (t1 , t 2 ) 2 t 2
1 h (1 )3 2 f 2 (t1 , t2 ) g (1 )2 c ( a bp ) 1 4 t (1 ) . j 1 2 2 2 (1 t1 )2 t12 6 (1 t1 ) 1 t1
(1 ) 3 4t1 (1 ) is always positive and hence, H 11 0 . Using Lemma 1, we see that (1 t1 ) 2 Again, the second principal minor is 2 f 2 (t1 , t 2 ) 2 f 2 (t1 , t 2 ) 2 f 2 (t1 , t 2 ) 2 f 2 (t1 , t 2 ) H 22 2 2 t 2 t1 t1t 2 t1 t 2
c j (a bp) 2 h (1 ) 3 c s 1 g (1 ) 2 p c c 1 4 t ( 1 ) 0 j l 1 1 t1 2 2 (1 t1 ) 2 6 (1 t1 ) 2 (1 t 2 ) 2 Consequently, the Hessian matrix for f 2 (t1 , t 2 ) is negative definite. Therefore, f 2 (t1 , t 2 ) is a nonnegative, differentiable and (strictly) concave function with respect to t1 and t 2 simultaneously. Moreover, the function g 2 (t1 , t 2 ) t1 t 2 is positive, differentiable and convex function, so the retailer’s total profit function per unit time TP2 ( p, t1 , t 2 ) is jointly pseudo-concave function in
(t1 , t 2 ) , and has only one maximum value. Consequently, the objective function TP2 ( p, t1 , t 2 ) , 38
for any fixed p , attains the global maximum value at the point (t1* , t 2 * ) .This completes the □
proof. Appendix E:
For any certain values of t1 0 and t2 0 , differentiate the objective function TP2 ( p, t1 , t 2 ) twice partially with respect to p , we get:
2TP2 ( p, t1 , t2 ) 2b 2 t1 t2 p
1 t1 δ ln(1 δt2 ) 0.
(E1)
Therefore, TP2 ( p, t1 , t2 ) is a concave function in p for any given t1 0, t2 0 and hence there exists a unique p such that TP2 ( p, t1 , t2 ) is maximized. References 1. Alfares, H. K., and Ghaithan, A. M. (2016). Inventory and pricing model with pricedependent demand, time-varying holding cost, and quantity discounts. Computers and Industrial Engineering, 94, 170-177. 2. Avinadav, T., Herbon, A., and Spiegel, U. (2013). Optimal inventory policy for a perishable item with demand function sensitive to price and time. International Journal of Production Economics, 144(2), 497-506. 3. Bhunia, A., and Shaikh, A. (2014). A deterministic inventory model for deteriorating items with selling price dependent demand and three-parameter Weibull distributed deterioration. International Journal of Industrial Engineering Computations, 5(3), 497-510. 4. Cambini, A and Martein, L. (2009). Generalized Convexity and Optimization: Theory and Applications. Springer-Verlag Berlin Heidelberg. 5. Chen, S. C., and Teng, J. T. (2015). Inventory and credit decisions for time-varying deteriorating items with up-stream and down-stream trade credit financing by discounted cash flow analysis. European Journal of Operational Research, 243(2), 566-575. 6. Chen, S.C., Min, J., Teng, J. T., and Li, F., (2016). Inventory and shelf-space management for fresh produce with freshness-and-stock dependent demand and expiration date. Journal of the Operational Research, 67(6), 884-896. 7. Das Roy, M., and Sana, S. S. (2017). Random sales price-sensitive stochastic demand: An imperfect production model with free repair warranty. Journal of Advances in Management Research, 14(4), 408-424. 8. Feng, L., Chan, Y. L., and Cárdenas-Barrón, L. E. (2017). Pricing and lot-sizing polices for perishable goods when the demand depends on selling price, displayed stocks, and expiration date. International Journal of Production Economics, 185, 11-20. 9. Ghare, P. M. and Schrader, G. P.(1963). A model for an exponentially decaying inventory. Journal of Industrial Engineering, 14, 238-243.
39
10. Ghoreishi, M., Weber, G. W., and Mirzazadeh, A. (2015). An inventory model for noninstantaneous deteriorating items with partial backlogging, permissible delay in payments, inflation-and selling price-dependent demand and customer returns. Annals of Operations Research, 226(1), 221-238. 11. Gupta, R. K., Bhunia, A. K., and Goyal, S. K. (2009). An application of genetic algorithm in solving an inventory model with advance payment and interval valued inventory costs. Mathematical and Computer Modelling, 49(5-6), 893-905. 12. Guria, A., Das, B., Mondal, S., and Maiti, M. (2013). Inventory policy for an item with inflation induced purchasing price, selling price and demand with immediate part payment. Applied Mathematical Modelling, 37(1-2), 240-257. 13. Hsu, P. H., Wee, H. M., and Teng, H. M. (2006). Optimal lot sizing for deteriorating items with expiration date. Journal of Information and Optimization Sciences, 27(2), 271-286. 14. Hsu, P. H., Wee, H. M., and Teng, H. M. (2007). Optimal ordering decision for deteriorating items with expiration date and uncertain lead time. Computers and Industrial Engineering, 52(4), 448-458. 15. Jaggi, C. K., Tiwari, S., and Goel, S. K. (2017). Credit financing in economic ordering policies for non-instantaneous deteriorating items with price dependent demand and two storage facilities. Annals of Operations Research, 248(1-2), 253-280. 16. Jain, R., and Singh, S. R. (2011). Inflationary implications on an inventory with expiration date, capital constraint and uncertain lead time in a multi-echelon supply chain. International Journal of Procurement Management, 4(4), 419-432. 17. Modak, N. M., Modak, N., Panda, S., and Sana, S. S. (2018). Analyzing structure of twoechelon closed-loop supply chain for pricing, quality and recycling management. Journal of Cleaner Production, 171, 512-528. 18. Maihami, R., and Kamalabadi, I. N. (2012). Joint pricing and inventory control for noninstantaneous deteriorating items with partial backlogging and time and price dependent demand. International Journal of Production Economics, 136(1), 116-122. 19. Maiti, A. K., Maiti, M. K., and Maiti, M. (2009). Inventory model with stochastic lead-time and price dependent demand incorporating advance payment. Applied Mathematical Modelling, 33(5), 2433-2443. 20. Mondal, B., Bhunia, A. K., and Maiti, M. (2003). An inventory system of ameliorating items for price dependent demand rate. Computers and industrial engineering, 45(3), 443-456. 21. Mukhopadhyay, S., Mukherjee, R. N., and Chaudhuri, K. S. (2004). Joint pricing and ordering policy for a deteriorating inventory. Computers and Industrial Engineering, 47(4), 339-349. 22. Mukhopadhyay, S., Mukherjee, R. N., and Chaudhuri, K. S. (2005). An EOQ model with two-parameter Weibull distribution deterioration and price-dependent demand. International Journal of Mathematical Education in Science and Technology, 36(1), 25-33.
40
23. Pal, B., Sana, S. S., and Chaudhuri, K. (2017). A stochastic production inventory model for deteriorating items with products’ finite life-cycle. RAIRO-Operations Research, 51(3), 669684. 24. Philip, G. C.(1974). A generalized EOQ model for items with Weibull distribution. AIIE Transactions, 6 (2), 159-162. 25. Roy, A. (2008). An inventory model for deteriorating items with price dependent demand and time-varying holding cost. Advanced Modelling and Optimization, 10(1), 25-37. 26. Roy, T., and Chaudhuri, K. S. (2007). An inventory model for a deteriorating item with price-dependent demand and special sale. International Journal of Operational Research, 2(2), 173-187. 27. Roy, A., Sana, S. S., and Chaudhuri, K. (2018). Optimal Pricing of competing retailers under uncertain demand-a two layer supply chain model. Annals of Operations Research, 260(1-2), 481-500. 28. Sana, S.S. (2011). Price-Sensitive demand for perishable items- an EOQ model. Applied Mathematics and Computation, 217(13),6248-6259. 29. Sridevi, G., Nirupama Devi, K., and SrinivasaRao, K. (2010). Inventory model for deteriorating items with Weibull rate of replenishment and selling price dependent demand. International Journal of Operational Research, 9(3), 329-349. 30. Taleizadeh, A. A. (2014). An EOQ model with partial backordering and advance payments for an evaporating item. International Journal of Production Economics, 155, 185-193. 31. Taleizadeh, A. A., Mohammadi, B., Cárdenas-Barrón, L. E., and Samimi, H. (2013). An EOQ model for perishable product with special sale and shortage. International Journal of Production Economics, 145(1), 318-338. 32. Taleizadeh, A. A., Jalali, M. S., and Sana, S. S. (2018). A game theoretic approach to optimal pricing of flights and passengers at congested airports. Journal of Modelling in Management, (just-accepted), 00-00. 33. Taleizadeh, A. A., Niaki, S. T. A., Aryanezhad, M. B., and Tafti, A. F. (2010). A genetic algorithm to optimize multiproduct multi-constraint inventory control systems with stochastic replenishment intervals and discount. The International Journal of Advanced Manufacturing Technology, 51(1-4), 311-323. 34. Taleizadeh, A. A., Niaki, S. T. A., and Aryanezhad, M. B. (2010). Optimizing multi-product multi-chance-constraint inventory control system with stochastic period lengths and total discount under fuzzy purchasing price and holding costs. International Journal of Systems Science, 41(10), 1187-1200. 35. Taleizadeh, A. A., Noori-daryan, M., and Cárdenas-Barrón, L. E. (2015). Joint optimization of price, replenishment frequency, replenishment cycle and production rate in vendor managed inventory system with deteriorating items. International Journal of Production Economics, 159, 285-295. 36. Taleizadeh, A. A., Pentico, D. W., Jabalameli, M. S., and Aryanezhad, M. (2013). An EOQ model with partial delayed payment and partial backordering. Omega, 41(2), 354-368. 41
37. Taleizadeh, A. A. (2017). Lot-sizing model with advance payment pricing and disruption in supply under planned partial backordering. International Transactions in Operational Research, 24(4), 783-800. 38. Tayal, S., Singh, S. R., and Sharma, R. (2014). Multi item inventory model for deteriorating items with expiration date and allowable shortages. Indian Journal of Science and Technology, 7(4), 463-471. 39. Tavakkoli, Sh., and Taleizadeh, A.A.,(2017). A lot sizing model for decaying item with full advance payment from the buyer and conditional discount from the supplier. Annals of Operations Research, 259,415-436. 40. Teng, J. T., Cárdenas-Barrón, L. E., Chang, H. J., Wu, J., and Hu, Y. (2016). Inventory lotsize policies for deteriorating items with expiration dates and advance payments. Applied Mathematical Modelling, 40(19-20), 8605-8616. 41. Tiwari, S., Cárdenas-Barrón, L. E., Goh, M., and Shaikh, A. A. (2018). Joint pricing and inventory model for deteriorating items with expiration dates and partial backlogging under two-level partial trade credits in supply chain. International Journal of Production Economics, 200, 16-36. 42. Wang, W. C., Teng, J. T., and Lou, K. R. (2014). Seller’s optimal credit period and cycle time in a supply chain for deteriorating items with maximum lifetime. European Journal of Operational Research, 232(2), 315-321. 43. Wu, J., and Chan, Y. L. (2014). Lot-sizing policies for deteriorating items with expiration dates and partial trade credit to credit-risk customers. International Journal of Production Economics, 155, 292-301. 44. Wu, J., Ouyang, L. Y., Cárdenas-Barrón, L. E., and Goyal, S. K. (2014). Optimal credit period and lot size for deteriorating items with expiration dates under two-level trade credit financing. European Journal of Operational Research, 237(3), 898-908. 45. Wu, J., Chang, C. T., Cheng, M. C., Teng, J. T., and Al-khateeb, F. B. (2016a). Inventory management for fresh produce when the time-varying demand depends on product freshness, stock level and expiration date. International Journal of Systems Science: Operations and Logistics, 3(3), 138-147. 46. Wu, J., Al-Khateeb, F. B., Teng, J. T., and Cárdenas-Barrón, L. E. (2016b). Inventory models for deteriorating items with maximum lifetime under downstream partial trade credits to credit-risk customers by discounted cash-flow analysis. International Journal of Production Economics, 171, 105-115. 47. Wu, J., Teng, J. T., and Chan, Y. L. (2018). Inventory policies for perishable products with expiration dates and advance-cash-credit payment schemes. International Journal of Systems Science: Operations and Logistics, 5(4), 310-326. 48. You, P. S. (2006). Ordering and pricing of service products in an advance sales system with price-dependent demand. European Journal of Operational Research, 170(1), 57-71. 49. Harris, F.W., (1913). How many parts to make at once: Reproduced from Factory, The Magazine of Management, 10(2), 135-136,152. 42
Highlights
We study two supply chain models by assuming the demand to be dependent on price.
