JOINT OPTIMIZATION INVENTORY MODEL WITH STOCHASTIC DEMAND AND CONTROLLABLE LEAD TIME BY REDUCING ORDERING COST AND SETUP COST

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1 Communications in Applied Analysis, 21, No , ISSN: JOINT OPTIMIZATION INVENTORY MODEL WITH STOCHASTIC DEMAND AND CONTROLLABLE LEAD TIME BY REDUCING ORDERING COST AND SETUP COST M. VIJAYASHREE 1 AND R. UTHAYAKUMAR 2 Department of Mathematics The Gandhigram Rural Institute Deemed University Gandhigram, Dindigul, Tamil Nadu, INDIA ABSTRACT: This paper explores the setup and order processing cost reduction in the single vendor and the single buyer integrated production inventory model. The mathematical model is derived to investigate the effects of the optimal decisions when capital investment strategies in setup and order processing cost reduction are adopted. This proposed model intends to derive the exact cost function for the entire supply chain including logarithmic investment functions and an efficient computational algorithm is constructed to find the best solution. We have developed the set up cost and ordering cost reductions in an integrated inventory production system in which demand during the lead time follows a normal distribution. The objective of this paper is to minimize the integrated total cost by optimizing the order quantity, setup cost, ordering cost, safety factor and number of deliveries simultaneously. An algorithm to find the optimal solutions is developed. Order cost and setup cost are considered as logarithmic functions of capital investment. Besides, an efficient algorithm is developed to determine the optimal solution, and our approach is illustrated through a numerical example. A

2 152 M. Vijayashree and R. Uthayakumar computer code using the software Matlab is developed to derive the optimal solution and numerical example is presented to illustrate the model. Finally, sensitivity analysis is carried out with respect to the key parameters and some managerial implications are also included. AMS Subject Classification: 90B05, 90C25, 90C30 Key Words: integrated inventory model, stochastic demand, controllable lead time crashing cost, cost reduction, capital investment, normal distribution case Received: October 17, 2016; Accepted: December 1, 2016; Published: February 18, doi: /caa.v21i2.2 Dynamic Publishers, Inc., Acad. Publishers, Ltd INTRODUCTION Inventory control is an important field in supply chain management, and a great deal of research efforts have been devoted to it over past few decades. Lead time plays an important role in today s logistics management. In order to satisfy customer demands in today s competitive markets, critical information needs to be shared along the supply chain. A high level of coordination between vendors and buyers decision making is also required. The concept of joint economic lot sizing JELS has been introduced to refine traditional methods for independent inventory control. Most of inventory models considered to date assume just one facility e.g., a buyer or vendor managing its inventory policy to minimize its own cost or maximize its own profit. This one-sided-optimal strategy is not suitable for global markets. The issue of Just-In-Time JIT has recently received great attention. Most JIT research has focused on the integration between vendor and buyer. Once a long-term relationship between both facilities has been developed, both parties can cooperate and share information to achieve improved benefits. Coordination among different business entities such as buyer, vendor, producer etc., are an important way to gain today s competitive advantages. It involves synchronization of different efforts or actions of the various units of an

3 Joint Optimization Model 153 organization to provide the requisite amount, timing, quality and sequence of efforts so that the planned objectives may be achieved with minimum conflict. The coordination inventory replenishment decision in a supply chain increase the efficient of the channel and improve the position of the companies involved. Benefits of the coordination can include lower inventory related costs, reduced lead time and higher product quality. Thus, via integrated inventory management, the competitive position of the whole supply chain can be improved. It is clear that this is especially important in industries facing high competitive pressure and in industries where internal logistics processes have already been efficient. In modern production management, controllable lead time and ordering cost reduction and setup cost reduction are keys to business success and have attracted considerable research attention. Ordering quantity, service level and business competitiveness can be shown to possibly be influenced directly or indirectly via lead-time and or ordering cost control. The integrated inventory models treat the ordering cost and/or lead time as constants. However, in some practical situations, lead time and ordering cost can be controlled and reduced in various ways. Ordering cost and setup cost reduction can be attained through worker training, procedural changes, and specialized equipment acquisition. In the literature, Porteus [32] first introduced the concept and developed a framework for investing in reducing EOQ model set-up cost. This development encouraged many researchers to examine setup/ordering cost reduction e.g. Keller et al. [20]; Nasri et al. [25]; Kim et al. [21]; Paknejad et al. [30]. Traditionally, the lead time of inventory model is hypothesized as known or with certain probability distribution, which therefore is not subject to control. But in many practical situations, lead time can be reduced by an additional crashing cost. That is, it is controllable. In fact, as pointed out in Tersine [33] lead time usually comprises several components, such as setup time, process time, and wait time, move time and queue time. In many practical situations, lead time can be reduced using an added crashing cost. In other words, lead time is controllable. Chang et al. [3] studied an integrated vendor-buyer inventory model with controllable lead time and ordering cost reduction. Zhang et al. [41] presented an integrated vendor-managed inventory model for a two-echelon system with order cost reduction. The integrated model was an extension of Woo et al. [42]

