A production-inventory model with permissible delay incorporating learning effect in random planning horizon using genetic algorithm

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1 J Ind Eng Int (205) DOI 0.007/s y OIGINAL ESEACH A production-inventory model with permissible delay incorporating learning effect in random planning horizon using genetic algorithm Mohuya B. Kar Shanar Bera 2 Debasis Das 2 Samarjit Kar 2 eceived 2 August 204 / Accepted 6 December 204 / Published online 20 October 205 The Author(s) 205. This article is published with open access at Springerlin.com Abstract This paper presents a production-inventory model for deteriorating items with stoc-dependent demand under inflation in a random planning horizon. The supplier offers the retailer fully permissible delay in payment. It is assumed that the time horizon of the business period is random in nature and follows exponential distribution with a nown mean. Here learning effect is also introduced for the production cost and setup cost. The model is formulated as profit maximization problem with respect to the retailer and solved with the help of genetic algorithm (GA) and PSO. Moreover, the convergence of two methods GA and PSO is studied against generation numbers and it is seen that GA converges rapidly than PSO. The optimum results from methods are compared both numerically and graphically. It is observed that the performance of GA is marginally better than PSO. We have provided some numerical examples and some sensitivity analyses to illustrate the model. Keywords Inventory Learning effect Inflation andom planning horizon Permissible delay in payment & Samarjit Kar ar_s_@yahoo.com Mohuya B. Kar mohuyaar@yahoo.com Shanar Bera berashanar@gmail.com Debasis Das debasis_opt@yahoo.co.in 2 Department of Computer Science, Heritage Institute of Technology, Kolata, WB 70007, India Department of Mathematics, National Institute of Technology, Durgapur, WB 73209, India Stoc-dependent demand Deteriorating items Genetic algorithm Introduction In the present competitive maret, the supplier influences the customers in many different ways to capture the maret. For this reason, they offer a delay period in payment without any extra charges. The wholesaler allows a time period after which retailers are to mae the payment without paying any interest. But after the delay period, if it is not payed, an interest charged for the rest of the period. Goyal (985) first established a single-item inventory model under permissible delay in payment. Later many researchers wored in this area. Chang et al. (2003) and Chung and Liao (2004) dealt with the problem of determining the EOQ for exponentially deteriorating items under permissible delay in payment. Das et al. (200a, b) presented an EPQ model for deteriorating items under permissible delay in payment. Teng et al. (20) developed an EOQ model for stoc-dependent demand under supplier s trade credit offer with a progressive payment scheme. Min et al. (202) developed an EPQ model with inventory-level-dependent demand and permissible delay in payment. ecently, Ouyang and Chang (203) proposed an optimal production lot with imperfect production process under permission delay in payment and complete baclogging Now a days, there is a stiff competition amongst the multi-nationals to influence the customers and to capture the maret. Thus, in the recent competitive maret, the inventory stoc is decoratively exhibited and displayed through electronic media to attract the customers and thus to boost the sale. For this reason, a group of researchers 23