We assume expiration rate dependent deterioration and shortages with partial backlogging.
We formulate the problem mathematically and prove its optimality.
A solution procedure has been ameliorated and two numerical examples are presented.
43
S0360-8352(19)30205-0 https://doi.org/10.1016/j.cie.2019.04.002 CAIE 5789
To appear in:
Computers & Industrial Engineering
Received Date: Revised Date: Accepted Date:
25 April 2018 27 March 2019 1 April 2019
Please cite this article as: Khan, A-A., Shaikh, A.A., Panda, G.C., Konstantaras, I., Taleizadeh, A.A., Inventory system withexpiration date: pricingand replenishment decisions, Computers & Industrial Engineering (2019), doi: https://doi.org/10.1016/j.cie.2019.04.002
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Inventory system with expiration date: pricing and replenishment decisions
Inventory system with expiration date: pricing and replenishment decisions Md. Al-Amin Khan Department of Mathematics, Jahangirnagar University, Savar, Dhaka-1342, Bangladesh, [email protected] Ali Akbar Shaikh Department of Mathematics, The University of Burdwan, Burdwan-713104, India, [email protected]
Gobinda Chandra Panda Department of Mathematics, Mahavir Institute of Engineering and Technology, BBSR, India752054, [email protected] Ioannis Konstantaras* Department of Business Administration, School of Business Administration, University of Macedonia, 156 Egnatia Str., Thessaloniki 54636, Greece, [email protected] Ata Allah Taleizadeh School of Industrial Engineering, College of Engineering, University of Tehran, Tehran, Iran, Email: [email protected]
Abstract Every product has a life cycle i.e., the product has a certain time period for useable conditions for mankind. After the certain time period, the product is not for useable conditions and which is known as expired product. However, price discount is a very much attractive strategy in the present business situations. Using these two concepts in together, we introduce two supply chain models by assuming the demand of the product to be dependent on price. Also, shortages are considered and these depend on the customer waiting time. In both models, we formulate the problem mathematically, considering price as a decision variable, and prove their optimality. Two proposed solution procedures have been ameliorated and then solved two numerical examples in order to validate our proposed models as well as to compare which model is more
1
economical. Two sensitivity analyses are performed to make a conclusion regarding of our paper.
Keyword: Inventory; expiration date; deterioration; discount; demand dependent on price; partial backlogging.
*Corresponding author: Tel. 00302310891695
Abstract Every product has a life cycle i.e., the product has a certain time period for useable conditions for mankind. After the certain time period, the product is not for useable conditions and which is known as expired product. However, price discount is a very much attractive strategy in the present business situations. Using these two concepts in together, we introduce two supply chain models by assuming the demand of the product to be dependent on price. Also, shortages are considered and these depend on the customer waiting time. In both models, we formulate the problem mathematically, considering price as a decision variable, and prove their optimality. Two proposed solution procedures have been ameliorated and then solved two numerical examples in order to validate our proposed models as well as to compare which model is more economical. Two sensitivity analyses are performed to make a conclusion regarding of our paper.
Keyword: Inventory; expiration date; deterioration; discount; demand dependent on price; partial backlogging. 1.
Introduction
Inventory control is an essential activity in business management. The function of inventory control starts from the very beginning i.e., from the processing of raw materials into complete products manufacturing, movement of products from warehouses to wholesalers, wholesalers to retailers and retailers to customers. In every stage, control or management of inventory is most
2
vital. Various inventory control models have been developed based on inventory control theories. Basically, it helps in bringing maximum profits to the companies or minimizes their costs. The proposed inventory model is developed considering some essential features related to the present marketing scenario. First, we have discussed about demand of the model as demand always plays a crucial role during the construction of any inventory model. Generally, from wholesaling to retailing business, the crucial factor affecting the demand is the price of a new or existing product, because we know that low-priced products have higher demand and high-priced products have lower demand. As a result, the decision of a customer during the purchase period is highly influenced by price of the goods. Consequently, demand depends on price is one of the most general cases in the inventory management system. That’s the reason why we have taken price-dependent demand as one of our basic assumptions. Again, a large part of the grocery products have limited lifetimes and when these products approach to their expiration dates, the deterioration rate gradually increases and finally reaches to 100% at their expiration dates. When the customers purchase these types of products, they always check the expiration date of the products and try to purchase which is further from the expiration date so that they can store the products for a long time. As a result, alongside the suitability of the price of a product, the expiration date of the product is also equally important to attract the customers during buying or purchasing. Products like fruits, vegetables, electronic gadgets, medicine, dried food, rice, milk, wheat and radioactive materials etc., which deteriorate or decline over time with an increasing rate and have also an expiry period after which they lose their usability and are not in a condition for sale. Accordingly, the deterioration rates for these types of products can be considered as an increasing function in terms of the expiration dates. So here we have considered the expiration date of the product as an assumption in our model. These two above factors such as price and the deterioration rate with the expiration date are most important features which simultaneously affect the purchase-decision of customers. As we have considered price-demand demand and have also paid attention to the expiration date or maximum lifespan of a product this may lead to higher demand and in consequence the probability of shortages. So another two faceted assumptions that we have considered in this model are: i) shortages may not occur and ii) shortages may occur with partially backorders. In most of the typical inventory models, one ordinary assumption is carrying cost per unit is constant during the entire cycle length. As the deterioration rate is gradually increasing, the 3
decision maker always wants to control this rate by providing better preserving facilities. Consequently, the holding cost per unit cannot be constant in the entire cycle length but, in practice, it is an increasing function of the storage time. In this work, the holding cost per unit has been considered as a linearly increasing function of the storage time. Another typical assumption in most of the inventory models is that the unit purchase cost is fixed whatever the order size. However, in practical scenario, it is a decreasing function according to the order size. So, finally, we assumed a discount environment where unit purchase cost is a decreasing step function based on increasing order size. In the recent past, many inventory models have been developed considering different assumptions. But to the best of our knowledge, combination of the factors such as pricedependent demand, time varying deterioration rate depending on the product’s maximum lifetime or expiration date and linearly increasing holding cost have not yet been considered under all-units quantity discount environment in a single model. 1.1. Literature Review There is a large volume of published studies describing the role of price-dependent demand in inventory models for deteriorating items. Mondal et al. (2003) first proposed a model for an ameliorating item where demand of the product is dependent on price. Then, Mukhopadhyay et al. (2004) introduced an ordering policy on a pricing inventory model for deteriorating items. Next onwards Mukhopadhyay et al. (2005) extended their previous problem introducing another feature that the deterioration rate follows two parameter Weibull distribution and demand of the product depends on price. Similarly, You (2006) formulated a model where demand depends on price with an advance sales system. Again, Roy and Chaudhuri (2007) designed a model for price-dependent demand for a deteriorating item with special sale of the product and Roy (2008) extended his previous problem by incorporating time dependent holding cost. Further, Maiti et al. (2009) built a price-dependent inventory problem with stochastic lead time in advance payment system. Next, Sridevi et al. (2010) has mentioned a price-dependent model under the Weibull distribution deterioration rate. Similarly, Sana (2011) designed a price sensitive model for perishable items while Maihami and Kamalabadi (2012) formulated a model where demand of the product is dependent on jointly time and price. Avinadav et al. (2013) have introduced a supply chain problem for a perishable item. Bhunia and Shaikh (2014) proposed an inventory problem with price effect on demand. After that, Ghorieshi et al. (2015) proposed a supply 4
problem with price affected demand in trade credit system. Then, Alfares and Ghaithan (2016) included quantity discounts offered to the customer with holding cost dependent on time. Recently, Jaggi et al. (2017) proposed an inventory problem for price-dependent demand with credit financing and non-instantaneous deterioration in two storage facilities. A considerable amount of literature has been published on inventory control models for deteriorating products. In this literature, Ghare and Schrader (1963) first introduced the concept of deterioration in the area of inventory control, while Philip (1974) has upgraded from constant deterioration to three parameters Weibull deterioration and proposed another model. Thereafter a lot of researches have been discussed by several authors with different types of deterioration. Much of the current literature on deterioration pays particular attention to inventory models on deteriorating items with expiration date. Hsu et al. (2006) first proposed an expiration rate dependent deteriorating inventory model. Hsu et al. (2007) further extended their previous problem considering the lead time is uncertain in nature. Later on, Jain and Singh (2011) modified Hsu et al. (2007) model in multi-echelon supply chain system by considering several factors like inflation, expiration date dependent deterioration, capital constraint and uncertain lead time. Next onwards, Wu et al. (2014) investigated a model for expiration rate dependent deterioration in two level trade credit systems. Then, Chen and Teng (2015) designed a timevarying deteriorating inventory model with upstream and downstream delay in payment system. Wu et al. (2016) introduced another concept of freshness of the product in stocking policy with expiration rate of the product in the inventory system. Teng et al. (2016) further designed lot size inventory policies where the deteriorating item having an expiration date and advance payment system is followed. Similarly, Feng et al. (2017) built an inventory model by considering same concept of Wu et al. (2016) except freshness of the product. Further, Wu et al. (2017) extended their previous model with expiration rate dependent deterioration and advance-cash-credit payment schemes. Recently, Tiwari et al. (2018) developed a profit maximizing inventory model under two level trade credit policy in a supply chain for a deteriorating product with pricedependent demand and time varying deterioration rate with maximum lifetime. Additionally, we have summarized the comparison between the present work and earlier published works in Table 1: 1.2. Research gap and our contribution
5
Alfares and Ghaithan (2016) introduced an inventory model for zero ending case and did not consider the effects of deterioration and shortages. In the present paper, we have generalized the Alfares and Ghaithan (2016) model by considering expiration rate dependent deterioration with partial backlogging shortages during shortage time of the product. Also, we proved the optimality for each model (zero ending inventory and partial backlogged shortages).