4 154 M. Vijayashree and R. Uthayakumar but it relaxed their assumption of a common cycle time for all buyers and the vendor. The capital investment in reducing buyer s ordering cost is assumed a logarithmic function of the ordering cost. This function is consistent with the Japanese experience as reported in Hall [12] and has been utilized in many researches e.g., Porteus [32]; Nasri et al. [25]; Paknejad et al. [30]; Chang et al. [3]; Hou [17]. The Japanese experience of using JIT production showed that the benefits associated with lead time control are clear. Therefore, reducing lead time is both necessary and beneficial. Many researchers focused on developing either setup cost or ordering cost reduction, Additionally, from the authors literature search, none of the researchers have considered integrated production inventory model by controlling setup and order processing cost simultaneously to reduce the integrated total inventory cost. Therefore, this paper intends to fill this remarkable gap in the literature. In this paper we consider the joint optimization model with stochastic demand and controllable lead time by reducing the ordering cost and setup costs. Here, we consider the lead time demand follows a normal distribution. Also assume that order processing cost and setup cost are logarithmic functions of capital investment. The objective of this study is to minimize the joint total cost by simultaneously optimizing the order quantity, setup cost, ordering cost, safety factor and number of deliveries. Besides, an efficient algorithm is developed to determine the optimal policy, and our approach is illustrated through a numerical example. From the results of numerical example, it can be shown that, the significant savings can be achieved through the reductions of order processing cost and setup cost. 2. LITERATURE REVIEW In recent years, the successful Japanese experience of employing just-in-time JIT production has triggered considerable attention. The ultimate goal of JIT from the production/inventory management standpoint is to produce small-lot sizes with high quality products. Investing capital in shortening lead time and improving quality are regarded as the most effective means of achieving this goal. By shortening lead time, we can lower the safety stock, reduce the out-of-stock loss, and improve the customer service level so as to gain

5 Joint Optimization Model 155 competitive advantages in business. With such characteristics, researchers have modified traditional inventory models to incorporate the implementation of lead time concepts. Liao et al. [22] first presented a probabilistic inventory model in which the order quantity is predetermined and lead time is the unique decision variable. Ben-Daya et al. [2] extended Liao et al. [22] model by considering both lead time and ordering quantity as decision variables where shortages are neglected. Ouyang et al. [29] generalized Ben-Daya et al. [2] model by allowing shortages with partial backorders. Moon et al. [24] and Hariga et al. [13] revised the Ouyang et al. [29] model by including the reorder point as one of the decision variables. Researchers Liao et al. [22], Ben-Daya et al. [2], Moon et al. [24] and Hariga et al. [13] in this line most often addressed have focused on the benefits of lead-time reduction where the quality-related issues and the benefits of cooperation between vendor and buyer are not taken into account. During the last few years, the concept of integrated vendor-buyer inventory management has attracted considerable attention, accompanying the growth of Supply Chain Management SCM. Research on the integrated vendor-buyer cooperative inventory problem primarily focused on the production shipment schedule in terms of the size and frequency of shipments transferred between both parties under perfect quality e.g., Goyal [6],[8], Lu [23], Hill [14], [15], Goyal et al. [9], and Kelle et al. [19]. The cooperation between vendor and buyers for improving the performance of inventory control has received a great deal of attention, and the integration approach has been studied for years. For example, Banerjee [1] developed a joint economic lot size model for a single-buyer, single-vendor system with a lot for-lot policy. Goyal [7] generalized Banerjee s model [1] by relaxing the assumption of the lot-for-lot policy. Lu [23] improved the policies proposed by Banerjee [1] and Goyal [8] by assuming that the delivery quantity to the buyer is an identical at each replenishment. Viswanathan [34] compared the above two policies and proved that there is no strategy that obtain the best solution for all possible problem parameters. Ha et al. [10] analyzed the integration between buyer and supplier by setting up a mathematical model in which the inventory cost of the vendor is taken as a discontinuous sawtooth function. Yang et al. [36] developed an integrated economic ordering policy for deteriorating items for vendor and buyer. Wu et

6 156 M. Vijayashree and R. Uthayakumar al. [35] considered multiple lot size deliveries in the model proposed in Yang et al. [39]. Ouyang et al. [28] presented a single-vendor single-buyer integrated production inventory model under the assumption that lead time demand is stochastic and the lead time can be reduced at an added cost. But whereas these studies focused on joint lot sizing and vendor-buyer coordination, the case of multiple buyers was overlooked, except in Lu [23]. In recent years, several authors have studied integrated inventory models for a single vendor and multiple buyers. Yang et al. [38] generalized Wu et al. [35] by considering a single vendor and multiple buyers. Yang et al. [40] extended the model proposed in Yang et al. [37] by taking into account raw material inventory. Yu et al. [40] considered raw materials procurement decisions and proposed an integrated inventory model for the supply chain in which a single vendor supplies a single deteriorating item to multiple buyers with a common replenishment cycle. Jalbar et al. [18] studied a multistage distribution/ inventory system with a central warehouse and multiple retailers. They developed a heuristic for computing near-optimal integer-ratio policies to minimize the overall cost in the system. The Japanese experience of using JIT production showed that the benefits associated with lead time control are clear. Therefore, reducing lead time is both necessary and beneficial. Inventory models incorporating lead time as a decision variable were developed by several researchers. Liao et al. [22] first devised a probability inventory model in which lead time was the unique decision variable. Later, several researchers e.g., Ben-Daya et al. [2], Ouyang et al. [29], Moon et al. [24] and Hariga et al. [13] developed various analytical inventory models to explore the lead time reduction problem. The underlying assumption in the above studies was that lead time could be decomposed into n mutually independent components, each with a different but fixed crashing cost independent of the ordered lot size. However, this view may not be realistic. In a real environment, to reduce lead time, managers may ask workers to work overtime, employ part-time workers, use special delivery, and so on. Intuitively, extra cost should be paid for these services, and these costs may depend on the ordered lot size. Generally, the larger the ordered lot size, the higher the cost needed to reduce the lead time. Therefore, it seems reasonable to consider that lead time crash cost depends not only on the amount of lead time to be shortened, but also on the ordered lot size.