2 556 J Ind Eng Int (205) have considered inventory control systems with stoc-dependent demand in their research such as Datta and Pal (992), Mandal and Phaujdar (989a, b), Giri et al. (996), Hou (2006), oy et al. (2009a), and others. They considered linear form of stoc-dependent demand. Production cost of a manufacturing system depends upon the combination of different production factors. These factors are (a) raw materials, (b) technical nowledge, (c) production procedure, (d) firm size, (e) quantity of product, and so on. Normally, the cost of raw materials is imprecise in nature. So far, cost of technical nowledge, that is labor cost, has been usually assumed to be constant. However, as the employees perform the same tas repeatedly, they learn how to provide repeatedly a standard level of performance efficiently. Therefore, the processing cost per unit product decreases in every cycle. Similarly, a part of the ordering cost may also decrease in every cycle. In the inventory control literature, this phenomenon is nown as the learning effect. Although different types of learning effects in various areas have been studied (cf. Chiu and Chen 2005, Kuo and Yang 2006, Alamri and Balhi 2007, etc.), it has rarely been studied in the context of inventory control problems. Classical inventory models are usually developed over the infinite planning horizon. According to Gurnani (985) and Chung and Kim (989), the assumption of infinite planning horizon is not realistic due to several reasons such as variation of inventory costs, change in product specifications and designs, technological changes, etc. Moreover, for seasonal products lie fruits, vegetables, warm garments, etc., business period is not infinite, rather fluctuates with each season. Hence, the planning horizon for seasonal products varies over years and may be considered as random with a distribution. Moon and Yun (993) developed an EOQ model with the random planning horizon. Also Das et al. (200a, b) presented an EPQ model with inflation random product life cycle. Till now, none has developed EPQ models considering the delay in payment under random planing horizon. Table represents the summary of related literature for inventory models with random planning horizon. Taing the above shortcomings into account, this paper presents some EPQ models for a deteriorating item considering delay in payment and linearly stoc-dependent demand with a random planing horizon, that is, the life time of the product is assumed as random in nature and follows an exponential distribution with a nown mean. Learning effect in setup cost and production cost is introduced, i.e., setup cost and production cost reduce successively with each cycle. Interest earned and interest paid vary with the permitted fixed period of permissible delay offered by the wholesaler. The model is formulated as a profit maximization problem and solved using Genetic Algorithm. The model is illustrated with numerical examples and some sensitivity analyses are performed. In addition to the above considerations, the rest of the article is organized as follows. Section Notations and assumptions presents assumptions and notations that are used throughout this article. Section Mathematical formulation formulates a mathematical model in order to maximize the total profit of the retailer. Two different solution procedures (GA and PSO) are discussed in Sect. Solution procedure. In Sect. Numerical example and sensitivity analysis, several numerical examples are given to illustrate the solution procedure and sensitivity analyses are performed. The paper is concluded in Sect. Conclusion and future scope. Finally the managerial insights are presented in Sect. Managerial insights. Notations and assumptions The mathematical model of the proposed production-inventory problem is developed based on the following assumptions and notations Table Summary of related literature for inventory models with random planning horizon Author(s) and year Single/two warehouse Demand rate Inflation Learning effect Delay in payment Method Moon and Yun (993) SW Constant No No No Classical method Maiti et al. (2006) TW Stoc dependent No No No Classical method oy et al. (2007) TW Stoc dependent No No No GA oy et al. (2009a) SW Stoc dependent Yes Yes No FGA oy et al. (2009b) SW Stoc dependent Yes No No GA Das et al. (202) TW Stoc dependent Yes No No GA Su et al. (204) SW Constant No No No Classical method Present paper SW Stoc dependent Yes Yes Yes GA,PSO 23

