An Integrated Production Inventory System for. Perishable Items with Fixed and Linear Backorders

Transcription

1 Int. Journal of Math. Analysis, Vol. 8, 2014, no. 32, HIKARI Ltd, An Integrated Production Inventory System for Perishable Items with Fixed and Linear Backorders M. Ravithammal Sathyabama University Chennai , Tamil Nadu, India R. Uthayakumar Gandhigram Rural Institute - Deemed University Dindigul Tamil Nadu, India S. Ganesh Sathyabama University Chennai , Tamil Nadu, India Copyright 2014, M. Ravithammal, R. Uthayakumar and S. Ganesh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The paper mainly deals with production for vendor manufacturer and fixed, linear backorders for buyer of fixed life time product. The decentralized model with and without coordination to analyze the benefit costs of both the buyer and vendor manufacturer for coordinating the supply chain is proposed in this paper. The quantity discount is implemented for coordinating the supply chain. The inventory situation with backorder cost, varying holding cost production and demand is also analyzed in this paper. The outcome of such situation on the savings percentage is found. The previous researchers had analyzed different models on this relevant topic without backorders and production. But this paper mainly features on model with production and two types of backorder cost. This paper also includes a detailed numerical example for more understanding of the proposed strategy. Keywords: Quantity discount, Production, Fixed backorder cost, Linear backorder cost

2 1550 M. Ravithammal, R. Uthayakumar and S. Ganesh 1. INTRODUCTION Inventory refers to the stock of various items such as raw material, work-in-progress, finished goods and spares parts. Inventory management is gaining attention for the past three decades for the reason that the inventory cost and inventory decision has an effect on the survival of business enterprises. The inventory decision becomes more complicated when the item stored has a limited shelf life. Analytical decisions and mathematical models are much needed in inventory management. The model described in this paper deals with vendor production and buyer backorders for fixed life time products with quantity discount. The buyer purchases the limited shelf period items from the vendor who is also a manufacturer. The real and perceived costs of the inability to fulfill an order is the backorder cost. The costs can include negative customer relations, interest expenses etc. The backorder cost is further classified as fixed and linear back order cost (figure1). Liu and Shi [7] developed (s, S) model for inventory with exponential lifetimes and renewal demands. Goyal and Gupta analyzed integrated inventory models: the buyer-vendor coordination. Yongrui Duan et al. [11] developed buyer-vendor inventory coordination with quantity discount incentive for fixed lifetime product. Fries [2] considered optimal order policies for a perishable commodity with fixed life time. Nandakumar and Morton [8] studied near myopic heuristic for the fixed life perishability problem. Liu and Lian [5] developed (s, S) model for inventory with fixed life time. Lian and Liu [6] studied continuous review perishable inventory systems. Cardenas-Barron [1] developed the derivation of EOQ/EPQ inventory models with two backorder costs using analytic geometry and algebra. Sphicas [9] analyzed EOQ and EPQ with linear and fixed backorder costs. Hung-Chi Chang [4] analyzed a comprehensives note on an economic order quantity with imperfect quality and quantity discounts. Wong et al. [10] developed coordinating supply chains with sales rebate contracts and vendor-managed Inventory. In this paper single - manufacturer, single - buyer supply chains with production for vendor and backorders (fixed and linear) for buyer are considered. In supply chain management some important mechanisms are implemented to coordinate vendor and buyer. They are quantity discount, revenue sharing, sales rebate and trade credit. Among this quantity discount is a commonly used scheme. The developed models analyze the benefit of coordinating supply chain by quantity discount strategy. If we ignore production for vendor and backorders for buyer then we get the model by Yongrui Duan et al. [11] which is considered a particular case in our model. The detailed description of the paper is as follows. In section 2, assumption and notations are given. Decentralized models with and without coordination are formulated in section 3. Analytically easily understandable solutions are obtained in these models. In section 4, a numerical example is given in detail to illustrate the models. Finally conclusion and summary are presented. 2. ASSUMPTIONS AND NOTATIONS 2.1 Assumptions (1) The model is allowed for single product and demand rate is constant and known. (2) Two types of back order cost are considered (Linear back order and fixed back order