7 Joint Optimization Model 157 Pan et al. [31] developed an integrated inventory model with controllable lead time. Recently, Ouyang et al. [27] investigated the influence of ordering cost reduction on modified continuous review inventory systems involving variable lead time with partial backorders. Subsequently, Ouyang et al. [26] proposed a modified lot-size reorder-point inventory model with imperfect production processes to study the effects of reducing lead time and set-up cost. The optimal policies derived in these two articles are buyer focused, and the lead time and ordering/set-up cost reduction were assumed to act independently. However, an independent relationship between lead time and ordering/set-up cost is just one possibility. In some practices, lead time and ordering/set-up cost reduction might be closely related. A lead time reduction could accompany a reduction in the ordering /set-up cost, and vice versa. In the recent year, Glock [5] developed the joint economic lot size problem. And also Glock [4] developed lead time reduction strategies in a single-vendor- single buyer integrated inventory model with lot size-dependent lead time and stochastic demand. Hoque [16] developed a vendor buyer integrated production inventory model with normal distribution of lead time. In this study, we assume that long-term strategic partnerships between buyer and vendor are well established. Therefore, buyer and vendor are willing to cooperate and share information with each other to benefit both parties. Based on this assumption, we consider an integrated production inventory model with shortage permitted and assumed that lead time is controllable. Ouyang et al. [28] model by simultaneously optimizing ordering lot size, reorder point, lead time and the number of lots delivered in one production cycle. Firstly, we assume that the lead time demand follows a normal distribution and try to find the optimal ordering policy. Therefore, to the best of our knowledge the author has considered the joint optimization model with stochastic demand and controllable lead time by reducing the ordering cost and setup cost. The purpose of the proposed model is to extend Ouyang et al. [28] model, we assume that the lead time demand follows a normal distribution and try to find the order quantity, lead time, setup and ordering cost reduction and number of deliveries in one production cycle. The main purpose this proposed model is reducing both setup cost and ordering cost. Further, numerical examples are provided to illustrate the

8 158 M. Vijayashree and R. Uthayakumar benefits of integration. The remainder of the chapter is organized as follows. Section 3 describes the notation and assumptions used throughout this study. We formulate a single-vendor single-buyer inventory model in section 4. The optimal solution for the lead time demand follows a normal distribution in section 4.1. A section 5 describes the solution procedure. An efficient algorithm is developed to obtain the optimal solution in section 6. A numerical example is provided in section 7 to illustrate the results. In section 8 sensitive analysis for the different parameter is done. Managerial implications are also included in section 9. Finally, we draw some conclusions and give suggestion for future research in section NOTATIONS AND ASSUMPTIONS To establish the mathematical model, the following notations and assumptions of the model are used as follows 3.1. NOTATIONS To develop the proposed model, we adopt the following variables and parameters Variables Q Buyer s order quantity; L Length of lead time for the buyer; m The number of deliveries of the product delivered from the vendor to the buyer in one production cycle, positive integer; S Vendor s setup cost per set-up; A Buyer s ordering cost per order; k Safety factor. Parameters D Expected demand per unit time for the buyer; P Vendor s production rate in units per unit time, P > D ; C v Unit production cost paid by the vendor; C b Unit purchase cost paid by the buyer;

9 Joint Optimization Model 159 A 0 Buyer s original ordering cost per order before any investment is made; S 0 Vendor s original setup cost per setup before any investment is made; h v Vendor s holding cost per item per unit time; h b Buyer s holding cost per item per unit time; R Reorder point of the buyers; φ The standard normal distribution; ϕ The standard normal cumulative distribution; π The unit backorder cost; X The lead time demand which has a cumulative distribution function c.d.f. F with finite mean DL and standard deviation σ L, where σ denotes the standard deviation of demand per unit time. x + Maximum value of x and 0, i.e. x + = max {x, 0} ; α Annual fractional cost of capital investment; IA Capital investment in order processing cost reduction; IS Capital investment in setup cost reduction; ITC Integrated total cost for the single vendor and the single buyer ASSUMPTIONS To develop the proposed model, we adopt the following assumptions 1. A single vendor and a single buyer are considered in this model. 2. The buyer orders a lot size Q and the vendor produces mq units with a finite production rate P P > D in one setup but ships in quantity Q to the buyer over m times. The vendor incurs a setup cost S for each production run and the buyer incurs an ordering cost A for each order quantity Q. 3. Inventory is continuously reviewed. The buyer places an order when the inventory position reaches the reorder point R. 4. The reorder point R = expected demand during lead time + safety stock SS, and SS = k standarddeviationofleadtimedemand, that is, R = DL+kσ L where k is a safety factor.