3 J Ind Eng Int (205) Fig. a Pictorial representation for the inventory model for case (a). b Pictorial representation for the inventory model for case (b). c Pictorial representation for the inventory model for case (c) Assumptions. Demand rate is stoc dependent. 2. Time horizon is random. 3. Time horizon accommodates first N cycles and ends during (N? ) cycles. 4. Setup time is negligible. 5. Production rate is nown and constant. 6. Shortages are not allowed. 7. A constant fraction of on-hand inventory gets deteriorated per unit time. 8. Lead time is zero. 9. Production cost and setup cost decrease due to the learning effect. Notations The following notations have been used throughout this paper N ¼ Number of fully accommodated cycles to be made during the prescribed time horizon. q(t) ¼ On-hand inventory of a cycle at time t,ðj ÞT t jt (j ¼,2,...,N). t ¼ Production period in each cycle. P ¼ Production rate in each cycle. D ¼ Demand rate in each cycle ¼ a bqðtþ. C ¼ Holding cost per unit item per unit time. C j 3 ¼ C 3 C 0 3 ed j is setup cost in j-th (j ¼ ; 2;...; N) cycle, d [ 0(d is the learning coefficient associated with setup cost). 23

4 558 J Ind Eng Int (205) p 0 e c j ¼ Production cost in j-th ðj ¼ ; 2;...; NÞ cycle, p 0 ; c [ 0 (c is the learning coefficient associated with production cost). m 0 p 0 e c j ¼ Selling price in j-th (j ¼ ; 2;...; N) cycle, p 0 ; c [ 0, m 0 [. I c ¼ Interest charged per dollar per unit time. I e ¼ Interest earned per dollar per unit time. M ¼ The period of cash discount for which the supplier does not charge any interest. T ¼ Duration of a complete cycle. i ¼ Inflation rate. r ¼ Discount rate. ¼ r i, the net discount rate of inflation is constant. P(N, T) ¼ Total profit after completing N fully accommodated cycles. H ¼ Total time horizon (a random variable) and h is the real time horizon. m p 0 e cðnþ ¼ educed selling price for the inventory items in the last cycle at the end of time horizon, p 0 ; c [ 0, m \. h ¼ Deterioration rate of the produced item. E fpðn; TÞg ¼ Expected total profit from N complete cycles in -th case, ¼ ; 2. E ftp L ðtþg ¼ Expected total profit from last cycle in -th case, ¼ ; 2. E ftpðtþg ¼ Expected total profit from the planning horizon in -th case, ¼ ; 2. Mathematical formulation In this section, we formulate a production-inventory model for deteriorating items under inflation over a random planning horizon incorporating learning effect using permissible delay period. Here it is assumed that there are N full cycles during the real-time horizon h and the planning horizon ends within ðn Þth cycle, i.e., between the time t ¼ and t ¼ðN ÞT. At the beginning of every jth ðj ¼ ; 2;...; N Þ cycle, production starts at t ¼ðj ÞT and continues up to t ¼ðj ÞT t, and inventory gradually increases after meeting the demand due to production. Production thus stops at t ¼ðj ÞT t and the inventory falls to zero level at the end of the cycle time t ¼ jt, due to deterioration and consumption. The retailer pays the payment at time period M without any interest. At the end of this period, he/she starts paying for the interest charged on the items in stocs. Also during the time, the account is not settled, and generated sales revenue is deposited in an interest bearing account. This cycle repeats again and again. For the last cycle, some amounts may be left after the end of planning horizon. This amount is sold at a reduced price in a lot. egarding the interest payed and interest earned based on the length of the jthð j NÞ cycle time T, two different cases may arise Case ðj ÞT M ðj ÞT t jt Case 2 ðj ÞT t ðj ÞT M jt Again for last cycle, according to the values of M, T, and t, we have three different subcases for each case, which are pictorially depicted in Figs.a, b, c, 2a, b, c, respectively. We discuss the detailed formulations in each subcases. Here, it is assumed that the planning horizon H is a random variable and follows exponential distribution with probability density function (p.d.f) as f ðhþ ¼ eh ; h 0 ðþ 0; otherwise Formulation for N full cycles The differential equations describing the inventory level q(t) in the interval ðj ÞT t jtð j NÞ, j ¼ ; 2;...; N are dqðtþ dt ¼ P ðabqðtþþ hqðtþ; ðj ÞT t ðj ÞT t ða bqðtþþ hqðtþ; ðj ÞT t t jt; ; ð2þ where P [ 0, a [ 0, b [ 0, h [ 0, and 0 \t \T, subject to the conditions that qðtþ ¼0att ¼ðjÞT, and qðtþ ¼0 at t ¼ jt. The solutions of the differential equations in (2) are 8 P a >< qðtþ ¼ h b ½ eðhbþfðjþttg ; ðj ÞT t ðj ÞT t a > ðh bþ ½eðhbÞðjTtÞ ; ðj ÞT t t jt Now at t ¼ðjÞT t, from (3) we get, P a e ðhbþt a ¼ e ðhbþðttþ h b ðh bþ ) t ¼ h b ln a P ðeðhbþt Þ Expected total profit from N full cycles ð3þ ð4þ From the symmetry of every full cycle, present value of the expected total profit from N full cycles, E fpðn; TÞg in thð ¼ ; 2Þ case is E fpðn; TÞg ¼ ESN EI EN EPCN EHCN ETOCN EI PN ð5þ where ESN, EI EN, EPCN, EHCN, ETOCN, and EI PN are present values of expected total sales revenue, expected total interest earned, expected total production cost, 23

