 American Journal of Oper ations Research, 2011, 1, 46-50 doi:10.4236/ajor.2011.12007 Published Online June 2011 (http://www.SciRP.org/journal/ajor) Copyright © 2011 SciRes. AJOR Optimal Policy and Simple Algorithm for a Deteriorated Multi-Item EOQ Problem Bin Zhang, Xiayang Wang Lingnan College, Sun Yat-Sen University, Guangzhou, China E-mail: bzhang3@mail.ustc.edu.cn, wangxy@mail.sysu.edu.cn Received March 29, 201 1; revised April 19, 2011; accepted May 12, 2011 Abstract This paper considers a deteriorated multi-item economic order quantity (EOQ) problem, which has been stu-died in literature, but the algorithms used in the literature are limited. In this paper, we explore the optimal policy of this inventory problem by analyzing the structural properties of the model, and introduce a simple algorithm for solving the optimal solution to this problem. Numerical results are reported to show the effi-ciency of the proposed method. Keywords: Inventory, EOQ, Deterioration, Multi-Item 1. Introduction Multi-item inventory problem with resource constraints is an important topic of inventory management . These constrained inventory models are still hot topics in aca-demic and practice fields, for example, see [2-6]. The re- source constraints typically arise from shipment capacity, warehouse capacity or budgetary limitation. Since some items such as fruits, vegetables, food items, drugs and fashion goods will deteriorate in the shipment or storage process, many works have been done for investigating inventory problems for deterioration items [7-11]. Since items’ deterioration often takes place during the storage period, some researchers have considered economic order quantity (EOQ) models for deteriorating ite ms, f or ex am-ples see [12,13]. Recently, Mandal et al.  present a constrained multi-item EOQ model with deteriorated items. In , the model is firstly formulated as the transcendental form and the polynomial form, i.e., without and with trunca-tion on the deterioration terms. These two versions of the model are both solved by applying non-linear program-ming (NLP) method (Lagrangian multiplier method). As  points out, the polynomial form is an approximation of the transcendental form. Secondly, the transcendental form is converted to the minimization of a signomial expression with a posynomial constraint, which is so lved by applying a modified geometric programming (MGP) method. However, we argue that the studied problem can be solved using a simple algorithm without any model approximation or conversion. In this paper, we prove that the deteriorated multi-item EOQ model is a special convex separable nonlinear knapsack problem studied in , which is characterized by positive marginal cost (PMC) and increasing marginal loss-cost ratio (IMLCR). PMC requires p ositive marginal cost of decision variable, and IMLCR means that the marginal loss-cost ratio is increasing in decision v ariable. Following , we explore the optimal policy for the problem, and develop a simple algorithm for solving it. The main purpose of this paper is twofold: 1) to explore the optimal policy of this inventory problem by analyz-ing the structural properties of the model; 2) to introduce a simple algorithm for solving the optimal solution to this problem. The reminder of this paper is organized as follows. We formulate the problem in the next section. In Section 3, we explore the structural properties of the problem, and provide the optimal policy and algorithm. Numerical results are reported in Section 4, and Section 5 briefly concludes this paper. 