10 160 M. Vijayashree and R. Uthayakumar 5. Shortages are allowed and completely backordered. 6. The demand X during the lead time L has a normal probability density function p.d.f.f x with finite mean DL and standard derivations σ L> The lead time L consists of n mutually independent components. The ith component has a normal duration b i, minimum duration a i and crashing cost per unit time c i. For convenience, we rearrange c i such that c 1 < c 2 < c 3 <... < c n. The components of lead time are crashed one at a time starting from the first component because it has the minimum unit crashing cost, and then the second component, and so on. 8. Let L 0 = n b i and L i be the length of lead time with components i=1 1, 2, 3,..., i crashed to their minimum duration, then L i can be expressed as L i = L 0 n b j a j, i = 1, 2,..., n; and the lead time j=1 crashing cost per cycle CL is given by i 1 CL = c i L i 1 L+ c j b j a j, L L i, L i 1. j=1 In addition, the length of lead time is equal for all shipping cycles, and the lead time crashing costs occur in each shipping cycle. 9. The extra costs incurred by the vendor will be fully transferred to the buyer if shortened lead time is required. 10. We assume that the capital investment, IA in reducing buyers ordering cost is a logarithmic function of the ordering cost A. That is, IA = 1 λ ln A 0 A for 0 < A A0. 1. We assume that the capital investment, IS in reducing vendors setup cost is a logarithmic function of the setup cost A. That is, IS = 1 δ ln S 0 S for0 < S S0. ThisfunctionisconsistentwiththeJapanese experience as reported in Hall [12], and has been utilized in many researchers e.g., Porteus [32], Nasri et al. [25], Kim et al. [21], Paknejad et al. [30], and Ouyang et al. [26].

11 Joint Optimization Model MODEL FORMULATION As we asserted in assumption 1, if the buyer order quantity Q, then the vendor produces mq at one set-up where m is an integer in order to reduce its set-up cost, and as soon as the buyer s lot size Q is produced, the lot is delivered to the buyer to reduce the inventory cost. Although the vendor manufactures mq in one production cycle, he will deliver quantity Q to the buyer over m times. Therefore, the expected cycle lengths are mq D and Q D for vendor and buyer respectively. The first step in the formulation to the total expected cost per unit time for the buyer is to identify the separate components of cost. The cost per order is A, so that the ordering cost per unit time is AD Q. Since the system is continuously reviewed by the buyer, when the inventory level drops to the buyer, when the inventory level drops to the reorder point R, an order of quantity Q is made. The expected net inventory level just before receipt of an order is R DL, and the expected net inventory level immediately after the successive order is Q+R DL. Hence, the average inventory over the cycle can be approximated by Q 2 +R DL. So that the buyer s expected holding cost per unit time is [ ] h b C Q b 2 +R DL. As mentioned earlier, the lead time demand X has a c.d.f. F with finite mean DL and standard deviation σ L, and the reorder point R = DL+kσ L. Shortage occurs when X > R, then, the expected shortage quantity at the end of cycle is given by EX R + = x RdF x. Because the unit backorder cost is πdex R+ Q from assumption 7, the lead time crashing cost per unit time is DCL Q. The resulting total cost per unit time for the buyer is therefore R TC b Q,R,L = ordering cost + holding cost + shortage cost + lead time crashing cost = AD Q Q +h bc b 2 +R DL πdex R+ + + DCL Q Q. On the other hand, as to the vendor, its total cost per unit time can be represented by TC v Q,m = setup cost + holding cost

12 162 M. Vijayashree and R. Uthayakumar Figure 1: The inventory pattern for the buyer and the vendor Since S is the vendor s set-up cost per set-up and the production quantity for a vendor in a lot will be mq, its expected set-up cost per unit time is given by SD mq. During the production period, when the first Q units have been produced, the vendor will deliver them to the buyer, after that the vendor will make the delivery on the average every Q D unit of time until the inventory level falls to zero see Fig 1 The average inventory for the vendor can be calculated as follows {[ Q mq P +m 1 Q m2 Q 2 ] [ ]} Q 2 D D 2P D m 1 mq = Q [ m 1 D 1+ 2D ]. 2 P P Hence, the vendor s expected holding cost per unit time is [ Q h v C v m 1 D 1+ 2D ]. 2 P P Therefore, the total cost per unit time for the vendor is

13 Joint Optimization Model 163 TC v Q,m = Setup cost + holding cost [ m = SD mq +h Q vc v 2 1 D P 1+ 2D P ]. Consequently, the integrated total cost per unit time for the vendor and buyer is given by TTC vb Q, R, L,m = TC b Q,R, L +TC v Q,m D [ A+ S ] Q m +πex R+ +CL + Q [h b C b +h v C v m 1 D 1+ 2D ] +h b C b R DL. 2 P P Building upon model 1 that is equation 1, we desire to study the effects of investment on setup and order processing cost reduction. Now we take setup and order processing cost, which are given in the equation 1 as decision variables. When setup and order processing cost are no longer considered to