6
Table 1: Summarized between earlier published research and our present work. Deterioration
Discount
Holding cost
Situation
PBO FBO PBO FBO FBO PBO FBO No FBO PBO No No No No No No No PBO No No FBO No PBO PBO FBO PBO
Payment Mode AP AP No No AP DP No DP AP No DP DP DP DP No No AP AP DP No AP No AP No AP DP
No No No No Constant No Constant Expiration No Expiration Expiration Expiration Expiration Expiration No Expiration Time-varying Time-varying Expiration No Expiration Expiration No Constant No Expiration
Yes Yes No Yes No No Yes No No No No No Yes No No No No No Yes Yes No No No No No No
Constant Constant Constant Constant Constant Constant Constant Constant Constant Constant Constant Constant Constant Constant Constant Constant Constant Constant Constant Time-varying Constant Constant Constant Constant Constant Constant
Single Single Double Single Single Single Single No Single Single No No No No No Single No Single No No Single No Single Single Single Single
Solution methodology Soft computing Soft computing Soft computing Soft computing Soft computing Mathematically Mathematically Mathematically Mathematically Mathematically Mathematically Mathematically Mathematically Mathematically Mathematically Mathematically Mathematically Mathematically Mathematically Mathematically Mathematically Mathematically Mathematically Mathematically Mathematically Mathematically
No
No
No
No
Constant
Single
Mathematically
PBO FBO
No No
Time-varying No
No No
Time-varying Constant
Single Single
Mathematically Mathematically
No
No
No
No
Constant
Single
Mathematically
FBO
No
No
Constant
Single
Mathematically
PBO
No
Expiration
No Linked to Order
Time varying
Double
Mathematically
Authors
Demand
Shortage
Gupta et al. (2009) Maiti et al. (2009) Taleizadeh et al. (2010) Taleizadeh et al. (2010) Guira et al. (2013) Taleizadeh et al. (2013) Taleizadeh et al. (2013) Wu et al. (2014) Taleizadeh (2014) Tayal et al. (2014) Wang et al. (2014) Wu and Chan (2014) Chen and Teng (2015) Sarkar et al. (2015) Taleizadeh et al. (2015) Chen et al. (2016) Teng et al. (2016) Wu et al. (2016a) Wu et al. (2016b) Alfares and Ghaithan (2016) Wu et al. (2017) Feng et al.(2017) Taleizadeh(2017) Pal et al. (2017) Tavakoli and Taleizadeh (2017) Tiwari et al. (2018)
Das Roy et al. (2018)
Constant Price-dependent Constant Constant Variable Constant Constant Constant Constant Price and expiration rate Credit period dependent Constant Depends on credit period Constant Price-dependent Stock dependent Constant Constant Constant Price-dependent Constant Price-dependent Constant Price-dependent Constant Price-dependent Retail price and advertising expenditures Time-dependent Price-dependent Retail price and product quality Price-sensitive stochastic
This Paper
Price dependent
Taleizadeh (2018) Pervin et al. (2018) Roy et al. (2018) Modak et al. (2018)
PBO: Partial Back-ordering, FBO: Full Back-ordering, AP: Advanced Payment, DP: Delayed Payment
7
We also proposed two solution algorithms and solved each model. In the present work, we have modified the model of Alfares and Ghaithan (2016) under the given considerations: (i)
Expiration date dependent deterioration has been introduced.
(ii)
Two separate cases have been considered (zero ending case and shortage case).
(iii)
Compared which model is more economical.
The remainder of this paper is organized as follows. The basic notation of the problem is presented in Section 2 while problem’s definition is briefly discussed along with the assumptions in Section 3. The mathematical formulation of the proposed models is given in Section 4 and the proofs of the optimality are described in section 5. An algorithm of the solution procedure is presented in Section 6 while Section 7 provides numerical results. Section 8represents the sensitivity analysis of our work and the main conclusions, comparisons and further research are described in Section 9. 2. Problem description To describe the problem, we define first the notations which have been used in our proposed work. Notations K
Units $/unit Constant Constant $/unit
Description replenishment cost per order constant part of demand rate (a 0) coefficient of the price in the demand rate (b 0) purchasing cost
$/unit $/unit
selling price shortage cost
$/unit
opportunity cost
(u )
Constant
rate of deterioration at any time u (0 (u) 1)
g
$/unit $/unit Week
constant part of holding cost coefficient of linearly time dependent holding cost the maximum lifetime or expired date
S
Constant
backlogging parameter
Units
the initial number of inventories (for the model with shortages)
R
Units
Q
Units
maximum number of partially backordered quantity (for the model with shortages) number of order placed per cycle
t1 t2
Week
time period when stock reaches to zero for shortages case
Week
time period of shortages
T
Week
cycle length (for shortages case T t1 t 2 )
a b cj p
cs cl
h
8
TP1 ( p, T )
$/week
the total profit of the problem for non shortages case
TP2 ( p, t1 , t 2 )
$/week
the total profit of the problem for partially backlogged shortages case
$/unit Week
Decision variable selling price time period when stock reaches to zero for shortages case
Week
time period of shortages
Week
cycle length for the model without shortages
p
t1 t2 T
3. Problem definition We assume that a supplier in order to stimulate demand, increase cash flow, and run smooth business offers a temporary price discounts on unit purchase costs of a deteriorating product based on order sizes as follows: purchase cost of a unit is discontinuous and it is a decreasing step function according to the order quantity. Since this discount directly affects the order size of the retailers, sometimes it leads to stock-out problems due to heavy demand and unavailability of adequate stock in the system. Therefore, we need to discuss both cases: with and without shortages. The model without any shortages will be discussed first and then the model with shortages. Our proposed works are developed under the following considerations: 1. The models are developed for a single deteriorating product.
2. Replenishment rate is infinite and instant. 3. It’s well known that if the price of an item is low then its demand appears highly in the market. From this view point, we assume the same as in Alfares and Ghaithan (2016), i.e., the demand is dependent linearly on price and can be expressed as: D( p) a bp
(1)
where a is the amount of demanded products when selling price p is zero, b is the steepness of the demand and a bp 0 . 4. The deterioration rate (u ) is a time varying increasing function. This rate depends on the product’s maximum lifetime or expired date . In addition, we have considered this rate similar to Wang et al. (2014) and Chen and Teng (2014) which is expressed bellow:
(u )
1 1 u
, 0 u T .
(2)
Remarkable point is that the replenishment cycle length T cannot be exceeding the maximum lifetimeτ because of after the maximum lifetime product cannot be sold. 9
5. Any replacement or repair cannot be taken into account for deteriorated products during the cycle length. We can say in other way, once a product has sold it cannot returnable or repairable. 6. Partially backlogged shortages are considered and it depends on the waiting time of the customers up to the arrival of next lot of the products. The partial backlogging rate L( y ) is a decreasing function of the customer waiting time and it can be express as: L( y )
1 , 1 y
(3)
where y indicates the customers’ waiting times up to arrival of the product and 0 . Notice that if 0 , the partial backlogging rate convert in fully backlogged shortages. 7. Holding cost of the products depends on the duration of the stocking time of each unit and also proportional to the purchase cost c j . Similar to Alfares and Ghaithan (2016), the holding cost consists of two parts: a constant part g and a linearly increasing with the storage time part h . Therefore, the holding cost function can be expressed as follows:
H (u) ci ( g hu) .
(4)
8. The unit purchase cost ( c j ) in the discount environment is a discontinuous decreasing step function according to the order size Q as follows:
qi , i 1, 2,3,........, n 1 (q1 q2 ........ qn qn1 ) are quantities which determine the n price breaks with the unit purchase cost ci , i 1, 2,3,........, n (c1 c2 ........ cn ) . 4. Description of mathematical modelling Here, we describe the modelling of the two problems. 4.1. Inventory model without shortages The inventory model starts with Q units of a deteriorating item and the stock level gradually decays because of customers’ demand and deterioration throughout the cycle length T and finally drops at zero at u T . After that, a new order will be placed then and there and the whole inventory system will be repeated (see Fig. 1). So, at any time u the level of inventory I (u ) can be expressed in the following governing differential equation:
10
Inventory
Q
Q
Time
T
Fig. 1. Pictorial presentation of proposed problem of without shortages case dI (u ) 1 I (u ) (a bp), 0 u T du 1 u
(5)
with conditions I (0) Q and I (T ) 0 .Using I (T ) 0 , we have: 1 u I (u ) (a bp)(1 u ) ln . 1 T
(6)
From Eq. (6) with the help of the condition I (0) Q , the order size is: 1 Q (a bp)(1 ) ln . 1 T So, the entire cycle length, T , can be calculated as follows: Q ( a bp )(1 ) . T (1 ) 1 e
(7)
(8)
The profit of the entire inventory system through the cycle length involves the following components which are stated below: 1) Ordering cost of (OC): K
1 2) Acquisition cost (AC): c j Q c j (a bp)(1 ) ln 1 T
11
T
3) Holding cost (HC): c j ( g hu ) I (u )du 0
T
T
1 u 1 u c j g (a bp) (1 u ) ln du c j h(a bp) u (1 u ) ln du 1 T 1 T 0 0 (1 ) 2 1 T 2 (1 )T c j g (a bp) ln 2 1 T 4 2 1 3 1 T 3 (1 )T 2 1 2 T c j h(a bp) ln 12 6 6 1 T 9 T
4) Sale Revenue (SR): p (a bp)du p(a bp)T 0
So, the profit function of the system can be expressed as: 1 TP1 ( p, T ) SR OC AC HC T (1 )2 1 T 2 (1 )T 1 2 ln ap bp T K c j (a bp)(1 ) ln c j g (a bp) 2 1 T 1 T 4 2 1 3 2 1 1 T 3 (1 )T 2 1 T T c j h(a bp) 6 ln 1 T 9 12 6
(9)
(10)
We need to determine the retailer’s optimum cycle length, T * , as well as selling price of the product, p * , in such a way that maximize retailer’s profit TP1 ( p, T ) . Now we investigate the model with partial backlogged shortages due to the waiting time of the customers. 4.2. Model with partial backlogged shortages In this case, an order of Q (S R) units of a deteriorating product which has fixed life time is placed at the beginning of a replenishment cycle. Shortly after, R units are utilized to fulfill the total accumulated backlogged demand; consequently, the remaining on hand inventory level becomes S . Then the inventory level gradually decreases not only for the customer’s demand but also the deterioration over the interval [0, t1 ] and finally drops at zero at time u t1 . After that, shortages are appeared which are accumulated depend upon waiting time of the customers in time interval [t1 , t1 t 2 ] . A new order is placed again at this instant and the entire inventory system is reiterated. The whole inventory situation over the entire cycle length can be presented in Fig. 2. The inventory level I (u ) satisfies the following two governing differential equations: 12
Inventory
S
t2 Time Backorders
t1 Lost Sales
Fig. 2. Graphical presentation of the proposed problem with partial backlogged shortages dI1 (u ) 1 I1 (u ) (a bp) , 0 u t1 du 1 u
(11)
dI 2 (u ) (a bp) , t1 u t1 t 2 du 1 (t1 t 2 u )
(12)
with the conditions I1 (0) S , I1 (t1 ) 0 and I 2 (t1 t 2 ) R . From Eqs. (11) and (12) and using the conditions I1 (t1 ) 0 , I 2 (t1 t 2 ) R , we have:
1 u and I 1 (u ) (a bp)(1 u ) ln 1 t1
I 2 (u )
(a bp)
ln1 (t1 t 2 u ) R .
(13)
(14)
Using the initial condition I1 (0) S , from Eq. (13) one can get:
1 . S (a bp)(1 ) ln 1 t1
(15)
The total backordered quantity, with the help of continuity of I (u ) at u t1 , is:
R
(a bp) ln 1 δ t2 . δ
(16)
13
Therefore, the total number of order quantities for each replenishment cycle is:
1 1 ln 1 t 2 . Q (a bp) (1 ) ln 1 t1
(17)
The cycle length of the shortage period is: 1 Q 1 ( a bp) (1 ) ln 1 t1 t2 e 1 .