2. Problem Formulation Consider a multi-item EOQ problem with a storage sp ace constraint, in which all items (1, ,in) deteriorate after certain periods. Before presenting the model, we list all notation as follows. Notice that the same notation used in  is presented. B. ZHANG ET AL. Copyright © 2011 SciRes. AJOR 47n = Total number of items Qi = Order quantity c0i = Purchasing cost c1i = Holding cost per unit quantity per unit time c3i = Set-up cost i= Constant rate of deterioration (01i) wi = Required storage quantity per unit time Di = Demand rate per unit time Ti = Time period of each cycle 1(,, )nTC TT= Total average cost function W = Available storage space Following , we set 201 3iiiiiii iacD cDc , 2010iiiiiiibcD cD , and 1iiiiccD. Then the deteriorated multi-item EOQ model can be expressed as follows (denoted as problem P). We refer the reader to  for the details of this model. 111min (,,)() iinTnniiii iiiiiTC TTaefTbcTT, (1) subject to 11() ((1))iinn Tiii iiiiDgTwe W , (2) 0iT, 1, ,in. (3) The order quantity is given by (1)iiTiiiDQe, 1, ,in. The first and second order derivatives of ()iifT,1, ,in, and the first order derivative of ()iigT,1, ,in, are calculated as follows: 2()(1 )iiTiii iiiiidf TabeTdT T, (4) 22223() 2(22)iiTiii iiiiiiidfTa beTTdT T , (5) () iiTii iiidg TDwedT. (6) 3. Structural Properties and Algorithm In this section, we first establish structural properties of problem P, and then we present an efficient procedure for solving the optim al solution to problem P. 3.1. Structural Properties Before presenting the structural properties of problem P, we give two basic equations, which will be used in our proofs. Since Taylor expansion of exponential function is 11!iikkTiikTek, then we have 2211 2iiTiiiii iTTTe , (7) for 0iT, 1, ,in. By comparing the definitions of ai and bi, we have iiab, 1, ,in. (8) Considering the objective function of problem P, we have the following proposition. Proposition 1. The cost 1(,, )nTC TT is strictly convex. Proof. Since the function 1(,, )nTC TT is separable, we only need to prove that ()iifT is strictly convex in iT, 1, ,in. According to Equations (7) and (8), we have. 222222 442(22 )221 1222iiTii iiiiiiii iiiii iiiiabeT TTTbTbb TT  . Substituting the above equation into Equation (5), we have. 244 423 022ii ii iii iiidfT bT bTdT T for 0iT. Thus, ()iifT and 1(,, )nTC TT are strictly convex. QED. The strictly convexity of 1(,, )nTC TT ensures that the optimal solution to problem P is unique. This prop-erty has also been indicated in  by using a more complicated proof procedure. Denote ProductLog( )z as the principal solution for x in xzxe, which has the same function name in Ma-thematica to stand for the Lambert W function. Let ˆiT, 1, ,in, be a value such that () 0iiidfTdT . Denote problem CP as problem P without the constraint in Equa-tion (2), then the following proposition characterizes the optimal solution to problem CP. Proposition 2. The optimal solution to problem CP is 1 ProductLogˆiiiiiabeT, 1, ,in. (9) Proof. Since ()0ii idfTdT  at the point ˆiT, we have ˆˆ(1) 0iiTii iiabe T . This equation can be rewritten as ˆ1ˆ(1)iiTiiiiaeTbe, and hence we have 1 ProductLogˆiiiiiabeT. From Equation (8), we know 1iiiiaebe . According B. ZHANG ET AL. Copyright © 2011 SciRes. AJOR 48 to , we know that ProductLog( )z is an increasing function for 1ize , then we have ProductLog( )iiiabe 1ProductLog( ) 1ie. Substituting this equation into ˆiT, we have hence ˆ0iT, which satisfies the positive constraint in Equation (3). Thus, ˆiT, 1,,in, is an optimal solution to problem CP. Since the optimal solu-tion is unique, we know that the optimal solution to problem CP is ˆiT, 1,,in. QED. Following , we define the marginal loss-cost ratio of item i (1,,in) as. 2()() ()iiii Tii iiiiii ii iiiidf TdTa ebTbrT dg TDwTdT . (10) Then we have the following proposition. Proposition 3. ()iirT is strictly increasing in ˆ(0, ]iiTT, 1, ,in. Proof. From Equation (7), we know 1iiTiiTe