14 164 M. Vijayashree and R. Uthayakumar be fixed parameters but decision variables, the setup and order processing cost are accomplished by varying the capital investment allocated to reduce the setup and order processing cost. The logarithmic function is one of many possible investment functions, and it may be an interesting research topic to consider a general investment function. Our problem is to minimize the sum of the investment in setup and order processing cost reduction, and the inventory relevant costs as expressed in 1 by simultaneously optimizing Q, A, S, m and L L i, L i 1, constrained on 0 < A A 0 and 0 < S S 0. That is, the objective of our problem is to minimize the following integrated total cost ITC vbas Q,R,L,A,S,m = ITC vb Q,L,R, m+αia+is = D [ A+ S ] Q m +πex R+ +CL + Q [h b C b +h v C v m 1 D 1+ 2D ] 2 P P 1 +h b C b R DL [ A0 +α aln A +bln ] S0, S for 0 < A A 0 and 0 < S S 0, where α is the annual fractional cost of capital investment e.g., interest rate. 5. THE LEAD TIME DEMAND FOLLOWS A NORMAL DISTRIBUTION CASE In this section, we assume that the lead time demand X is normally distributed with finite mean Q and standard deviation σ L and R = DL + kσ L. Therefore, the expected shortage quantity at the end of the cycle is given by EX R + = x RdF x = σ Lψk, where ψk = R ϕk k[1 ϕk], and ϕ, ϕ denote the standard normal probability density function p.d.f and c.d.f respectively. Consequently, by considering the safety factor k instead of R, the expression of the integrated total cost 1 can

15 Joint Optimization Model 165 be rewritten as ITC NvbAS Q,k,L,A,S,m = D [ A+ S ] Q m +πσ Lψk+CL + Q [h b C b +h v C v m 1 D 1+ 2D ] 2 P P +h b C b R DL [ A0 +α aln A +bln ] S0 S 2 where the subscript N in IT C denotes the normal distribution case. The problem is to find the order quantity Q, lead time L, ordering cost A, setup cost S, safety factor k and number of deliveries m that minimize the integrated total cost ITC. 6. SOLUTION PROCEDURE To solve this non linear programming problem, we temporarily ignore the restriction 0 < A A 0, 0 < S S 0 and try to solve the optimal solution of ITC NvbAS Q,k,L,A,S,m. First for fixed m, we take the first order partial derivative of ITC NvbAS Q,k,L,A,S,m with respect to Q, k, A, S and L L i,l i 1 and obtain ITC NvbAS Q,k, L,A,S,m Q = D A+ S Q 2 m +πσ Lψk+CL h b C b +h v C v m 1 D P 1+ 2D P, 3 ITC NvbAS Q,k, L,A,S,m A ITC NvbAS Q,k, L,A,S,m S = D Q αa A, 4 = D Qm αb S, 5 ITC NvbAS Q,k, L,A,S,m k ITC NvbAS Q,k, L,A,S,m L = D Q πσ L[ϕk 1]+h b C b σ L, 6 = D Q πσψk 2 L c i h bc b kσ 2 L. 7

16 166 M. Vijayashree and R. Uthayakumar Further, for fixed Q,k, L,A,S,m is concave function in L L i,l i 1, because ITC NvbAS Q,k, L,A,S,m L 2 = D 1 4Q πσψkl h bc b kσl 3 2 < 0. 8 Hence, for fixed Q,k, A,S,m the minimum integrated total cost per unit time will occur at the end points of the interval L L i,l i 1. On the other hand, by setting equations 3-6. Equal to zero, we obtain Q = 2D A+ S m +πσ Lψk+CL, h b C b +h v C v m 1 D P 1+ 2D P 9 A = Qαa D, 10 S = Qmαb D, 11 ϕk =1 h bc b Q πd. 12 For fixed m and L L i,l i 1, by solving equations 9-12, we can obtain the values Q, k,a,s denote these values by Q, k,a,s, respectively. The following propositions assert that, for fixed m and L L i,l i 1, when the constraint 0 < A A 0, 0 < S S 0 is ignored, the point Q, k,a,s is the optimal solution such that the integrated total cost per unit time has a minimum value. Proposition 1. For fixed m and L L i,l i 1, the Hessian matrix for ITC NvbAS Q,k,L,A,S,m is the positive definite at point Q, k,a,s. Proof. See Appendix. Next in order to examine the effect of m on the integrated total cost per unittime,wetakethefirstandsecondorderpartialderivativesofitc NvbAS Q, k,l,a,s,m with respect to m and obtain ITC NvbAS Q,k,L,A,S,m m 2 ITC NvbAS Q,k,L,A,S,m 2 m = DS Qm 2 + Qh [ vc v 1 D ], 13 2 P = 2DS > Qm3

17 Joint Optimization Model 167 Therefore ITC NvbAS Q,k,L,A,S,m is convex in m, for fixed Q, k,a,s and L L i 1, L i. As a result, the search for the optimal deliveries m is reduced to find a local minimum. Now we consider the constraints 0 < A A 0 and 0 < S S 0. From equations 10 and 11 we note that A and S are positive, as a,b,α,d is positive. Moreover, if Si < S 0 and A i < A 0, then Q i, A i,k i,s i is an interior optimal solution for a given m and L L i 1, L i.however, if A i A 0 andsi < S 0 then the optimal order processing cost is the original order processing cost, i.e., A i = A 0. On the other hand, if A i < A 0 andsi S 0 then the optimal setup cost is the original setup cost, i.e., Si = S 0. Finally, if Si S 0 anda i A 0, then it is unrealistic to invest in changing the current setup and ordering cost. For special case take A i = A 0 and Si = S 0 the corresponding Q, k, can be obtained by solving equations 9 and 12. Because the evaluation of each equation equations. 9 and 12 requires the knowledge of the value of others; we can prove the convergence of the procedure adopting a graphical technique similar to that used in Hadley et al. [11]. Therefore, we develop the following iterative algorithm to find the optimal values for delivery lot size, safety stock, setup cost, order processing cost, and number of deliveries in one production cycle. Further, based on the convexity and concavity behavior of the objective function with respect to the decision variable, the following algorithm is designed to find the optimal values of order quantity Q, L ordering cost A, Setup cost S, lead time L, safety factor k and total number of deliveries m which minimizes the integrated total cost ITC NvbAS Q,k,L,A,S,m. Therefore we establish the following iterative algorithm to obtain the optimal solution. 7. ALGORITHM Step 1. Let m = 1. Step 2. For each L i, i = 1, 2, 3...n, perform Start with A i1 = A 0, S i1 = S 0 k i1 = 0implies ψk i1 = , which willbeobtainedbycheckingthestandardnormaltableφk i1 = and ϕk i1 = 0.5.