(18)
The profit of the entire inventory system involves the following components: 1) Ordering cost (OC): K
1 1 ln 1 t 2 2) Acquisition cost (AC): c j Q c j (a bp) (1 ) ln 1 t1 t1
3) Holding cost (HC): c j ( g hu ) I 1 (u )du 0
1 1 u 1 u du c j h(a bp) u (1 u ) ln du c j g (a bp) (1 u ) ln 1 t1 1 t1 0 0 (1 ) 2 1 t1 2 (1 )t1 c j g (a bp) ln 2 1 t 4 2 1 (1 ) 3 1 t13 (1 )t1 2 (1 ) 2 t1 c j h(a bp) ln 6 1 t 9 12 6 1 t1 t 2 c (a bp) ln(1 t 2 ) 4) Shortage cost (SC): c s I 2 (u )du s t2 t1
t1
t
t1 t 2
5) Lost sale cost (LSC): cl
t1
ln(1 t 2 ) 1 1 (a bp)du cl (a bp) t 2 1 (t1 t 2 u )
t1
1 6) Sale Revenue (SR): p (a bp)du pR (ap bp 2 ) t1 ln(1 t 2 ) 0 So, the profit function of the system can be expressed as: TP2 ( p, t1 , t 2 )
1 SR OC AC HC SC LSC t1 t 2
(19)
14
1 2 (ap bp ) t1 ln(1 t 2 ) K 1 1 c j (a bp ) (1 ) ln ln(1 t 2 ) 1 t 1 (1 ) 2 1 t1 2 (1 )t1 1 ln c j g (a bp ) t1 t 2 2 1 t1 4 2 3 2 2 3 c h(a bp ) (1 ) ln 1 t1 (1 )t1 (1 ) t1 1 t 9 j 6 12 6 1 cs ln(1 t 2 ) cl (a bp ) t 2
(20)
Our objective is to find the optimum selling price per unit p * , t1* and t 2 * in order to maximize retailer’s profit per unit time TP2 ( p, t1 , t 2 ) . 5. Theoretical results Here, we present the concavity of the objective function for the problem with no shortage case first and then for the problem with partial backlogged shortages. To investigate the concavity for both models, some results have been used from Cambini and Martein (2009). Based on the Theorems 3.2.9 and 3.2.10 in Cambini and Martein (2009), the function of the form
( x)
f ( x) , x Rn g ( x)
(21)
is (strictly) pseudo-concave, if f (x) is a non-negative, differentiable and (strictly) concave function but g (x) is a positive, differentiable and convex function. 5.1 Inventory model with no shortage case Here we describe the optimality of the first problem by using the results of the Theorems 3.2.9 and 3.2.10 in Cambini and Martein (2009). The problem’s optimal policy can be obtained in two folds. Firstly, the optimal replenishment T * is found which optimizes the retailer’s total profit for any certain selling price p 0 . Secondly, the optimal p * is determined which maximizes the retailer’s total profit per unit time using the optimal replenishment T * . Theorem 1. For a certain p 0 , TP1 ( p, T ) is a pseudo-concave function of T and so there exists a unique T such that TP1 ( p, T ) is maximized. Proof. See Appendix A. 15
Lemma 1. For all u 0 ,
1 3 4u (1 ) is always positive. 1 u 2
Proof. See Appendix B. According to the necessary condition (maximization or minimization) to find the optimum cycle length T * , calculate first order derivative with respect to T of the objective function TP1 ( p, T ) for any given p , set the result to zero and after rearranging terms, we get
1 K c j (1 ) ln a bp 1 T
c j gT 2 4
T (1 ) (1 ) 2 1 g h 2 6 1 T
2T 3 (1 )T 2 c jh 0 12 9
(22)
Theorem 2. For any given positive values of cycle length T 0 , TP1 ( p, T ) is a concave function in p and so there exists a unique p such that TP1 ( p, T ) is maximized. Proof. See Appendix C. The necessary condition to find the optimal price p * can be found for any T by setting the first order partial derivative of the objective function with respect to p i.e., Eq. (C1) equal to zero, we get: T 2 (1 )T 1 (1 ) (1 ) 2 1 g h c gb j 2 6 2 1 T 4
a 2bp T c j b(1 ) ln
T 3 (1 )T 2 1 2 T 0 c j hb 12 6 9
(23)
5.2. Inventory model with partial backlogged shortages Now we investigate the concavity of TP2 ( p, t1 , t 2 ) with respect to p , t1 and t 2 for the model with partial backlogged shortages in two folds. Firstly, the joint concavity of TP2 ( p, t1 , t 2 ) in (t1 , t 2 ) is observed for any certain p 0 and then optimal (t1 , t 2 ) is determined. Secondly, the concavity of *
*
TP2 ( p, t1 , t 2 ) is investigated in p for fixed (t1 , t2 ) . For any certain p 0 , the first order partial derivatives of TP2 ( p, t1 , t 2 ) with respect to t1 and
t 2 are:
16
TP2 (t1 , t2 p) 1 TP2 ( p, t1 , t2 ) t1 t1 t2 (1 τ ) 2 t (1 τ ) 1 τ p c c g 1 j j 1 τ t1 2 a bp 2(1 τ t1 ) 2 t1 t2 (1 τ )3 t12 (1 τ )t1 (1 τ ) 2 c j h 6(1 τ t ) 3 6 6 1
TP2 (t1 , t 2 p) t 2
c c 1 a bp 1 TP2 ( p, t1 , t 2 ) p c j cl s cl s t1 t 2 t1 t 2 (1 t 2 )
(24)
(25)
The necessary condition to find the optimal t1* and t 2 * are:
TP2 (t1 , t 2 p) t1
0 and
TP2 (t1 , t 2 p) t 2
0 i.e.
(26)
2 2t 3 (1 )t1 2 c ln(1 t 2 ) t K c j g 1 c j h 1 p c j cl s (a bp) 4 12 9
1 t t g (1 ) (1 ) 2 1 2 1 c j (1 )ln 2 6 1 t 1 1 u 1
(27)
t1 2 (1 )t1 (1 ) 2 cs t1 (1 ) t 2 p cl c j g c j h 0 2 6 6 2 3
and cs p cl
t12 (1 )t1 cs ln(1 t2 ) t1 t2 K t p c c c g j l j 1 (1 t2 ) (a bp) 2 4
t13 (1 )t12 (1 ) 2 t1 1 g (1 ) (1 ) 2 c j (1 ) ln cjh 1 0. 2 6 12 6 1 t1 9
(28)
* * On solving Eqs. (27) and (28), we can obtain the optimal values of t1 and t 2 , say t1 and t 2 .
Theorem 3. For certain p 0 , TP2 ( p, t1 , t 2 ) is jointly pseudo-concave function in (t1 , t 2 ) and so there exists a unique pair of values (t1 , t 2 ) such that TP2 ( p, t1 , t 2 ) is maximized. Proof. See Appendix D. Theorem 4. For any given t1 0, t2 0 , TP2 ( p, t1 , t 2 ) is a concave function of p and so there exists a unique p such that TP2 ( p, t1 , t 2 ) is maximized. Proof. See Appendix E. The necessary condition to find the optimal selling price per unit p * is:
17
TP2 ( p, t1 , t2 ) 0 i.e., p t 2 (1 τ )t1 c ln(1 δt2 ) c (a 2bp)t1 a 2bp c j b cl s b b cl s t 2 c j gb 1 δ δ δ 2 4 1 τ t13 (1 τ )t12 (1 τ ) 2 t1 (1 τ ) (1 τ ) 2 c j b(1 τ ) ln h c j hb 1 g 0 2 6 12 6 9 1 τ t1
(29)
6. Solution procedure In the discount environment, suppose that there are n price breaks ( n quantity discounts) which is offered from supplier to retailer. Here, we have considered the purchasing cost ( c j ) is a decreasing step function measured by these n price breaks. Suppose that qi , i 1, 2,3,........, n 1
(q1 q2 ........ qn qn1 ) are quantities which determine the n price breaks with the corresponding unit purchase cost ci , i 1, 2,3,........, n (c1 c2 ........ cn ) . There are two possibilities of the optimal order quantity: 1) the optimal order size (Q) may lie between any two price breaks, qn Q qn1 , or 2) the optimal order size (Q) may occur on any of the price breaks
Q q n . For the first case, the optimal order size can be found from Eq. (7) by using the necessary conditions (22) and (23) for the model without shortages and from Eq. (17) by using the necessary conditions Eqs. (27), (28) and (29) for the model with shortages. For the second case, if the optimal order size is any price breaks, then Q is fixed. So from Eq. (8) (without shortages model) we see that T depends on p and also from Eq. (18) (with shortages model) we see that t 2 depends on p and t1 . Hence, in this situation, the derived necessary conditions for both models are not suitable and thus we need to calculate new necessary conditions for both cases. 6.1 Necessary conditions when optimal order size (Q) occurs on any of the price breaks Let us derive the necessary condition for the model without shortages when the optimal order size (Q) occurs on any of the price breaks (q n ) first, and then the necessary conditions for second model with shortages case. 6.1.1 The model without shortages If we substitute each T with the help of Eq. (8), the total profit function TP1 contains only one independent variable namely p . Differentiating Eq. (7) with respect to p , one can get
18
dT b(1 T ) 1 ln dp (a bp) 1 T
.
(30)
The necessary condition to find the optimal selling price per unit p * can be found by calculating the first order derivative
dTP1 with the help of Eq. (30). Equating the result to zero and after dp
simplifying the terms, the necessary condition can be written as: 2 1 1 T (1 ) c j (1 ) ln 1 g 2 h 6 1 T 1 T 1 2 a 2 bp T b (1 T ) ln 2 3 2 1 T 2T T (1 )T K c jh c j g 4 12 (a bp) 9
1 c j bT (1 ) ln 1 T
1 (1 ) h 1 g 2 6
2
(31)
T 3 (1 )T 2 1 2 T T 2 (1 )T h 0 g 2 12 6 9 4
6.1.2 The model with shortages If we substitute each t 2 with the help of Eq. (18), the total profit function TP2 contains only two independent variables, namely, p and t1 . Consequently, in the shortages case, only two decision variables exist under which the total profit function TP2 ( p, t1 ) has to be maximized. Differentiating Eq. (17)with respect to p, anyone can find
1 1 t 2 b(1 t 2 ) ln(1 t 2 ) . (1 ) ln p (a bp) 1 t1
(32)
Again, differentiating Eq. (17) with respect to t1 , we find:
t2 1 τ (1 δ t2 ) . t1 1 τ t1
(33)
To find the optimal price of the product p * , the necessary condition can be found for any given
t1 by calculating the first order partial derivative
TP2 with the help of Eq. (32). Setting this p
result with equal to zero and after rearranging terms, we have:
19
1 τ 1 b(1 δt2 ) (1 τ ) ln ln(1 δt2 ) 1 τ t1 δ 2 p c cs t K c (1 τ ) ln 1 τ 1 g (1 τ ) h (1 τ ) l j 1 δ (a bp ) 2 6 1 τ t1 cs 1 t1 t2 p c j cl ln(1 δt2 ) δ δ (1 δt2 ) 2 2 2 3 c g t1 (1 τ )t1 c h t1 (1 τ )t1 1 τ t1 j j 4 2 9 12 6 (a 2bp)t a 2bp c b b c cs ln(1 δt2 ) b c cs t 1 j l l 2 δ δ δ 1 τ (1 τ ) (1 τ ) 2 (t1 t2 ) c j b(1 τ ) ln h 0. 1 g 2 6 1 τ t1 2 2 2 3 c g t1 (1 τ )t1 c h t1 (1 τ )t1 1 τ t1 j j 4 2 9 12 6
(34)
* Again, the necessary condition for finding the optimal t1 can be found for any given p by
calculating the first order partial derivative
TP2 with the help of Eq. (33). Setting this result t1
equal to zero and after rearranging terms, the necessary condition is: 1 (t1 t 2 ) p c j 1 t1
t1 2 (1 )t1 (1 ) 2 (1 ) (1 ) 2 t1 (1 ) 1 g h c g c h j j 2 6 2 6 6 2 3
cs cs ln(1 t2 ) t1 t2 K p cl t1 p c j cl 1 t2 (a bp) 1 (1 ) (1 ) 2 1 (1 t2 ) c j (1 ) ln h 0. 1 g 1 t1 2 6 1 t1 2 3 2 2 c j g t1 (1 )t1 c j h t1 (1 )t1 (1 ) t1 2 12 6 4 9
(35)
We develop two algorithms for the aforementioned described models in order to obtain the optimal solutions under this discount environment. Additionally, we assume that the value of Q should be considered to the nearest integer value, while the values of p, t1 , t 2 , T and TPi should be
20
considered up to3-decimals. Now we present the flowchart and algorithm for both models which are given below. 6.1.3 Flowchart for the two models Before going to the discussion of solution algorithms, we have supplied two flowcharts, in figures 3 and 4 of our solution technique for each model in order to elucidate the solution algorithms. Inside the flowchart, we have used the equations numbers as well as mathematical expression. The first flowchart is for inventory model for without shortages case and the second one is for the inventory model with partially backlogged shortages case. 6.2 Algorithm for the model without shortages Step 1:
Initialize TPmax ( p, T ) 0 and i n .