5 J Ind Eng Int (205) Fig. 2 a Pictorial representation for the inventory model for case 2(a). b Pictorial representation for the inventory model for case 2(b). c Pictorial representation for the inventory model for case 2(c) expected holding cost, expected total ordering cost, and expected total interest paid, respectively, from N full cycles and their expressions are derived in Appendix [see equations (22), (25), (3), (6), (3), (9), (28), (34), respectively]. Formulation for last cycle Duration of the last cycle is [, h], where h is the realtime horizon corresponding to the random time horizon H. Here two different cases may arise depending upon the last cycle length. Case-I \h t and Case-II t \h ðn ÞT. The differential equations describing the inventory level q(t) in the interval \t h are dqðtþ ¼ P ðabqðtþþ hqðtþ t t dt ða bqðtþþ hqðtþ; t t ðn ÞT subject to the conditions that, qðþ ¼0 and qfðn ÞTg ¼0 ð6þ The solutions of the differential equations in (6) are given by 8 P a >< e ðhbþðtþ ; t t h b qðtþ ¼ a > e ðhbþfðnþttg ; t t ðn ÞT h b ð7þ 23

6 560 J Ind Eng Int (205) Expected total profit from last cycle Present value of expected total profit from last cycle, E ftp L ðtþg, inthð ¼ ; 2Þ case is E ftp L ðtþg ¼ ES L ESP L EI E L EHC L EPC L EOC L EI P L ; ð8þ where ES L ; ESP L ; EI E L ; EHC L ; EPC L ; EOC L ; EI P L are present values of expected sales revenue, expected reduced selling price, expected interest earned, expected holding cost, expected production cost, expected ordering cost, and expected interest paid, respectively, from the last cycle and their expressions are derived in Appendix 2 [see equations (45), (49), (60), (68), (4), (44), (48), (6), (69), respectively]. Expected total profit from the system Now, expected total profit from the complete time horizon, E ftpðtþg, inthð ¼ ; 2Þ case is E ftpðtþg ¼ E fpðn; TÞg E ftp L ðtþg ð9þ Problem formulation When the resultant effects of inflation and discounting () are crisp in nature, then our problem is to determine T from Max E ðtpþ; subject to T 0 ð0þ ¼ ; 2 Solution procedure Genetic algoritm (GA) The discovery of genetic algorithms (GA) by Holland (975) is further described by Goldberg (998). GA is a randomized global search technique that solves problems imitating processes observed from natural evolution. GA continually exploits new and better solutions without any pre-assumptions such as continuity and unimodality. GA has been successfully adopted in many complex optimization problems and shows its merits over traditional optimization methods, especially when the system under study has multiple local optimal solutions. A GA normally starts with a set of potential solutions (called initial population) of the decision maing problem under consideration. Individual solutions are called chromosomes. Crossover and mutation operations happen among the potential solutions to get a new set of solutions and it continues until terminating conditions are encountered. Michalewicz (992) proposed a GA named contractive mapping genetic algorithm (CMGA) and proved the asymptotic convergence of the algorithm by Banach s fixed-point theorem. In CMGA, movement from old population to new taes place only when average fitness of new population is better than the old one. The algorithm is presented below. In the algorithm, p c ; p m are probability of crossover and probability of mutation, respectively, T is the generation counter, and P(T) is the population of potential solutions for generation T. M is iteration counter in each generation to improve P(T) and M 0 is the upper limit of M. Initializing (P()) function generates the initial population P() (initial guess of solution set). Objective function value due to each solution is taen as fitness of the solution. Evaluating (P(T)) function evaluates fitness of each member of P(T). GA algorithm. Set generation counter T ¼, iteration counter in each generation M ¼ Initialize probability of crossover p c, probability of mutation p m, upper limit of iteration counter M 0, population size N. 3. Initialize (P(T)). 