18 168 M. Vijayashree and R. Uthayakumar 2. Substitute ψk i1, A i1 and S i1 into Eq. 9 and evaluate Q i1. 3. Utilizing Q i1 determine the value of k i2, A i2, S i2, from equations 12, 10 and Repeat steps until no change occurs in the values of Q i, k i, S i, A i. Denote the solution by Q i, k i, A i S i. Step 3. Compare S i with S 0 and A i with A 0 respectively. 1. If S i < S 0 and A i < A 0, then the solution is optimal for the given L i. We denote the optimal solution by Q 0 i,k0 i,s0 i, A0 i, Q 0 i,k 0 i,s0 i, A0 i = Q i, k i, A i,s i, then go to step If S i < S 0 and A i A 0 then for this given L i, let S 0 i = S 0 and utilize equations 9 replace S by S 0 and utilize equations 12 and 10 to determine the new Q i, k i, A i by a procedure similar to the one in step 2, the result is denoted by ˆQ i,ˆk i,  i. If  i < A 0, then the optimal solution is obtained, i.e., if Q 0 i,k0 i,s0 i, A0 i = ˆQi,ˆk i, A 0, Ŝi then go to step 4, otherwise go to step If S i S 0 and A i < A 0 then for this given L i, Let A 0 i = A 0 and utilize equations 9 replace A by A 0 and utilize equations 12 and 11 to determine the new Q i,k i, S i by a procedure similar to the one in step 2, the result is denoted by Q i, k i, S i. If S i < S 0, then the optimal solution is obtained, i.e., if Q 0 i,k0 i,s0 i, i A0 = Qi, k i, A i, S 0, then go to step 4, otherwise go to step If S i S 0 and A i A 0 and go to step 4. Step 4. For given L i, let Si 0 = S 0 and A 0 i = A 0 utilize Eq. 2 replace S by S 0 and A by A 0 to determine the corresponding optimal solution Q 0 i,k0 i by a procedure similar to the one step 2. Step 5. Utilize equation 2 to calculate the corresponding ITC NvbAS Q 0 i, k 0 i,s0 i, A0 i, m. Step 6. Find min i=0,1,...,n ITC NvbAS Q 0 i,k 0 i,s0 i, A0 i, m. Step 7. If

19 Joint Optimization Model 169 ITC NvbAS Q, L,k,A,S,m = min i=0,1,...,n ITC NvbAS Q 0 i,k 0 i,s 0 i, A 0 i, m, then ITC NvbAS Q m,l m,k m,a m,s m,m is the optimal solution for a fixed m. Step 8. Set m = m+1 and repeat steps 1 and 6 to get Step 9. If ITC NvbAS Q m,l m,k m,a m,s m,m. ITC NvbAS Q m,l m,k m,a m,s m,m ITC NvbAS Q m+1,l m+1,k m+1,a m+1,s m+1,m+1, then go to step 8, otherwise go to step 9. Step 10. Set Q,k, L,A,S,m = Q m+1,l m+1,k m+1,a m+1,s m+1,m+1, then Q,k, L,A,S,m, is the minimum integrated total cost of the logarithmic investment function case, and Q, k, L,A,S,m is the optimal solution. 8. NUMERICAL EXAMPLES The values of the parameters in appropriate units are considered as follows. In order to illustrate the above solution procedure, let us consider an inventory system with the data used in Ouyang et al. [28]: D = 600 units/year, A 0 = $200/ order, c b = $100/units π = $50/units, σ = 7/units/week and the lead time has three components with data shown in table 1 as well as the summarized lead time components information is given in table 1. Besides, for integrated the vendor and the buyer cooperative inventory system, we take P = 2000 units/year,s 0 = $1500/setup, α = 0.2,c v = $70/units,h b = 0.2, h v = 0.2,a = 1800, b = We assume that the lead time demand follows a normal distribution and the capital investment in reducing ordering cost IA and setup cost IS can

20 170 M. Vijayashree and R. Uthayakumar Lead time component i Normal duration b i days Minimum duration a i days Unit crashing cost c i days Table 1: Lead time components with data Lead time in week CL Table 2: Summarized lead time data L m = 1 m = 2 m = 3 m = 4 Q A S k ITC Q A S k ITC Q A S k ITC Q A S k ITC Table 3: Summarized lead time data be described by a logarithmic function. Applying the solution procedure of the proposed algorithm, the computational results are demonstrated in table 2. The optimal solutions from table 2 can be read off as order quantity Q = 73units, number of deliveries m = 2, ordering cost A = 44, setup cost S = 170 safety factor k = 1.66 and the corresponding integrated total cost ITC = A graphical representation is presented to show the convexity of ITCQ, k,a, S L, m in figure 1 and the graphical representation of the integrated total cost for different number of deliveries m is shown in figure 2.