Step 2:
Set j 1and initialize p ( j ) p1 , where p1 is the solution of a 2bp 0 .
Step 3:
Solve Eq. (22) to obtain T ( j ) by using the values of K , a, b, g , h, m and ci . Substitute the ( j) ( j) value of T ( j ) in Eq. (23) and solve for p1 . Set p ( j 1) p1 .
Step 4:
If p ( j 1) p ( j ) 10 4 , then set ( p, T ) p ( j 1) , T ( j ) and go to Step 5. Otherwise go to step 3 with j j 1 .
Step 5:
Substitute the values of p and T in Eq. (7) in order to calculate the corresponding order size Q . i) If Q belongs to the right unit purchase cost range (qn Q qn1 ) , then this solution is feasible. Calculate TPi ( p, T ) from Eq. (10) with the help of the optimal values of p and T . Now if TPi ( p, T ) TPmax ( p, T ) , set TPmax ( p, T ) TPi ( p, T ) . Go to Step 8. ii) If Q does not belong to the right unit purchase cost range (qn Q qn1 ) , then obtained solution is infeasible and go to Step 6.
21
Begin
Initialize
TPmax ( p, T ) 0 and i n
Input all the values of given parameters
Set j 1 and p ( j ) p1 , where p1 is the solution of a 2bp 0
Solve Eq. (22) for T
( j)
and using this T
( j)
( j 1)
( j)
Set p
p1
solve Eq. (23) for p1
( j)
.
Set j j 1
No
p ( j 1) p ( j ) 10 4
Set Q qn
Yes
Set ( p, T ) p
( j 1)
,T ( j)
Q T (1 ) 1 e ( a bp )(1 )
Calculate the corresponding Q from Eq. (7)
Plug T into Eq. (31) and solve for p No
q n Q q n 1
Calculate T from Eq. (8) with derived p
Yes Calculate
TPi ( p, T ) from Eq. (10)
TPi ( p, T ) TPmax ( p, T )
Yes
TPmax ( p, T ) TPi ( p, T )
No Print maximum profit TPmax ( p, T ) with optimal
End
p * and T *
No
i2
Yes
Set i i 1
Fig.3. Flowchart of the solution technique for without shortages case 22
Begin
Initialize
TPmax ( p, t1 , t 2 ) 0 and i n
Input all the values of given parameters
Set j 1 and p ( j ) p1 , where p1 is the solution of a 2bp 0
Solve Eqs. (27) and (28) for t1
( j)
solve Eq. (29) for p1
, t2
( j)
( j)
and using this t
1
. Set p
( j 1)
p1
( j)
, t2
( j)
( j)
No
p ( j 1) p ( j ) 10 4
Set Q qn
Yes
Set ( p, t1 , t 2 ) p ( j 1) , t1
( j)
, t2
( j)
1 Q 1 ( a bp) (1 ) ln 1 t1 t2 e 1
Calculate the corresponding Q from Eq. (17)
No
q n Q q n 1 Yes Calculate
Set j j 1
Plug t 2 into Eqs. (34)& (35) and solve for p & t 1
Calculate t 2 from Eq. (18) with derived p & t 1
TPi ( p, t1 , t 2 ) from Eq. (20)
TPi ( p, t1 , t 2 ) TPmax ( p, t1 , t 2 )
Yes
TPmax ( p, t1 , t 2 ) TPi ( p, t1 , t 2 )
No Print maximum profit TPmax ( p, t1 , t1 ) with optimal
No
i2
Yes
Set i i 1
*
p * ,t1 and t 2 * Fig.4. Flowchart of the solution technique for with shortages case End
23
Step 6:
Set Q qn in Eq. (8) and calculate the corresponding cycle length T in terms of p . Plug the cycle length into Eq. (31) and solve for p .Calculate the corresponding optimal cycle length from Eq. (8) and finally calculate the corresponding TPi ( p, T ) from Eq. (10) with the help of the optimal values of p and T . If TPi ( p, T ) TPmax ( p, T ) , set
TPmax ( p, T ) TPi ( p, T ) . Go to Step 7. Step 7:
If i 2 , go to Step 2 with i i 1 ; otherwise go to Step 8.
Step 8:
Find the total profit per unit time is TPmax ( p, T ) by using the optimal value of p and T .
6.3Algorithm for the model with partially backlogged shortages Step 1:
Initialize TPmax ( p, t1 , t 2 ) 0 and i n .
Step 2:
Set j 1and initialize p ( j ) p1 , where p1 is the solution of a 2bp 0 .
Step 3:
Solve Eqs. (27) and (28) to obtain t1 , t 2
( j)
( j)
by
substituting all the input values
( K , a, b, g , h, m, cl , cs ) and ci . Substitute the values of t1( j ) , t 2 ( j ) in Eq. (E3) and solve for p1 . Set p ( j 1) p1 . ( j)
Step 4:
( j)
If p ( j 1) p ( j ) 10 4 , then set ( p, t1 , t 2 ) p ( j 1) , t1 , t 2 ( j)
( j)
and go to Step 5. Otherwise
go to step 3 with j j 1 . Step 5:
Substitute the values of p,t1 and t 2 in Eq. (17) in order to calculate the corresponding order size Q . i) If Q belongs to the right unit purchase cost range (qn Q qn1 ) , then this solution is feasible. Calculate TPi ( p, t1 , t 2 ) from Eq. (20) with the help of the optimal values of
p,t1 and t 2 . Now if TPi ( p, t1 , t 2 ) TPmax ( p, t1 , t 2 ) , set TPmax ( p, t1 , t 2 ) TPi ( p, t1 , t 2 ) . Go to Step 8. ii) If Q does not belong to the right unit purchase cost range (qn Q qn1 ) , then this solution is infeasible and go to Step 6. Step 6:
Set Q qn in Eq. (18) and calculate the corresponding t 2 in terms of p and t1 .Plug the expression of t 2 into Eqs. (34)and (35) and solve for p and t1 . Calculate the
24
corresponding t 2 from Eq. (18) and finally calculate the corresponding TPi ( p, t1 , t 2 ) from Eq.
(20)
with the
help
of the
optimal
values
of
p,t1 and t 2 . If
TPi ( p, t1 , t 2 ) TPmax ( p, t1 , t 2 ) , set TPmax ( p, t1 , t 2 ) TPi ( p, t1 , t 2 ) .Go to Step 7. Step 7:
If i 2 , go to Step 2 with i i 1 ; otherwise go to Step 8.
Step 8:
Find the total profit per unit time is TPmax ( p, t1 , t 2 ) with the optimal values of p,t1 and t 2 .
7. Numerical illustration and comparisons In this section, we have illustrated two numerical examples for both cases with the help of our developed algorithms in order to study the applicability of our theoretical results and also to achieve managerial insights of our proposed problems. The data of the numerical examples are taken from Alfares and Ghaithan (2016) with some additional data to adapt our models. Example 1: Model without shortages Let us consider the following example Quantity
0 q1 Q q2 100
100 q2 Q q3 200
200 q3 Q q4
Purchase cost
c1 $5 /unit
c2 $4.75 /unit
c3 $4.5 /unit
Along with
K $520 /order, a 100, b 1.5, g $0.2 /unit/week , h $0.05 /unit/week, 4
weeks. Step 1: Initialize TPmax ( p, T ) 0 and i 3 . Iteration 1: i 3 Step 2: Set j 1and initialize p (1) p1
a 33.33333 . 2b
Step 3: c3 $4.5 (200 Q ) The solutions of Eqs. (22) and (23) by executing the Step 3 to Step 4 in the computational algorithm 6.2, we can obtain p $37.204 , T 2.398 weeks. Step 5: With the help of the values of p and T , the corresponding lot size from Eq. (7) is Q 144 units and the retailer’s total profit from Eq. (10) is TP3 ( p, T ) $1085.265 . Since Q does not belong to the following range (200 Q ) , this is not a feasible solution and hence goes to step 6. Step 6: c3 $4.5 , Q 200 units Setting Q 200 units in Eq. (8) and then calculate the corresponding cycle length T in terms of p .Plug this derived cycle length into Eq. (31) and solve for p gives: p $36.606 . Then the corresponding cycle length from Eq. (8) is T 2.941 weeks. Finally, with the help of the optimal values of p and T from Eq. (10), we get 25
TP3 ( p, T ) $1067.524 .
Since TP3 ( p, T ) $1067.524 0 TPmax ( p, T ) ,
set
TPmax ( p, T ) $1067.524 .Go to step 7. Step 7: Set i 2 and go to step 2. Iteration 2: i 2 a Step 2: Set j 1and initialize p (1) p1 33.33333 . 2b Step 3: c3 $4.75 (100 Q 200) The solutions of Eqs. (22) and (23) by executing the Step 3 to Step 4 in the computational algorithm 6.2, we can obtain p $37.379 , T 2.361 weeks. Step 5: Using these values of p and T , the corresponding lot size from Eq. (7) is Q 140 units and the retailer’s total profit from Eq. (10) is TP2 ( p, T ) $1066.409 .Since Q belongs to the following range (100 Q 200) , this solution is feasible and TP2 ( p, T ) $1066.409 TPmax ( p, T ) $1067.524 . Therefore, TPmax ( p, T ) $1067.524 which has found in step 6 of iteration 1 and hence go to step 8. * Step 8: The optimal solution is p * $36.606 , T 2.941weeks, TPmax ( p, T ) $1067.524 .
T * 2.941weeks, Q * 200 units, Therefore the optimal solution is p * $36.606 , TPmax ( p, T ) $1067.524 .The concavity can be investigated from the Fig. 5 of the total profit of the problem against p . 1070 1060 1050
Total Profit
1040 1030 1020 1010 1000 990 980 32
34
36
38
40
42
44
p
Fig. 5. TP1 ( p, T ) versus p .