4. Evaluate (P(T)). 5. While (M\ M 0 ). 6. Select N solutions from P(T) for mating pool using roulette-wheel selection process Michalewicz (992). Let this set be P 0 ðtþ. 7. Select solutions from P 0 ðtþ, for crossover depending on p c. 8. Mae crossover on selected solutions. 9. Select solutions from P 0 ðtþ, for mutation depending on p m. 0. Mae mutation on selected solutions for mutation to get population P ðtþ.. Evaluate (P ðtþ). 2. Set M ¼ M. 3. If average fitness of P ðtþ [ average fitness of P(T), then 4. Set PðT Þ ¼P ðtþ. 5. Set T ¼ T. 6. Set M ¼ End if 8. End while 9. Output Best solution of P(T). 20. End algorithm. The above model is solved by using GA approach, discussed in article-2. Our GA consists of parameters, population size = 50, probability of crossover = 0.6, probability of mutation = 0.2, and maximum generation = 50. A real 23

7 J Ind Eng Int (205) number presentation is used here. In this representation, each chromosome X is a string of n numbers of GA, which denote the decision variable. For each chromosome X, every gene, which represents the independent variables, is randomly generated between their boundaries until it is feasible. In this GA, arithmetic crossover and random mutation are applied to generate new offsprings. Perticle swarm optimization (PSO) PSO is a population-based stochastic optimization technique developed by Eberhart and Kennedy in 995, inspired by social behavior of bird flocing or fish schooling. A swarm of m particles moving about in an n- dimensional real-valued search space, the ith particle is a n- dimensional vector, denoted as X i ¼ðx i ; x i2 ; x i3 ;...; x in Þ; i ¼ ; 2; 3;...; n. The ith particle s velocity is also a n- dimensional vector, denoted as V i ¼ðv i ; v i2 ; v i3 ;...; v in Þ; i ¼ ; 2; 3;...; n. Denote the best position of the ith particle as P besti ¼ðp i ; p i2 ; p i3 ;...; p in Þ; i ¼ ; 2; 3;...; n, and the best position of the colony P gbest ¼ðp g ; p g2 ; p g3 ;...; p gn Þ; i ¼ ; 2; 3;...; n. Each particle of the population modified its position and velocity according to the following mathematical expression V t i ¼ w Vi t c rand ðp besti Xi t Þc 2 rand ðp gbest Xi t Þ X t i ¼ X t i Vt i where t is the current generation number, w is the inertia weight factor, c and c 2 are learning factors, determining the influence of P besti and P gbest, rand and and are random numbers uniformly distributed in the range (0, ), Vi t and Xt i are the current velocity and position of the particle, respectively, P besti is the best solution this particle has reached, and P gbest is the current global best solution of all the particles. The first part of the first formula is the inertia velocity of particle, which reflects the memory behavior of particle; the second part (the distance between the current position and the best position of the ith particle) is cognition part, which represents the private thining of the particle itself; the third part (the distance between the current position of the ith particle and the best position of the colony) is the social part, showing the particle s behavior stem from the experience of other particles in the population. The particles find the optimal solution by cooperation and competition. Numerical example and sensitivity analysis Let us consider a numerical example with the following numerical data Table 2 The sensitivity analysis of the demand parameter when h ¼ 0, and ¼ 00 a b Case- Case-2 C 3 ¼ 50; C3 0 ¼ 40; C ¼ 075; P ¼ 95; m 0 ¼ 8; ¼ 00; r ¼ 0; i ¼ 005; ¼ 005; c ¼ 005; p 0 ¼ 4; I e ¼ 05; I c ¼ 02; m ¼ 08; d ¼ 05 in appropriate units. According to the proposed computational procedures (GA and PSO), the results listed in Table are