for 0.iTThe convexity of ()iifT guarantees iiTiiabe (1)0iiT for ˆ(0, ]iiTT. Using iiab in Equa-tion (8) and the above two equations, we have (2) (2)[(1)][(1)][(1)][(1)]0iiii iiii iiTiiiiiiTTiiiiiiiiTTiiii iiibeT aTabeTaT beabeTbT e  , for ˆ(0, ]iiTT. Hence we have 3()[(2 )(2 )]0ii iiTTiiiii iiiiidr TebeTaTdT DT w. Thus, ()iirT is strictly increasing in ˆ(0, ]iiTT, 1, ,in. QED Since this proposition illustrates that ()iirT is strictly increasing in ˆ(0, ]iiTT, 1, ,in, we let ()iiTr , (,0]ir, be the inverse function of ()iirT . Although it is difficult to write ()iiTr in a closed form, the strict monotony of ()iiTr ensures that ()iiTr can be easily determined by applying a bi-section search procedure over ˆ(0, ]iiTT, for any given (,0]ir . 3.2. Optimal Policy and Algorithm We now demonstrate that the deteriorated multi-item EOQ model is a special convex separable nonlinear knapsack problem studied in . Firstly, Proposition 1 illustrates that P is a convex problem; Secondly, from Equation (6), we know () 0iiidg TdT , for 0iT,1, ,in, which means there are positive marginal costs in problem P; Finally, Proposition 3 ensures that the marginal loss-cost ratio ()iirT is increasing in ˆ(0, ]iiTT, 1,,in. Therefore, the theoretical results and solution procedure proposed in  are both applicable for problem P. Denote by *iT, 1, ,in, the optimal solution to problem P. By directly applying the theoretical results in , we can summarize the optimal policy for the dete-riorated multi-item EOQ problem in the following pro- position. Proposition 4. The optimal policy of problem P is (a) *ˆiiTT, 1, ,in, if 1ˆ()niiigTW; (b) *1()niiigTW and **() ()ii kkrT rT, , 1,,ik n, specify the optimal solution *iT, 1, ,in, if 1ˆ()niiigTW. This proposition is obtained by directly applying the theoretical results in  to problem P, since problem P has PMC and IMLCR. Based on Propositions 2-4, the idea of the algorithm proposed in  can be used for solving problem P. The basic idea of the algorithm is as follows: If the constr ain t in Equ a tion (2) is inactive, i.e. , 1ˆ()niiigTW, then the optimal solution to problem P equals to the optimal solution to the unconstrained prob-lem, i.e., *ˆiiTT, 1, ,in; Otherwise, Proposition 4(b) means that obtaining the exact value of ** *() ()ii kkrrT rT, , 1,,ik n, is the key to solving the optimal solution to problem P. The optimal value *r can be searched by applying a binary search method over the interval (,0]rM, where M is a sufficient large value such that ()1(( 1))iinTMiiiiDwe W. Main steps of the above solution procedure for solving the optimal solution to problem P are summarized in the following algorithm. The Algorithm 1*ˆ1: Solve , 1,,, from Equation (9);ˆ 2: If (), t h e n ˆ let , 1,,, go to 8; 3: Let ,0; 4: Le t ()2; 5: Calculate iniiiiiLULUiiStepT inStepg TWTTin StepSteprMrSteprr rStepT T 11(), 1,,; 6: If (), then let ; If (), then let ;nii Linii Uiri nStepg TWrrgT Wr r B. ZHANG ET AL. Copyright © 2011 SciRes. AJOR 49*** Go to 4; 7: Let , 1,,; 8: Calculate (,,), and output.iiinStepStepTT inStepTC TT In comparison with the two methods in , there are two main advantages of our algorithm: 1) It is a polyno-mial algorithm of O(n) order, which ensures that the al-gorithm is applicable for large-scale problems; 2) It does not need any approximation or conversion of the original model, thus it always solves the optimal solution to prob-lem P. 4. Numerical Results In this section, numerical experiments are provided to show the efficiency of the proposed algorithm for solv-ing problem P. The instances of problem P are all ran-domly generated. We use the notation x ~ (,)U to denote that x is uniformly generated over [,]. The parameters of test instances are generated as follows: iw~ (1,10)U, 0ic~ (1,10)U, 