Example 2: Model with partially backlogged shortages The data are same those are used in Example 1. Additionally, we consider that
0.4, cs 3 and cl 5 for the backlogged shortages. Then the optimal solution of the problem can be founded using algorithm 5.2 as follows: 26
Step 1: Initialize TPmax ( p, t1 , t 2 ) 0 and i 3 . Iteration 1: i 3 a Step 2: Set j 1and initialize p (1) p1 33.33333 . 2b Step 3: c3 $4.5 (200 Q ) The values of p,t1 and t 2 from the Eqs. (27), (28) and (29) can be found by executing the Step 3 to Step 4 in the computational algorithm 6.3, we can obtain p $36.928 , t1 2.27 weeks, t 2 0.491week. Step 5: With the help of the values of p,t1 and t 2 , the corresponding lot size from Eq. (17) is Q 155 units and the retailer’s total profit from Eq. (20) is TP3 ( p, t1 , t 2 ) $1117.701 . Since Q does not belong to the following range (200 Q ) , this is not a feasible solution and hence goes to step 6. Step 6: c3 4.5 , Q 200 Setting Q 200 in Eq. (18) and then calculate the shortages period t 2 in terms of p and t1 . Plugging the shortages period into Eqs. (34) and (35) and solving we get p $36.462 , t1 2.699 weeks and the corresponding shortages period from Eq. (18), t 2 0.594 week. With the help of these derived values and all given values, from Eq. (20), TP3 ( p, t1 , t 2 ) $1107.94 . Since TP3 ( p, t1 , t 2 ) $1107.94 0 TPmax ( p, t1 , t 2 ) , set
TPmax ( p, t1 ,2) $1107.94 .Go to step 7. Step 7: Set i 2 and go to step 2. Iteration 2: i 2 Step 2: Set j 1and initialize p (1) p1 Step 3: c3 $4.75 (100 Q 200)
a 33.33333 2b
The values of p,t1 and t 2 from the Eqs. (27), (28) and (29)can be found by executing the Step 3 to Step 4 in the computational algorithm 6.3, we can obtain p $37.083 , t1 2.227 weeks, t 2 0.504 week. Step 5: Using these values of p,t1 and t 2 , the corresponding lot size from Eq. (17) is Q 151 units and the retailer’s total profit from Eq. (20) is TP2 ( p, t1 , t 2 ) $1100.798 . Since Q belongs to the following range (100 Q 200) , this solution is feasible and TP2 ( p, t1 , t 2 ) $1100.798 TPmax ( p, t1 , t 2 ) $1107.94 .Therefore,
TPmax ( p, t1 , t 2 ) $1107.94 which has found in step 6 of iteration 1 and hence go to step 8. Step 8: The optimal solution is TPmax ( p, t1 , t 2 ) $1107.94 .
p * $36.462 , t1* 2.699 weeks, t 2 0.594 week, *
27
Therefore
the
optimal
is p * $36.462 ,
solution
t1 2.699 weeks, *
t 2 0.594 week, *
T * t1 t 2 3.293 weeks, S * 176 units, R * 24 units, Q * 200 units and TPmax ( p, t1 , t 2 ) $1107.94 . *
*
The concavity can be investigated from the Fig. 6 of the total profit against p and t1 . 1100
1200
1050 1100 1000 1000
Total Profit
950 900
900
800
850 800
700 750 600 3
700 2.5 t1
2
30
32
36
34
40
38
42
p
Fig. 6.Concavity of TP2 ( p, t1 , t 2 ) against p and t1 .
Moreover, we can observe the optimal solution easily from the two line diagrams, namely
p and TP2 ( p, t1 , t 2 ) versus t1 presenting in Fig. 7(a) and Fig. 7(b)
TP2 ( p, t1 , t 2 ) versus respectively.
1120
1110 1100
1100
1090 1080
Total Profit
Total Profit
1080 1070 1060 1050
1060
1040
1020
1040 1000
1030 1020 32
33
34
35
36 p
37
38
Fig. 7(a). TP2 ( p, t1 , t 2 ) versus p
39
40
980 2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3
t1
Fig. 7(b). TP2 ( p, t1 , t 2 ) versus t1
28
Closing this section, we compare our numerical results with those in the Alfares and Ghaithan (2016) paper. These can be found in the following table 2: Table 2: Comparisons of our results with Alfares and Ghaithan (2016) Proposed model Alfares and Ghaithan (2016) Inventory model with partial backlogged The Alfares and Ghaithan (2016) results are shortage gives better result than Alfares and given below: Ghaithan (2016). If we consider the expiration date is large number (say, TP=1207.20, T=4.423,Q=200,p=36.52. 9999 weeks)and considered partial backlogged shortage with the length of the waiting time of the customer then we get better results. In addition, we are giving the both result TP= 1217.78,t1= 4.092, T= 4.445, Q=200, R=14.935, p= 36.51897.
From the above table, we can conclude that the proposed model is more profitable in the perspective of profit maximization. 8.
Sensitivity analysis
Here, we have described two sensitivity analyses of the optimal solutions with respect to different parameters for both models which are presented in Table 3 and Table 4. Changing the value of one parameter at a time and keeping same values of the rest parameters, we observed the effects of different parameters on the objective function and different variables i.e., on the price of the product (p), zero ending time (t1 ) , shortages period (t 2 ) , cycle length of the model (T), maximum stock level (S), highest shortage level (R), order quantities (Q) and total profit of the whole inventory system (TP) based on the used examples. We changed the value ofone parameter in a relative steps of 10% (-20%, -10%, 10%, +20%)but once at a time. Table 3: Sensitivity analysis of different parameters for example 1 Parameter
Original Value
K
520
a
100
b
1.5
*
Changes Value
p*
T*
Q*
TP1
416 468 572 624 80 90 110 120 1.2
37.200 37.292 36.726 36.845 30.948 34.151 39.985 43.353 44.949
2.186 2.278 2.948 2.955 2.572 2.458 2.753 2.585 2.902
127 134 200 200 121 131 200 200 200
1112.120 1088.827 1049.863 1032.246 549.514 790.710 1384.560 1734.529 1480.968
29
g
0.2
h
0.05
4
c1 , c2 , c3
5, 4.75, 4.5
1.35 1.65 1.80 0.16 0.18 0.22 0.24 0.04 0.045 0.055 0.06 3.2 3.6 4.4 4.80 4, 3.8, 3.6 4.5, 4.275, 4.05 5.5, 5.225, 4.95 6, 5.7, 5.4
40.315 33.570 31.847 36.595 36.600 37.419 37.459 36.627 36.616 36.595 37.386 37.447 37.410 36.628 36.643 36.200 36.403 37.709 38.036
2.921 2.960 2.383 2.940 2.940 2.334 2.308 2.942 2.941 2.940 2.343 2.165 2.268 3.026 3.103 2.916 2.928 2.296 2.239
200 200 138 200 200 138 136 200 200 200 139 133 137 200 200 200 200 134 127
1251.100 917.613 793.570 1083.805 1075.664 1060.310 1054.310 1071.286 1069.405 1065.644 1064.118 1040.495 1054.595 1079.896 1090.328 1149.024 1108.212 1031.367 997.265
From Table 3, we can make the following observations: (i) The total profit for the first model TP1* decreases with respect to the augmenting values of
K , b, g , h, c1 , c2 and c3. It indicates that the retailer will earn fewer amounts if the values of the said parameters increase. Also, it increases for the rest of the parameters a and . So, retailer can earn more money if these two parameters’ values increase. Additionally, the demand parameter a has the greatest positive impact on increasing the retailer’s total profit because the higher value of a helps the retailer to earn the revenue to a great extent by increasing the demand. (ii) Optimal order size (Q*) rises when the values of the parameters K , a and are changed from -20% to -10% but remains constant for the changing from +10% to +20%. It is also observed that the order size declines with respect to the parameters b, h, c1 , c2 and c3 , if we increase the values of these parameters up to+20% and remains constant for -20% to -10% whereas we observed that order size remains constant if we change the value of g from -20% to -10% and decreases if we change the value from +10% to +20%. It indicates that if the value of g i.e., constant part of the unit holding cost increases, the decision maker (retailer) wants to reduce the total holding cost by making a small order size.
30
(iii) The optimal cycle length for the first model (T * ) rises with respect to the augmenting values of the parameters K and whereas it declines if we increase the value of the parameter a. This reveals that if the ordering cost increases, retailer wants to decrease it by increasing order size and also if maximum lifetime of the product increases, retailer makes a large order; consequently, the optimal cycle length increases. On the other hand, if the demand parameter a increases, retailer can sell his/her products so quickly because of customers’ high demand. It is to be noted that for rest of the parameters the cycle length is increasing first and then decreases. (iv) Optimal price of the product ( p* ) increases if we change the values of the parameters
a, g , c1 , c2 and c3 whereas it decreases if we change the values of b and . So, this reveals that if the unit purchase cost and unit carrying cost increase, retailer wants to cover his/her costs by increasing unit selling price of the product and also if the value of b increases, as a result, product’s demand decreases and then retailer makes the decision to decrease the selling price in order to boost customers demand. Additionally, it is to be noted that when we change the parameter value of K it proliferates first then declines whereas if we change the parameter value of h it decreases initially and then increases i.e., exactly reverse of the parameter K. Table 4: Sensitivity analysis of different parameters for example 2 Parameter
Original Value
K
520
a
100
b
1.5
g
0.2
h
0.05
*
New Values
p*
t1
416 468 572 624 80 90 110 120 1.2 1.35 1.65 1.80 0.16 0.18 0.22 0.24 0.04 0.045
36.945 36.347 36.576 36.691 30.574 33.062 39.849 43.228 44.807 40.170 33.428 30.900 36.472 36.467 36.456 36.450 36.483 36.472
2.064 2.698 2.700 2.702 2.397 2.887 2.533 2.385 2.700 2.699 2.701 2.703 2.713 2.706 2.692 2.686 2.704 2.702
t2
*
0.434 0.577 0.613 0.63 0.722 0.733 0.493 0.416 0.485 0.54 0.647 0.7 0.559 0.577 0.612 0.629 0.585 0.589
*
S*
R*
Q*
TP2
119 176 175 175 111 174 177 179 179 178 174 173 177 176 175 175 176 176
18 24 25 25 22 26 23 21 21 22 26 27 23 24 25 25 24 24
137 200 200 200 133 200 200 200 200 200 200 200 200 200 200 200 200 200
1140.559 1123.774 1092.196 1076.543 588.405 828.368 1421.220 1768.092 1515.206 1288.512 960.879 838.926 1119.877 1113.881 1102.053 1096.220 1110.470 1109.203
31
4
0.4
cl
5
cs
3
c1 , c2 , c3
5, 4.75, 4.5
0.055 0.06 3.2 3.6 4.4 4.8 0.32 0.36 0.44 0.48 4 4.5 5.5 6 2.4 2.7 3.3 3.6 4, 3.8, 3.6 4.5, 4.275, 4.05 5.5, 5.225, 4.95 6, 5.7, 5.4
36.451 36.440 36.383 36.427 36.489 36.510 36.456 36.459 36.465 36.469 36.452 36.457 36.466 36.470 36.448 36.455 36.468 36.474 36.094 36.278 36.643 36.824
2.697 2.695 2.510 2.610 2.779 2.850 2.658 2.680 2.716 2.730 2.693 2.696 2.702 2.705 2.690 2.695 2.704 2.708 2.714 2.707 2.692 2.684
0.599 0.603 0.651 0.62 0.572 0.554 0.688 0.638 0.556 0.523 0.609 0.602 0.587 0.58 0.616 0.605 0.583 0.574 0.491 0.541 0.648 0.703
176 176 174 175 177 177 172 174 177 179 175 176 176 176 175 175 176 177 179 178 174 172
24 24 26 25 23 23 28 26 23 21 25 24 24 24 25 25 24 23 21 22 26 28
200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200
1106.681 1105.426 1085.561 1097.718 1116.637 1124.116 1114.191 1110.839 1105.407 1103.175 1108.798 1108.364 1107.524 1107.116 1109.241 1108.580 1107.319 1106.717 1178.092 1142.728 1073.723 1040.073
From Table 4, we can make the following observations: (i) The
total
profit
for
the
second
model TP2* declines
with
respect
to
K , b, g , h, , cl , cs , c1, c2 and c3. Among these mentioned parameters the demand parameter b has the greatest negative effect on the retailer’s total profit since if b increases, customers’ demand decreases and as a result retailer’s total revenue also abates. On the other hand, total profit increases with respect to the parameters a and . Moreover, the demand parameter a has the greatest positive effect on increasing the retailer’s total profit because the higher value of a helps the retailer to earn the revenue to a great extent by increasing the demand. (ii) Optimal order size (Q*) increases if we increase the values from -20% to -10% of K and a but it is fixed for all other changes of the values of K and a as well as for all changes of the rest parameters. Hence, if the ordering cost increases, retailer wants to decrease it by increasing order size and if the demand parameter a increases, customers’ demand also increases and consequently, decision maker makes a large order size. (iii) Maximum shortage level (R*)increases with respect to the changes of the parameters
K , b, g , c1 , c2 and c3 whereas it decreases with respect to the changes of the parameters
32