21 Joint Optimization Model 171 Figure 2: Graph representing the convexity of ITC Figure 3: Graphical representation of the optimal solution in ITC

22 172 M. Vijayashree and R. Uthayakumar 9. SENSITIVITY ANALYSIS We now study the effects of changes in the system parameters Demand, Vendor s setup cost and Buyer s ordering cost on the optimal order quantity Q, lead time L, setup cost S, ordering cost A, safety factor k and the total number of deliveries m in order to minimize the integrated total cost ITC of the given example. 10. EFFECTS OF DEMAND ON THE OPTIMAL SOLUTION In order to study how various demand D affect the optimal solution of the proposed model, the demand sensitivity analysis is performed by changing the parameter D by + 50%, + 25%, -25% and -50% and keeping the remaining parameters unchanged. The results of the demand on optimal solution analysis are shown in table3 and the corresponding curves of the minimum integrated total cost are plotted in figure 3 as well. D m L weeks A S k Q ITC +50% % % % Table 4: Effects of demand on optimal solution 11. EFFECTS OF VENDOR S SETUP COST ON THE OPTIMAL SOLUTION In order to study how various vendor s setup cost S affect the optimal solution of the proposed model, the vendor s setup cost sensitivity analysis is performed by changing the parameter S by + 50%, + 25%, -25% and -50% and keeping the remaining parameters unchanged. The results of the vendor s setup cost

23 Joint Optimization Model 173 Figure 4: Curve representing minimum IT C for various demand on optimal solution analysis are shown in table 4 and the corresponding curves of the minimum integrated total cost are plotted in figure. 4 as well. S m L weeks A S k Q ITC +50% % % % Table 5: Effects of vendor s setup cost on optimal solution 12. EFFECTS OF BUYER S ORDERING COST ON OPTIMAL SOLUTION In order to study how various buyer s ordering cost A affect the optimal solution of the proposed model, the buyer s ordering cost sensitivity analysis

24 174 M. Vijayashree and R. Uthayakumar Figure 5: Curve representing minimum IT C for various vendors setup cost is performed by changing the parameter of A by + 50%, + 25%, -25% and -50% and keeping the remaining parameters unchanged. The results of the buyer s ordering cost on optimal solution analysis are shown in table 5 and the corresponding curves of the minimum integrated total cost are plotted in figure. 5 as well. A m L weeks A S k Q ITC +50% % % % Table 6: Effects of buyer s ordering cost on optimal solution

25 Joint Optimization Model 175 Figure 6: Curve representing minimum IT C for various buyer s ordering cost 13. MANAGERIAL IMPLICATIONS In this section, we present some managerial insights of the proposed model based on the numerical results and sensitivity analyses. 1. Table 3 shows that when demand D decreases, the total number of deliveries m and the integrated total cost ITC also decrease. This may occur in real life business because, in practice, once the demand of the buyer decreases, ordering quantity of the buyer also decreases, ordering quantity of the buyers also decrease, so that the number of deliveries m, lead time L and the integrated total cost automatically decreases. 2. Table 4 shows that when the vendor s setup cost S decreases, the integrated total cost IT C also decrease, without affecting the total number of deliveries m and lead time L. 3. Table 5 demonstrates that when the buyer s ordering cost A decreases, the integrated total cost IT C also decreases, without affecting the total number of deliveries m and lead time L.

26 176 M. Vijayashree and R. Uthayakumar 4. The proposed model can be used in industries like textiles, footwear, automobiles computer and printer. 14. CONCLUSION Traditional integrated production inventory models have not considered the reduction of delivery lot size, setup and order processing cost simultaneously. The setup and order processing cost can be reduced by adding certain capital investments, and this will affect the lot size decisions. The logarithmic relationship of a setup and order processing cost to investment discussed is not only an interesting special case but, also a practical one. For this practical point of view, this paper have developed joint optimization models with stochastic demand and controllable lead time by reducing setup and order processing cost. In this paper, we considered the joint optimization inventory model with stochastic demand and controllable lead time by reducing ordering and setup cost. In our proposed model, the capital investment in setup and order processing cost reduction is assumed to be a logarithmic function. Specifically, an efficient algorithm is developed to find the optimal solutions of order quantity, number of deliveries per order, setup and order processing cost. Then, an algorithm procedure is developed in order to find that the integrated total cost is minimized, and our approach is illustrated through a numerical example. From the numerical results we can say that, if the setup and ordering cost could be reduced efficiently, the integrated total cost could be automatically minimized. A solution procedure is suggested for solving the proposed model and numerical examples are used to illustrate the benefit of integration. In short, this study intends to fill a remarkable gap left by the recent integrated inventory model. There are several extensions of this work that could constitute future research related to this field. One immediate probable extension could be to discuss the lead time demand is distribution free model. We can consider multi-echelon supply chains such as; single buyer multiple-vendor, multiple-buyer single vendor and multiple -buyer multiple-vendor systems.