3 Integrated production inventory system 1551 cost). And lead time is zero. (3) Production rate is greater than the demand rate. (P > D) (4) Manufacturer produced new and fresh product and supply to the buyer. 2.2 Notations D - Annual demand of the buyer L - Life time of product P - Production rate for manufacturer k 1, k 2 - Manufacturer and buyer s setup costs per order, respectively h 1, h 2 - Manufacturer and buyer s holding costs, respectively p 1, p 2 - Delivered unit price paid by the manufacturer and the buyer respectively B - Size of back orders in units. - Back order cost per unit (fixed back order cost) π 1 - Back order cost per unit, per unit of time (linear back order cost) - Manufacturer s order multiple in the absence of any coordination - Manufacturer s order multiple under coordination - Buyer s order multiple under coordination. buyer s new order quantity - Denotes the per unit dollar discount to the buyer if he orders every time 3. MODEL FORMULATION In this section we analyzed decentralized models with and without coordination. In the coordination strategy quantity discount is offered by the manufacturer to the buyer. 3.1 EOQ model without coordination with fixed and linear backorders Without coordination the buyer s total cost is formulated as follows and (1) For optimality and Now and Total minimum cost of buyer ( ) Without coordination, the buyer s order size is with the annual cost ( ). The manufacturer s order size should be

4 1552 M. Ravithammal, R. Uthayakumar and S. Ganesh. Since he faced constant demands with the intervals. Thus the total annual cost for the manufacturer is given by = In the absence of coordination the manufacturer s problem can be developed as (2) s.t { where and ensure that items are not overdue before they are sold up by the buyer. (3) Theorem 1 If then where { } (4) is the optimum of (3) and is the least integer greater than or equal to. Proof is strictly convex in m because Assume that be the optimum of then { { } }. Applying into then we get, Take Since, holds. If and is a convex function, then, else. Therefore if, then. 3.2 EOQ model with coordination In coordination strategy, quantity discount is offered by the manufacturer to the buyer, if the buyer orders at a discount factor where. Now the manufacturer s order quantity is, where and is the buyer s new order quantity. The total annual cost for the manufacturer is developed as follows

5 Integrated production inventory system 1553 where is buyer s quantity discount. Thus with coordination the problem can be developed as (5) subject to { ( ), (6) The first constraint indicates that items are not overdue before they are sold up by the buyer and the second constraint indicates that the buyer s cost under coordination is not greater than the absence of any coordination. Theorem 2 Let is true where and be the optimum of (3) and (6). Proof If the second constraint of (5) must be an equation then takes smallest value and also to be minimized. i.e., ( ) ( ) (7) If, ( ) ( ) Hence if equation (6) is equivalent to (3), which indicate (3) is a special case of (6). Thus holds. Put (7) into (5), we get ( ) (8) Here is convex function in, Since is convex in. Consider be the minimum of. For optimality, we get Now substitute the values of and into we have [ ] [ ] [ ] (10) (9)

6 1554 M. Ravithammal, R. Uthayakumar and S. Ganesh then first constraint of (6) is equivalent to. Substitute the values of and into, we get ( [ ]) ( ) { [ ]} (11) Thus (5) is equivalent to [ ] { } subject to { (12) Since is a strictly increasing function for. Here is convex because [ ], and is strictly concave because [ ]. Proposition 1 Consider the minimum of be for, then { [ ] [ ] (13) Proof Since the minimum of be for, then { } Now, we have Similarly, we have [ ] (14) [ ] (15) Thus [ ] when [ ] [ ] (By (14)

7 Integrated production inventory system 1555 and (15)). Otherwise when [ ]. Also note that if [ ], then. Therefore (14) holds. Proposition 2 Let and be solution of (10) then i) If or and, then for. ii) If and, then If, for and if and, for. where, and [ ] Proof To solve, we get and Because is a quadratic function, the following conclusions hold. If, then for every n and If then and are real solutions of. In view of, i) If, then for ; ii) If then for ; iii) If and, then view of n is positive integer, for. Theorem 3 If then. Proof (I) If [ ] then [ ]. Because holds for, and is a decreasing function of n. To prove [ [ ] ]