, , cl and cs .It is also remarkable that the time varying holding cost parameter h has no impact on the optimal value of maximum shortages. (iv) Highest stock level (S*) increases when we change the values of , a, E, , cl and cs whereas it decreases with respect to the changes of the parameters b, g , c1 , c2 and c3 . Among these mentioned parameters the demand parameter a has the greatest impact on stock level in a positive way as the higher value of a aggrandizes the demand and hence retailer wants to stock in a large extent. It is also remarkable that the optimal value of the highest stock level has no change if we change the value of h. (v) The shortages period of the inventory system (t 2 * ) increases if we change the values of the parameters K , b, g , h, , c1 , c2 and c3. Again, it decreases if we change the values of
, cl and cs . In addition, both the demand parameters a and b have the greatest effect on the optimal shortages period. It reflects the fact that for a higher value of a or a lower value of b , the demand augments significantly and then products are consumed faster. As a result, the duration of the shortages period waxes. (vi) Inventory
ending
time
(t1* ) increases
when
we
change
the
parameters
K , , , cl , cs , c1 , c2 and c3 whereas decreases if we change the values of the parameters h and g. In addition, it is to be noted that if we change the values of the parameters a and b ,
initially, the value of t1 increases and then decreases. (vii) Optimal price of the product ( p* ) increases if we change the values of the parameters
a, g , , cl , cs , c1 , c2 and c3 whereas it decreases if we change the value of the parameters
b and . Additionally, it is to be noted that when we change the value of K , selling price rises first then declines whereas if we change the value of h it declines initially and then rises. Among the all parameters the demand parameter b has the foremost important impact on the selling price of the product. For the lower values of b the optimal selling price increases most. Though the parameter b has the greatest impact on increasing p * but the parameter b has more effect on increasing the total profit. It is important to mention that a higher selling price does not always give a higher total profit. 9. Conclusion and future research directions 33
Every product has a fixed lifetime i.e., the products will be damaged or destroyed or deteriorated totally after certain time periods. After this time period, this type product cannot be perfect in usable condition for mankind. So, manufacturers/suppliers want to sell this type of products within their lifetime. If they cannot able to sell these products within maximum lifetime period, then they totally lose the invested amount. Consequently, to shun this situation and sell their products within the expiration date times they offer price discounts on unit purchase cost based on the order size. In this connection, if anyone wants to buy something in a shopping mall or other wholesale markets, he/she can observe price discount on the purchased amount or purchased quantity. So, it has a huge impact in the retail or wholesale markets. Under this discount environment, in this work, we have introduced two inventory models, namely, (i) zero ending inventory model and (ii) partial backlogging shortages inventory model for an expiration rate dependent deteriorating goods with variable demand depends on price. Shortages are accumulated with a decreasing rate of the customers waiting time. Two solution algorithms along with the flowcharts in order to solve our proposed problems have been constructed. The optimality for each model has been investigated mathematically. To observe the applicability of the algorithms, we solved two numerical examples by using the proposed algorithms, validated the obtained results graphically by using MATLAB software and conducted two sensitivity analyses. Our basic observations are that the demand parameters a and b for both models have the greatest effect on increasing the retailer’s total profit which reflects the fact that the decision maker (retailer) can increase the profit by following proper marketing policies for boosting customers’ demand. Another way retailer can increase his/her total profit by giving meticulous concentration to negotiate the unit purchase cost with the manufacturer/supplier. Alfares and Ghaithan (2016) studied an inventory model for zero ending case and didn’t consider the effects of deterioration and shortages. Comparing our results with those obtained in Alfares and Ghaithan (2016),we observe that the retailer’s optimal cycle length, T * , in our models are lower than the corresponding one in the model of Alfares and Ghaithan (2016). In the model of Alfares and Ghaithan (2016), the products are non-deteriorating and do not have any limited lifespan or expiration date. In contrast to the Alfares and Ghaithan (2016) model, the products in our models do not have only maximum lifetime but also a gradually increasing deterioration rate which reaches to 100% at their expiration dates. In our examples, the maximum lifetime of the products 34
has been considered as 4 weeks. That’s why the retailer wants to vend his/her products within this period and consequently, T * is lower than the optimal cycle length of Alfares and Ghaithan (2016). The optimal selling price ( p * ) in our model without shortages is higher than the corresponding one in the model of Alfares and Ghaithan (2016). But p * for the model with shortages is lower than the optimal selling price in Alfares and Ghaithan (2016). As our models are applicable for deteriorating products, the retailer tries to cover his/her loss by increasing the selling price per unit from the very beginning of the cycle. That’s why p * is higher than that of Alfares and Ghaithan (2016). If the retailer lessens the selling price in order to boost the customers’ demand, then the products have been consumed so faster and hence shortages have been occurred. This situation has been reflected in the proposed model with shortages. It is interesting to mention that, if we consider the products with a higher maximum lifetime (say,
9999 weeks), then the model without shortages gives the same results of Alfares and Ghaithan (2016) and the shortage case gives better result than Alfares and Ghaithan (2016) in the perspective of profit maximization. This model is more applicable for products like fruit, vegetables, electronic gadgets, gasoline, alcohol, medicine, dried food, rice, wheat, volatile substances and radioactive materials, etc. which deteriorate or decline over time and have also an expiry period after which they lose their usability and are not in a condition for sale. In the perspective of the retail marketing, suppliers buy these products directly from the manufacturer/producer and they transport the products to their local area and sale to their retailers or any shopping mall. Retailers sell the products directly to the customers. The proposed models are directly applicable between the manufacturer/producer to suppliers and suppliers to retailers. One of the interesting extensions of the current work would be possible by incorporating multiple equal sized advance payments scheme or multiple equal sized delay payments scheme. Relaxing zero ending inventory by non-ending inventory a profit maximization problem could be another possible extension. Also, anyone can extend this model by taking an imprecise value of inventory parameters (for instance, fuzzy-valued inventory parameters and defuzzyfy by any defuzzification technique, fuzzy-valued inventory cost and then apply alpha-cut and convert into interval or directly interval-valued inventory cost).
35
Acknowledgments The authors thank the three anonymous referees for their constructive comments and suggestions that improved the paper. Appendix A: Let us assume the following auxiliary functions from the equation (10), (1 )2 1 T 2 (1 )T 1 f1 (T ) ap bp 2 T K c j (a bp )(1 ) ln ln c j g (a bp ) 2 1 T 1 E T 4 2
(A1)
1 3 1 T 3 (1 )T 2 1 2 T c j h(a bp) ln 12 6 6 1 T 9
g1 (T ) T .
(A2)
Hence, we can write the total profit function in the following way TP1 ( p, T )
f1 (T ) . Now taking g1 (T )
the derivatives of f1 (T ) with respect to T for any given p , we have c j g (a bp ) (1 ) 2 1 2 f1 (T ) ap bp c j (a bp) T (1 ) 1 T 2 1 T
(A3)
c j h(a bp ) 1 3 2 2T 2 (1 )T 1 6 1 T
and f1 (T ) c j (a bp)
To
show
1 3 1 T
2
1 3 1 T
2
that
1
1 T the
2
c j h(a bp) 1 3 c j g (a bp) (1 )2 . 1 4 T (1 ) 2 6 1 T 2 1 T 2
function f1 (T ) is
4T (1 ) is
always
strictly
non-negative.
concave, Using
we
need
Lemma
1,
to we
(A4)
insure
that
see
that
4T (1 ) is always positive. As a result, f1 (T ) 0 for all T 0 and hence
f1 (T ) is differentiable, non-negative function and concave. Moreover, the function g1 (T ) T is positive, differentiable and convex function. Accordingly, the total profit function TP1 ( p, T ) is a strictly pseudo-concave function of T for any certain p , and hence there exists a unique optimal solution T * .
□
Appendix B: 36
Let us consider that 3 1 (u ) 4u (1 ). 1 u 2
(B1)
Therefore, at the point u 0 the value of is 3 1 (0) (1 ) 0 1 2
2 1
(B2)
3
Also, (u )
1 u 3
4 0 . This indicates that (u ) is an increasing function of u .
Consequently, (u ) is always positive for all u 0 .
□
Appendix C: For a settled cycle length T , differentiate the objective function TP1 ( p, T ) twice partially with respect to the decision variable p , we get (1 ) 2 1 T 2 (1 )T 1 ln a 2bp T c j b(1 ) ln c j gb 2 1 T 1 T 4 2 TP 1 1 3 1 T 3 (1 )T 2 1 2 T p T c j hb 6 ln 1 T 9 12 6
(C1)
2TP 2b 0 . (C2) p 2 Therefore, TP1 ( p, T ) is the concave function of p for any given value of T 0 and with this value there exists a unique p such that TP1 ( p, T ) is maximized. This completes the proof.
□
Appendix D: Let us assume the following auxiliary functions from the equation (20), 1 1 1 f 2 (t1 , t2 ) (ap bp 2 ) t1 ln(1 t2 ) K c j (a bp) (1 ) ln ln(1 t2 ) 1 t1
(1 ) 2 1 t12 (1 )t1 c j g (a bp) ln 2 1 t1 4 2 (1 )3 1 t13 (1 )t12 (1 )2 t1 c j h(a bp) ln 12 6 1 t1 9 6
(D1)
c ln(1 t2 ) cl s (a bp) t2
37
and
g 2 (t1 , t 2 ) t1 t 2 . Hence, from Eq. (20) we can write TP2 ( p, t1 , t 2 )
(D2)
f 2 (t1 , t 2 ) .To use results of Cambini and g 2 (t1 , t 2 )
Martein (2009), we need to show that f 2 (t1 , t 2 ) is a non-negative, differentiable and (strictly) joint concave function with respect to t1 and t 2 . The second order partial derivatives with respect to t1 and t 2 are:
1 h (1 ) 3 g (1 ) 2 c ( a bp ) 1 4 t ( 1 ) (D3) j 1 2 1 t1 2 2 (1 t1 ) 2 6 (1 t1 ) 2 t1 2 f 2 (t1 , t 2 ) 2 f 2 (t1 , t 2 ) (D4) 0 t 2 t1 t1t 2
2 f 2 (t1 , t 2 )
c p c j cl s t 2 2 f 2 (t1 , t 2 ) 2 t Therefore, the Hessian matrix for f 2 (t1 , t 2 ) is: H ii 2 1 f 2 (t1 , t 2 ) t t 2 1 and the first principal minor is: 2 f 2 (t1 , t 2 ) 2
H11
(a bp) 2 (1 t 2 )
(D5)
2 f 2 (t1 , t 2 ) t1t 2 , 2 f 2 (t1 , t 2 ) 2 t 2
1 h (1 )3 2 f 2 (t1 , t2 ) g (1 )2 c ( a bp ) 1 4 t (1 ) . j 1 2 2 2 (1 t1 )2 t12 6 (1 t1 ) 1 t1
(1 ) 3 4t1 (1 ) is always positive and hence, H 11 0 . Using Lemma 1, we see that (1 t1 ) 2 Again, the second principal minor is 2 f 2 (t1 , t 2 ) 2 f 2 (t1 , t 2 ) 2 f 2 (t1 , t 2 ) 2 f 2 (t1 , t 2 ) H 22 2 2 t 2 t1 t1t 2 t1 t 2
c j (a bp) 2 h (1 ) 3 c s 1 g (1 ) 2 p c c 1 4 t ( 1 ) 0 j l 1 1 t1 2 2 (1 t1 ) 2 6 (1 t1 ) 2 (1 t 2 ) 2 Consequently, the Hessian matrix for f 2 (t1 , t 2 ) is negative definite. Therefore, f 2 (t1 , t 2 ) is a nonnegative, differentiable and (strictly) concave function with respect to t1 and t 2 simultaneously. Moreover, the function g 2 (t1 , t 2 ) t1 t 2 is positive, differentiable and convex function, so the retailer’s total profit function per unit time TP2 ( p, t1 , t 2 ) is jointly pseudo-concave function in
(t1 , t 2 ) , and has only one maximum value. Consequently, the objective function TP2 ( p, t1 , t 2 ) , 38
for any fixed p , attains the global maximum value at the point (t1* , t 2 * ) .This completes the □
proof. Appendix E:
For any certain values of t1 0 and t2 0 , differentiate the objective function TP2 ( p, t1 , t 2 ) twice partially with respect to p , we get:
2TP2 ( p, t1 , t2 ) 2b 2 t1 t2 p
1 t1 δ ln(1 δt2 ) 0.