obtained for different values of M ¼ 40; 42; 44; 45; 46; 48 of case and 2. From the above numerical illustration, it is observed that for fixed a, b, and h as M increases, total profit also increases in both cases. esults are as per expectation. Sensitivity analysis Total profit t Total profit t Table 3 The sensitivity analysis of the deterioration parameter when a ¼ 55, and ¼ 00 b h Case- Case-2 Total profit t Total profit t Sensitivity analyses are performed using GA to study the effect of changes in different values of a, b, h,, d,, and c which are executed through the Tables 2, 3, 4, 5, 6, 7. Itis observed that if h and are fixed for different values of a as b increases, total profit increases. If a and are fixed for different values of b as h increases, total profit decreases. And for fixed values of a, b, and h as increases, total 23

8 562 J Ind Eng Int (205) Table 4 The sensitivity analysis of when a ¼ 55, b ¼ 02 and h ¼ 0 Case- Case-2 Total profit t Total profit t Table 7 The sensitivity analysis of c when a ¼ 55, b ¼ 02 and h ¼ 0 c Case- Case-2 Total profit t Total profit t Table 5 The sensitivity analysis of the learning coefficient d associated with setup cost a d Case- Case-2 Total profit t Total profit t Table 6 The sensitivity analysis of when a ¼ 55, b ¼ 02 and h ¼ 0 Case- Case-2 Total profit t Total profit t profit decreases. It is also observed that for different values of a as d increases, total profit increases, and for fixed values of a, b, and h as and c increase, total profit decreases. All these observations agree with the reality. Comparison of results using GA and PSO From Table 8, it is observed that in all cases, GA gives the better results than PSO. For comparison, we consider ten different generations in 50 runs of both the algorithms and Table 8 Optimal solutions of illustrated examples for case and 2 using GA and PSO M a b h Case- Case-2 perform a t-test to study the convergence. The result of t- test is shown in Table 9. In view of that the performance of GA is acceptable. Also it is clear that there is no significant difference in the mean with the two optimization algorithms. In addition, PSO can also provide a more stable and reliable solution, because it yields significantly smaller standard deviation. Conclusion and future scope Total profit t Total profit t GA PSO GA PSO GA PSO GA PSO GA PSO GA PSO In this paper, a realistic production-inventory model for deteriorating items has been considered under inflation and permissible delay in payments with stoc-dependent demand, over a random planning horizon. Also learning effect on production and setup costs is incorporated in an economic production quantity model. The model is formulated as a nonlinear programing problem and solved numerically by both GA and PSO and the compaired. Sensitivity analyses are also performed for different 23

9 J Ind Eng Int (205) parameters to study the effect of the decision variable. Here, for the first time, trade credit is allowed in an inventory model with random planning horizon which is obtained in the case fashionable goods. Finally, for future research, one can incorporate more realistic assumptions in the proposed model considering shortages, variable deterioration rate, stochastic nature of demand, and production rate. The similar problems can be formulated with multi-items with budget and space constraints. Managerial insights From the Table 8, it is observed that profit increases with trade credit. Therefore, a retailer will try to avoid the maximum trade credit from the supplier. In Table 2, higher Table 9 Optimal solutions of the Model for different generations of case- and case-2 using GA and PSO M Case-I Case-II T Total Profit T Total Profit GEN GA PSO GEN GA PSO GEN GA PSO GEN GA PSO GEN GA PSO GEN GA PSO GEN GA PSO GEN GA PSO GEN GA PSO GEN GA PSO Mean of GA is and SD of GA is Mean of PSO is and SD of PSO is and t - cal = , t - tab = at 8 d.f. with % level of significance So we failed to reject H0 at % level of significance a and b furnish more profits. Specially, as b increases, profit increases. Hence, a retailer will adopt more display of goods for more sales and profits. From Tables 5 and 7, the profit increases with d and c. Thus, manager should always employ experience worers to get the benefit of their experience. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http//crea tivecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a lin to the Creative Commons license, and indicate if changes were made. Appendix [Calculation for expected sales revenue for N full cycles] present value of holding cost of the inventory for the jthð j NÞ cycle, ðhc j Þ, is Z ðjþtt Z jt HC j ¼ C qðtþe t dt C qðtþe t dt ðjþt ðjþtt C ðp aþ ¼ e t C ðp aþ ðh bþ ðh bþðh b Þ e ðhbþt C a ðh bþðh b Þ e T e ðhbþtðhbþt C a e T e t e ðjþt ðþ ðh bþ Total holding cost from N full cycles, (HCN), is HCN ¼ XN j¼ HC j C ðp aþ ¼ e t C ðp aþ ðh bþ ðh bþðh b Þ e ðhbþt C a ðh bþðh b Þ e T e ðhbþtðhbþt C a e T e t e NT ðh bþ e T ð2þ So, the present value of expected holding cost from N complete cycles, (EHCN), is 23

10 564 J Ind Eng Int (205) EHCN ¼ X Z ðnþt HCN f ðhþ dh C ðp aþ ¼ e t C ðp aþ ðh bþ ðh bþðh b Þ e ðhbþt C a ðh bþðh b Þ e T e ðhbþtðhbþt C a e T e t e T ðh bþ e ðþt ð3þ Present value of production cost for the jthð j NÞ cycle, ðpc j Þ, is PC j ¼ p 0 e cj P ¼ p 0e cj P Z ðjþtt ðjþt e t e t dt e ðjþt ð4þ Present value of total production cost from N full cycles, (PCN), is PCN ¼ XN j¼ PC j ¼ p 0 PeT ð e t e Þe ðctþ NðcTÞ e ðctþ ð5þ Present value of expected total production cost from N full cycles, (EPCN), is EPCN ¼ X Z ðnþt PCN f ðhþ dh ¼ p 0 PeT ð e t Þe ðctþ e T ð e ðcttþ Þ ð6þ Present value of ordering cost for the jthð j NÞ cycle, C j 3, is C j 3 ¼fC 3 C3 0 edj ge ðjþt ; C 3 ; C3 0 ; d [ 0 ð7þ Present value of total ordering cost from N full cycles, (TOCN), is TOCN ¼ XN C j 3 j¼ ¼ C 3 e NT e T C 0 3 ed e NðdTÞ e ðdtþ ð8þ Present value of expected total ordering cost from N full cycles, (ETOCN), is ETOCN ¼ X Z ðnþt TOCN f ðhþ dh C 3 e T ¼ ð e ðþt Þ e T C0 3 ed ð e ðdttþ Þ ð9þ Present value of selling price for the jthð j NÞ cycle, ðs j Þ, is Z jt S j ¼ m 0 p 0 e cj fa bqðtþge t dt ðjþt ah Pb ¼ m 0 p 0 e cj e t ðh bþ bðp aþ e ðhbþt ðh bþðh b Þ ab e ðhbþðtt Þt e T ðh bþðh b Þ ah e t e T e ðjþt ðh bþ ð20þ Present value of total selling price from N full cycles, (SN), is SN ¼ XN S j j¼ ah Pb ¼ m 0 p 0 e cj e t ðh bþ bðp aþ e ðhbþt ðh bþðh b Þ ab e ðhbþðtt Þt e T ðh bþðh b Þ ah e t e e e T c NðcTÞ ðh bþ e ðctþ ð2þ Present value of expected total selling price from N full cycles, (ESN), is ESN ¼ X Z ðnþt SN f ðhþ dh ah Pb ¼ m 0 p 0 ðh bþ ð et Þ bðp aþ ðh bþðh b Þ ð eðhbþt Þ ab ðh bþðh b Þ ðeðhbþðttþt e T Þ ha ðh bþ ðet e T Þ e ðctþ e ðcttþ ð22þ 23

11 J Ind Eng Int (205) Calculation for interest earned and interest paid in jth cycle The retailer pays the payment at M without any interest, and at the end of this period, He/She starts paying for the interest charges on the items in stocs. Also during the time the account is not settled, generated sales revenue is deposited in an interest bearing account. egarding the interest pay and interest earned based on the length of the jth ð j NÞ cycle time T, two different cases may arise Case- ðj ÞT M ðj ÞT t jt Present value of Interest earned in jth cycle, I E j is given by Z ðjþtm I E j ¼ I e m 0 p 0 e cj a bqðtþ ðjþt ðj ÞT M t e t dt ah Pb ¼ I e m 0 p 0 e cj ah Pb M ðh bþ ðh bþ 2 ðem Þ ð23þ MbðP aþ ðh bþðh b Þ bðp aþ ðh bþðh b Þ 2 feðhbþm g e ðjþt Total interest earned during N full cycles, I EN is I EN ¼ XN j¼ I E j ah Pb ah Pb ¼ I e m 0 p 0 M ðh bþ ðh bþ 2 ðem Þ MbðP aþ ðh bþðh b Þ bðp aþ ðh bþðh b Þ 2 fe ðhbþm eðctþn g e ðctþ ec ð24þ Present expected value of total interest earned during N full cycles, EI EN; is EI EN ¼ X Z ðnþt I ENf ðhþ dh ah Pb ah Pb ¼ I e m 0 p 0 M ðh bþ ðh bþ 2 ðem Þ MbðP aþ ðh bþðh b Þ bðp aþ ðh bþðh b Þ 2 fe ðhbþm e ðctþ g e ðcttþ ð25þ Present value of interest paid to the wholeseller for the jth ð j NÞ cycle, ði P j Þ; is Z jt I P j ¼ I c p 0 e cj qðtþe t dt ðjþtm P a ¼ I c p 0 e cjðjþt e M e t h b e ðhbþm e ðhbþt h b a e ðhbþtðhbþt e T h b h b e t e T ð26þ Total interest paid during N full cycles, I PN; is given by I PN ¼ XN j¼ I P j P a ¼ I c p 0 e M e t h b e ðhbþm e ðhbþt h b a e ðhbþtðhbþt e T h b h b e e t e T ðctþn e ðctþ ec ð27þ Present expected value of total interest paid during N full cycles, EI PN; is EI PN ¼ X Z ðnþt I PNf ðhþ dh P a ¼ I c p 0 e M e t h b e ðhbþm e ðhbþt h b a e ðhbþtðhbþt e T h b h b e t e T e ðctþ e ðcttþ Þ ð28þ Case-2 ðj ÞT t \ðj ÞT M jt Present value of Interest earned in jth cycle, I 2 E j ; is 23