27 Joint Optimization Model 177 ACKNOWLEDGMENTS The first author research work is supported by DST INSPIRE Fellowship, Ministry of Science and Technology, Government of India under the grant no. DST/INSPIRE Fellowship/2011/413A dated and UGC-SAP, Department of Mathematics, The Gandhigram Rural Institute - Deemed University, Gandhigram , Tamilnadu, India. REFERENCES [1] A. Banerjee, A joint economic-lot-size model for purchaser and vendor. Decision Sciences, , [2] Ben-M. Daya and A. Raouf, Inventory models involving lead time as a decision variable. Journal of the Operational Research Society , [3] H.C. L. ChangY. K. OuyangS. Wu and C.H. Ho, Integrated vendor-buyer cooperative inventory models with controllable lead time and ordering cost reduction. European Journal of Operational Research , [4] C.H. Glock, Lead time reduction strategies in a single vendor-single buyer integrated inventory model with lot size-dependent lead time and stochastic demand. International Journal of Production Economics, , [5] C.H. Glock, The joint economic lot size problem: A review. International Journal of Production Economics, , [6] S.K. Goyal, An integrated inventory model for a single supplier-single customer problem. International Journal of Production Research, , [7] S.K. Goyal, A joint economic-lot-size model for purchaser and vendor: A comment. Decision Sciences, ,

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31 Joint Optimization Model 181 [41] T. Zhang, L. Liang, Y. Yu and Yan, Yu, An integrated vendor-managed inventory model for a two-echelon system with order cost reduction. International Journal of Production Economics, , APPENDIX We want to prove the Hessian Matrix of ITC NvbAS Q,k,L,A,S,m at the point Q, k,a,s for a fixed m and L L i, L i 1 is positive definite. where and We first obtain the Hessian Matrix H as follows m 1 = m 3 = 2 ITC NvbAS Q,k,L,A,S,m Q 2 2 ITC NvbAS Q,k,L,A,S,m Q k 2 ITC NvbAS Q,k,L,A,S,m Q A 2 ITC NvbAS Q,k,L,A,S,m Q S 2 ITC NvbAS Q,k,L,A,S,m A Q 2 ITC NvbAS Q,k,L,A,S,m A k 2 ITC NvbAS Q,k,L,A,S,m A 2 2 ITC NvbAS Q,k,L,A,S,m A S ITC NvbAS Q,k, L,A,S,m Q 2 ITC NvbAS Q,k, L,A,S,m k 2 ITC NvbAS Q,k, L,A,S,m A 2 ITC NvbAS Q,k, L,A,S,m S 2 2 ITC NvbAS Q,k, L,A,S,m Q k H = T T m 1 m 2 m 3 m 4,m 2 =,m 2 = 2 ITC NvbAS Q,k,L,A,S,m k Q 2 ITC NvbAS Q,k,L,A,S,m k 2 2 ITC NvbAS Q,k,L,A,S,m k A 2 ITC NvbAS Q,k,L,A,S,m k S 2 ITC NvbAS Q,k,L,A,S,m S Q 2 ITC NvbAS Q,k,L,A,S,m S k 2 ITC NvbAS Q,k,L,A,S,m S A 2 ITC NvbAS Q,k,L,A,S,m S 2 = 2D Q 3 A+ S m +πσ Lψk+CL = Dπσ Lϕk, Q = αa A 2, = αb S 2, = 2 ITC NvbAS Q,k, L,A,S,m k Q T T,,,

32 182 M. Vijayashree and R. Uthayakumar = Dπσ Lϕk 1 Q 2, 2 ITC NvbAS Q,k, L,A,S,m Q A = 2 ITC NvbAS Q,k, L,A,S,m A Q 2 ITC NvbAS Q,k, L,A,S,m = 2 ITC NvbAS Q,k, L,A,S,m Q S S Q 2 ITC NvbAS Q,k, L,A,S,m = 2 ITC NvbAS Q,k, L,A,S,m k A A k 2 ITC NvbAS Q,k, L,A,S,m = 2 ITC NvbAS Q,k, L,A,S,m k S S k 2 ITC NvbAS Q,k, L,A,S,m = 2 ITC NvbAS Q,k, L,A,S,m A S S A 2 ITC NvbAS Q,k, L,A,S,m = 2 ITC NvbAS Q,k, L,A,S,m A S S A H 11 = 2 ITC NvbAS Q,k,L,A,S,m Q 2. H 11 > 0, provided. 2D Q 3 A+ S m +πσ Lψk+CL > 0, = D Q 2, = D Q 2 m, = 0, = 0, = 0, = 0, 2 ITC NvbAS Q,k,L,A,S,m Q H 22 = 2 Q k 2 ITC NvbAS Q,k,L,A,S,m k Q k 2 = 2D Q 3 A+ S m +πσ Lψk+CL. Dπσ Lϕk 1 2 ITC NvbAS Q,k,L,A,S,m 2 ITC NvbAS Q,k,L,A,S,m Q 2 2 Dπσ Lϕk Q 2D 2 πσ Lϕk A+ S m +πσ Lψk+CL Dπσ 2 Lφk 1 = Q 4 Q 4. H 22 > 0, provided 2D 2 πσ Lϕk A+ S m +πσ Lψk+CL Dπσ 2 Lφk 1 Q 4 > Q 4. Hence for a fixed m and L L i, L i 1, the Hessian Matrix is positive definite and ITC NvbAS Q,k,L,A,S,m is a convex in Q, k,a