(E1)
Therefore, TP2 ( p, t1 , t2 ) is a concave function in p for any given t1 0, t2 0 and hence there exists a unique p such that TP2 ( p, t1 , t2 ) is maximized. References 1. Alfares, H. K., and Ghaithan, A. M. (2016). Inventory and pricing model with pricedependent demand, time-varying holding cost, and quantity discounts. Computers and Industrial Engineering, 94, 170-177. 2. Avinadav, T., Herbon, A., and Spiegel, U. (2013). Optimal inventory policy for a perishable item with demand function sensitive to price and time. International Journal of Production Economics, 144(2), 497-506. 3. Bhunia, A., and Shaikh, A. (2014). A deterministic inventory model for deteriorating items with selling price dependent demand and three-parameter Weibull distributed deterioration. International Journal of Industrial Engineering Computations, 5(3), 497-510. 4. Cambini, A and Martein, L. (2009). Generalized Convexity and Optimization: Theory and Applications. Springer-Verlag Berlin Heidelberg. 5. Chen, S. C., and Teng, J. T. (2015). Inventory and credit decisions for time-varying deteriorating items with up-stream and down-stream trade credit financing by discounted cash flow analysis. European Journal of Operational Research, 243(2), 566-575. 6. Chen, S.C., Min, J., Teng, J. T., and Li, F., (2016). Inventory and shelf-space management for fresh produce with freshness-and-stock dependent demand and expiration date. Journal of the Operational Research, 67(6), 884-896. 7. Das Roy, M., and Sana, S. S. (2017). Random sales price-sensitive stochastic demand: An imperfect production model with free repair warranty. Journal of Advances in Management Research, 14(4), 408-424. 8. Feng, L., Chan, Y. L., and Cárdenas-Barrón, L. E. (2017). Pricing and lot-sizing polices for perishable goods when the demand depends on selling price, displayed stocks, and expiration date. International Journal of Production Economics, 185, 11-20. 9. Ghare, P. M. and Schrader, G. P.(1963). A model for an exponentially decaying inventory. Journal of Industrial Engineering, 14, 238-243.
39
10. Ghoreishi, M., Weber, G. W., and Mirzazadeh, A. (2015). An inventory model for noninstantaneous deteriorating items with partial backlogging, permissible delay in payments, inflation-and selling price-dependent demand and customer returns. Annals of Operations Research, 226(1), 221-238. 11. Gupta, R. K., Bhunia, A. K., and Goyal, S. K. (2009). An application of genetic algorithm in solving an inventory model with advance payment and interval valued inventory costs. Mathematical and Computer Modelling, 49(5-6), 893-905. 12. Guria, A., Das, B., Mondal, S., and Maiti, M. (2013). Inventory policy for an item with inflation induced purchasing price, selling price and demand with immediate part payment. Applied Mathematical Modelling, 37(1-2), 240-257. 13. Hsu, P. H., Wee, H. M., and Teng, H. M. (2006). Optimal lot sizing for deteriorating items with expiration date. Journal of Information and Optimization Sciences, 27(2), 271-286. 14. Hsu, P. H., Wee, H. M., and Teng, H. M. (2007). Optimal ordering decision for deteriorating items with expiration date and uncertain lead time. Computers and Industrial Engineering, 52(4), 448-458. 15. Jaggi, C. K., Tiwari, S., and Goel, S. K. (2017). Credit financing in economic ordering policies for non-instantaneous deteriorating items with price dependent demand and two storage facilities. Annals of Operations Research, 248(1-2), 253-280. 16. Jain, R., and Singh, S. R. (2011). Inflationary implications on an inventory with expiration date, capital constraint and uncertain lead time in a multi-echelon supply chain. International Journal of Procurement Management, 4(4), 419-432. 17. Modak, N. M., Modak, N., Panda, S., and Sana, S. S. (2018). Analyzing structure of twoechelon closed-loop supply chain for pricing, quality and recycling management. Journal of Cleaner Production, 171, 512-528. 18. Maihami, R., and Kamalabadi, I. N. (2012). Joint pricing and inventory control for noninstantaneous deteriorating items with partial backlogging and time and price dependent demand. International Journal of Production Economics, 136(1), 116-122. 19. Maiti, A. K., Maiti, M. K., and Maiti, M. (2009). Inventory model with stochastic lead-time and price dependent demand incorporating advance payment. Applied Mathematical Modelling, 33(5), 2433-2443. 20. Mondal, B., Bhunia, A. K., and Maiti, M. (2003). An inventory system of ameliorating items for price dependent demand rate. Computers and industrial engineering, 45(3), 443-456. 21. Mukhopadhyay, S., Mukherjee, R. N., and Chaudhuri, K. S. (2004). Joint pricing and ordering policy for a deteriorating inventory. Computers and Industrial Engineering, 47(4), 339-349. 22. Mukhopadhyay, S., Mukherjee, R. N., and Chaudhuri, K. S. (2005). An EOQ model with two-parameter Weibull distribution deterioration and price-dependent demand. International Journal of Mathematical Education in Science and Technology, 36(1), 25-33.
40
23. Pal, B., Sana, S. S., and Chaudhuri, K. (2017). A stochastic production inventory model for deteriorating items with products’ finite life-cycle. RAIRO-Operations Research, 51(3), 669684. 24. Philip, G. C.(1974). A generalized EOQ model for items with Weibull distribution. AIIE Transactions, 6 (2), 159-162. 25. Roy, A. (2008). An inventory model for deteriorating items with price dependent demand and time-varying holding cost. Advanced Modelling and Optimization, 10(1), 25-37. 26. Roy, T., and Chaudhuri, K. S. (2007). An inventory model for a deteriorating item with price-dependent demand and special sale. International Journal of Operational Research, 2(2), 173-187. 27. Roy, A., Sana, S. S., and Chaudhuri, K. (2018). Optimal Pricing of competing retailers under uncertain demand-a two layer supply chain model. Annals of Operations Research, 260(1-2), 481-500. 28. Sana, S.S. (2011). Price-Sensitive demand for perishable items- an EOQ model. Applied Mathematics and Computation, 217(13),6248-6259. 29. Sridevi, G., Nirupama Devi, K., and SrinivasaRao, K. (2010). Inventory model for deteriorating items with Weibull rate of replenishment and selling price dependent demand. International Journal of Operational Research, 9(3), 329-349. 30. Taleizadeh, A. A. (2014). An EOQ model with partial backordering and advance payments for an evaporating item. International Journal of Production Economics, 155, 185-193. 31. Taleizadeh, A. A., Mohammadi, B., Cárdenas-Barrón, L. E., and Samimi, H. (2013). An EOQ model for perishable product with special sale and shortage. International Journal of Production Economics, 145(1), 318-338. 32. Taleizadeh, A. A., Jalali, M. S., and Sana, S. S. (2018). A game theoretic approach to optimal pricing of flights and passengers at congested airports. Journal of Modelling in Management, (just-accepted), 00-00. 33. Taleizadeh, A. A., Niaki, S. T. A., Aryanezhad, M. B., and Tafti, A. F. (2010). A genetic algorithm to optimize multiproduct multi-constraint inventory control systems with stochastic replenishment intervals and discount. The International Journal of Advanced Manufacturing Technology, 51(1-4), 311-323. 34. Taleizadeh, A. A., Niaki, S. T. A., and Aryanezhad, M. B. (2010). Optimizing multi-product multi-chance-constraint inventory control system with stochastic period lengths and total discount under fuzzy purchasing price and holding costs. International Journal of Systems Science, 41(10), 1187-1200. 35. Taleizadeh, A. A., Noori-daryan, M., and Cárdenas-Barrón, L. E. (2015). Joint optimization of price, replenishment frequency, replenishment cycle and production rate in vendor managed inventory system with deteriorating items. International Journal of Production Economics, 159, 285-295. 36. Taleizadeh, A. A., Pentico, D. W., Jabalameli, M. S., and Aryanezhad, M. (2013). An EOQ model with partial delayed payment and partial backordering. Omega, 41(2), 354-368. 41
37. Taleizadeh, A. A. (2017). Lot-sizing model with advance payment pricing and disruption in supply under planned partial backordering. International Transactions in Operational Research, 24(4), 783-800. 38. Tayal, S., Singh, S. R., and Sharma, R. (2014). Multi item inventory model for deteriorating items with expiration date and allowable shortages. Indian Journal of Science and Technology, 7(4), 463-471. 39. Tavakkoli, Sh., and Taleizadeh, A.A.,(2017). A lot sizing model for decaying item with full advance payment from the buyer and conditional discount from the supplier. Annals of Operations Research, 259,415-436. 40. Teng, J. T., Cárdenas-Barrón, L. E., Chang, H. J., Wu, J., and Hu, Y. (2016). Inventory lotsize policies for deteriorating items with expiration dates and advance payments. Applied Mathematical Modelling, 40(19-20), 8605-8616. 41. Tiwari, S., Cárdenas-Barrón, L. E., Goh, M., and Shaikh, A. A. (2018). Joint pricing and inventory model for deteriorating items with expiration dates and partial backlogging under two-level partial trade credits in supply chain. International Journal of Production Economics, 200, 16-36. 42. Wang, W. C., Teng, J. T., and Lou, K. R. (2014). Seller’s optimal credit period and cycle time in a supply chain for deteriorating items with maximum lifetime. European Journal of Operational Research, 232(2), 315-321. 43. Wu, J., and Chan, Y. L. (2014). Lot-sizing policies for deteriorating items with expiration dates and partial trade credit to credit-risk customers. International Journal of Production Economics, 155, 292-301. 44. Wu, J., Ouyang, L. Y., Cárdenas-Barrón, L. E., and Goyal, S. K. (2014). Optimal credit period and lot size for deteriorating items with expiration dates under two-level trade credit financing. European Journal of Operational Research, 237(3), 898-908. 45. Wu, J., Chang, C. T., Cheng, M. C., Teng, J. T., and Al-khateeb, F. B. (2016a). Inventory management for fresh produce when the time-varying demand depends on product freshness, stock level and expiration date. International Journal of Systems Science: Operations and Logistics, 3(3), 138-147. 46. Wu, J., Al-Khateeb, F. B., Teng, J. T., and Cárdenas-Barrón, L. E. (2016b). Inventory models for deteriorating items with maximum lifetime under downstream partial trade credits to credit-risk customers by discounted cash-flow analysis. International Journal of Production Economics, 171, 105-115. 47. Wu, J., Teng, J. T., and Chan, Y. L. (2018). Inventory policies for perishable products with expiration dates and advance-cash-credit payment schemes. International Journal of Systems Science: Operations and Logistics, 5(4), 310-326. 48. You, P. S. (2006). Ordering and pricing of service products in an advance sales system with price-dependent demand. European Journal of Operational Research, 170(1), 57-71. 49. Harris, F.W., (1913). How many parts to make at once: Reproduced from Factory, The Magazine of Management, 10(2), 135-136,152. 42
Highlights
We study two supply chain models by assuming the demand to be dependent on price.
We assume expiration rate dependent deterioration and shortages with partial backlogging.
We formulate the problem mathematically and prove its optimality.
A solution procedure has been ameliorated and two numerical examples are presented.
43