8 1556 M. Ravithammal, R. Uthayakumar and S. Ganesh (16) Since are all positive and then (16) holds. (II) If then. Since are all positive. (III) If then. Here is a decreasing function so ( ). From (1) to (3), if. Remarks (i) Without coordination the manufacturer s minimized total cost is ( ) orders with a regular interval ( ).He place and his order size is ( ). (ii) Theorem (2) ensures that the optimum total cost with coordination is less than without coordination, the manufacturer will get more benefit if the buyer orders every time. (iii) Theorem (3) proves the buyer s order size is greater with coordination to compare with without coordination,. 4. NUMERICAL EXAMPLE In this section numerical examples are presented to illustrate the model. The buyer s saving in percentage The vendor s saving in percentage The vendor s saving in percentage if he does not share the saving with the buyer Example Given D = 10,000 units per year, = 30$ per unit, α = 0.5, L = 0.25 year, k 1 = 300$ per order, k 2 = 100$. The different values of P, h 1, h 2, π, π 1 and computational results are as specified in Table 1

9 Integrated production inventory system 1557 Table 1 Sample computational result for above example P h 1 h 2 π π 1 d(k) Q B SP v1 SP b SP v The computational result indicates that, when the backorder cost increases for both vendor and buyer the savings percentage increases decreases. When the backorder cost for buyer increases in different production quantities the savings percentage decreases. When the backorder cost for buyer increases and vendor keeping production constant, the savings percentage decreases. When production for vendor, backorder cost for buyer and demand increases, the savings percentage decreases.

10 0,01 0,02 0,03 0,04 0,05 0,1 0,2 0, M. Ravithammal, R. Uthayakumar and S. Ganesh ,010,020,030,040,051,5 SPv1 SPb SPv2 Fig1: Representation of linear and fixed backorder cost Fig 2: Main effects for example when P, π fixed π 1 increases SPv1 SPb SPv ,1 1,2 1,3 1,4 1,5 2 2,5 3 SPv1 SPb SPv2 Fig 3: Main effects for example when Fig 4: Main effects for example when P, π 1 fixed and π increases P fixed π, π 1 increases 5. CONCLUSION The model developed and illustrated proves that the savings percentage decreases when holding cost, backorder cost, demand and production increases in combination or individually for the buyer. The backorder cost both linear and fixed has an enormous impact on savings percentage. Irrespective of changes in production and demand, the increase in backorder cost results in decrease in savings percentage in spite of coordination and quantity discount. The inventory items with limited shelf period manufactured by vendor and supplied buyer has to be managed in an effective way by exercising proper control on backorder cost. Analytically easily understandable solutions are obtained. It has been proved that the buyer s order size is higher with coordination than non coordination. We prove that the coordination quantity discount strategy can achieve system optimization and win-win outcome. As a result both the vendor and the buyer benefit in the long run. Numerical example is presented to illustrate the model.

11 Integrated production inventory system 1559 REFERENCES [1] L.e. Cardenas-Barron, The derivation of EOQ/EPQ inventory models with two backorder costs using analytic geometry and algebra, Applied Mathematical Modelling. 35 (2011) [2] B. Fries, Optimal order policies for a perishable commodity with fixed life time, Operations Research. 23(1975) [3] S.K. Goyal, Y.P. Gupta, Integrated inventory models: the buyer-vendor coordination, European Journal of Operational Research. 41(1989) [4] Hung-Chi Chang, A comprehensives note on an economic order quantity with imperfect quality and quantity discounts, Applied Mathematical Modelling. 35(2011) [5] L. Liu, Z. Lian, (s, S) model for inventory with fixed life time, Operations Research. 47 (1999) [6] Z. Lian, L. Liu, Continuous review perishable inventory systems: models and heuristics, IIE Transactions. 33(2001) [7] L. Liu, D. Shi, (s, S) model for inventory with exponential lifetimes and renewal demands, Naval Research Logistics. 46(1999) [8] P. Nandakumar, T.E. Morton, Near myopic heuristic for the fixed life perishability problem, Management Science. 39(1993) [9] G.P. Sphicas, EOQ and EPQ with linear and fixed backorder costs: Two cases identified and models analyzed without calculus, International Journal of Production Economics. 100(2006) [10] W.K. Wong, J. Qi, S.Y.S. Leung, Coordinating supply chains with sales rebate contracts and vendor-managed Inventory, International Journal of Production Economics. 120(2009) [11] Yongrui Duan, Jianwen Luo, Jiazhen Huo, Buyer-vendor inventory coordination with quantity discount incentive for fixed lifetime product. International Journal of Production Economics. 128(2010) Received: June 11, 2014