1ic~ (0.5,1.0)U, 3ic~ (40,100)U, i~ (0.01,0.10)U, iD~ (200,500)U, 1, ,in, and 100Wn. In this numerical study, we set n=100 and 1000, re-spectively. For each problem size n, 100 test instances are randomly generated. The statistical results on number of iterations of the binary search and computation time (in milliseconds) are reported in Table 1, where 95% C.I. stands for 95% confidence interval. From Table 1, we can conclude that the proposed al-gorithm can solve large-scale deteriorated multi-item EOQ models very quickly in few iteration times. Since the ranges of parameters are large, the standard devia-tions of number of iterations and computation time are quite low, reflecting the fact that the algorithm is quite effective and robust. We also use our algorithm to solve the illustrative exam-ple studied in , which outputs the optimal solution: *10.2899T, *20.2176T, *1102.6397Q, *298.6801Q, *2587.1382TC  with *= 0.5925r. Unfortunately, this result cannot be directly compared with that solved by , because there is something wrong with the values of *iT and *iQ, 1,2i, shown in Tables 2 and 3 of , since they violate the basic equation Table 1. Performance of our algorithm for solving the ran-domly generated instances. Number of iterations Computation time Problem size n 100 1000 100 1000 Mean 27.9 31.8 217.0 2619.3 Std. Dev. 1.7 1.7 8.9 38.8 95% Lower27.6 31.4 215.2 2611.6 C.I. Upper28.3 32.1 218.8 2627.0 (1)iiTiiiDQe, 1, 2i. For example, in the Table 2 of , when *10.2414712T and *20.2419020T, (1)iiTiiiDQe gives 185.3365Q, 2109.7828Q. In addition, their mistake can also be verified by our proved optimal pol-icy ** **11 22() ()rT rT. For example, the values of **()iirT , 1, 2i for the MGP solution presented in Table 3 of  are **11( )0.6017rT  , and **21( )0.5857rT  , which does not satisfy ** **11 22() ()rT rT. From the above analysis, we illustrate that the solution provided in  does not satisfy the optimal policy proved in this paper, which is easily used to verify the optimality of a solution to problem P. Thus, the numeri-cal results in  are incorrect. Since some comparison of NLP and MGP given by  were established based on the numerical results, especially the results in Tables 2 and 3 of , we argue that the comparison of NLP and MCP in  are questionable. 5. Conclusions In this paper, we explore the structural properties of de-teriorated multi-item EOQ model and propose a simple algorithm for solving the optimal solution by proving that the studied problem is a special convex separable nonlinear knapsack problem. In addition, it is obvious that the basic idea and obtained results in this paper can be simply modified for solving the classical constrained multi-item EOQ problem. 6. Acknowledgements The authors would like to thank the two reviewers for their insightful comments, which helped to improve the manuscript. This work is supported by national Natural Science Foundation of China (No. 70801065), the Fun-damental Research Funds for the Central Universities of China and Natural Science Foundation of Guangdong Province, China (No. 10451027501005059). 7. References  G. Hadley and T. M. Whitin, “Analysis of Inventory Systems,” Prentice-Hall, Englewood Cliffs, 1963.  B. Zhang, X. Xu and Z. Hua, “A Binary Solution Method for the Multi-Product Newsboy Problem with Budget Constraint,” International Journal of Production Eco-nomics, Vol. 117, No. 1, 2009, pp. 136-141. doi:10.1016/j.ijpe.2008.10.003  S. Sharma, “Revisiting the Shelf Life Constrained Multi- Product Manufacturing Problem,” European Journal of Operational Research, Vol. 193, No. 1, 2009, pp. 129-139. doi:10.1016/j.ejor.2007.10.045  B. Zhang and S. F. Du, “Multi-Product Newsboy Prob- B. ZHANG ET AL. 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