12 566 J Ind Eng Int (205) I 2 E j ¼ I e m 0 p 0 e cj Z ðjþtm ðjþt a bqðtþ fðj ÞT M t e t dt ¼ I e m 0 p 0 e cjðjþt ah Pb ðh bþ M Pb ðh bþ ðt MÞe t ah Pb ðh bþ 2 ðem Þ ðh bþ 2 Þ ðet bðp aþ ðh bþðh b Þ M ba ðh bþðh b Þ ðt MÞe ðhbþt ð e ðhbþt Þ Pb ðh bþðh b Þ ðt MÞe ðhbþt bðp aþ ðh bþðh b Þ 2 Þ ðeðhbþt ba ðh bþðh b Þ 2 eðhbþt ðe ðhbþm e ðhbþt Þ ð29þ Total interest earned during N full cycles, I 2 EN; is I 2 EN ¼ XN j¼ I 2 E j ah Pb ¼ I e m 0 p 0 ðh bþ M Pb ðh bþ ðt MÞe t ah Pb ðh bþ 2 ðem Þ ðh bþ 2 ðet Þ bðp aþ ðh bþðh b Þ M ba ðh bþðh b Þ ðt MÞe ðhbþt ð e ðhbþt Þ Pb ðh bþðh b Þ ðt MÞe ðhbþt bðp aþ ðh bþðh b Þ 2 ðeðhbþt Þ ba ðh bþðh b Þ 2 eðhbþt ðe ðhbþm e ðhbþt Þ eðctþn e ðctþ ec ð30þ Present expected value of total interest earned during N full cycles, EI 2 EN; is EI 2 EN ¼ X Z ðnþt I 2 ENf ðhþ dh ah Pb ¼ I e m 0 p 0 ðh bþ M Pb ðh bþ ðt MÞe t ah Pb ðh bþ 2 ðem Þ ðh bþ 2 ðet Þ bðp aþ ðh bþðh b Þ M ba ðh bþðh b Þ ðt MÞe ðhbþt ð e ðhbþt Þ Pb ðh bþðh b Þ ðt MÞe ðhbþt bðp aþ ðh bþðh b Þ 2 ðeðhbþt Þ ba ðh bþðh b Þ 2 eðhbþt ðe ðhbþm e ðhbþt Þ e ðctþ e ðcttþ ð3þ Present value of interest paid to the wholeseller for the jthð j NÞ cycle, ði 2 P j Þ; is Z jt I 2 P j ¼ I c p 0 e cj qðtþe t dt ðjþtm ¼ I c p 0 e cjðjþt a h b h b e ðhbþtðhbþm e T e M e T ð32þ Present expected value of total interest paid during N full cycles, EI 2 PN; is I 2 PN ¼ XN j¼ I 2 P j a ¼ I c p 0 h b h b e ðhbþtðhbþm e T e M e T eðctþn e ðctþ ec ð33þ Present expected value of total interest paid during N full cycles, EI 2 PN, is 23

13 J Ind Eng Int (205) EI 2 PN ¼ X Z ðnþt I 2 PNf ðhþ dh a ¼ I c p 0 e ðhbþtðhbþm e T h b h b e M e T e ðctþ e ðcttþ Þ ð34þ Appendix 2 [Calculation for expected sales revenue for last cycle] part- I ( \h t ) present value of holding cost of the inventory for the last cycle is HC L ¼ C qðtþe t dt ¼ C ðp aþ e e h h b eðhbþ h b e ðhbþ e ðhbþh Present value of production cost is PC L ¼ p 0 e cðnþ P ¼ p 0e cðnþ P e t dt e e h ð35þ ð36þ Present value of ordering cost ¼fC 3 C3 0 edðnþ g e NT Present value of sales revenue is S L ¼ m 0 p 0 e cðnþ fa bqðtþge t dt ah Pb ¼ m 0 p 0 e e cðnþ e h ðh bþ bðp aþeðhbþ e ðhbþ e ðhbþh ðh bþðh b Þ ð37þ Part-II ( t \h ðn ÞT) Present value of holding cost of the inventory for the last cycle is HC L2 ¼ C qðtþe t dt ¼ C ðp aþ h b eðhbþ e e ðtþ e ðhbþ e ðhbþðtþ ðh b Þ c a e ðhbþðnþt e ðhbþðtþ e ðhbþh h b ðh b Þ e ðtþ e h ð38þ Present value of production cost is PC L2 ¼ p 0 e cðnþ P ¼ p 0e cðnþ P Z t e t dt e e ðt Þ ð39þ Present value of ordering cost= fc 3?C 0 3 edðnþ g.e Present value of sales revenue is S L2 ¼ m 0 p 0 e cðnþ fa bqðtþge t dt ah Pb ¼ m 0 p 0 e cðnþ e ðh bþ ð et Þ bðp aþ e ðhbþt ðh bþðh b Þ ah e ðtþ e h ðh bþ baeðhbþðnþt ðh bþðh b Þ e ðhbþðtþ e ðhbþh ð40þ Present value of expected holding cost for the last cycle is EHC L ¼ X where ¼ X Z ðnþt Z t ¼ EHC L EHC L2 HC L f ðhþdh HC L f ðhþdh X Z ðnþt t ; HC L2 f ðhþdh ð4þ 23

14 568 J Ind Eng Int (205) EHC L ¼ X and Z t HC L f ðhþdh ¼ c ðp aþ h b h b ðh b Þ ð Þ et e ðþt ð Þ ðh b Þðh b Þ e ðhbþt e ðþt ð42þ Z ðnþt EHC L2 ¼ X HC L2 f ðhþdh t cðpaþ ¼ hb ðet e T Þ ðet Þ e ðhbþt hb c a e ðhbþtðhbþt et ðe t e T Þ hb hb ðhbþðhbþ e ðhbþtðhbþt e ðþt e ðþt e ðþt ðþ e ðþt ð43þ Present value of expected production cost for the last cycle is EPC L ¼ X ¼ X Z ðnþt Z t PC L f ðhþdh Z ðnþt PC L f ðhþdh X PC L2 f ðhþdh t ¼ p 0e c P ð e t Þ ð e ðttcþ Þ ð Þ e ðþt p 0e c P ð e ðttcþ Þ ð e t Þðe t e T Þ ð e ðttcþ Þ ð44þ Present value of expected sales revenue from the last cycle is ES L ¼ X ¼ X Z ðnþt Z t Z X ðnþt t ¼ ES L ES L2 S L f ðhþdh S L f ðhþdh S L2 f ðhþdh ; ð45þ where ES L ¼ X and ES L2 ¼ X Z t S L f ðhþdh ah Pb ¼ m 0 p 0 ð e t Þ ðh bþ bðp aþ e ðþt ðh bþðh b Þ fð e t Þ e ðhbþt h b e c e ðcttþ ð46þ Z ðnþt t S L2 f ðhþdh ah ¼ m 0 p 0 ðh bþ Pb ðh bþ ð et Þ bðp aþ e ðhbþt ðe t e T Þ ðh bþðh b Þ ah e ðþt e ðþt ðh bþð Þ bae ðhbþt e ðhbþt ðe t e T Þ ðh bþðh b Þ e ðhbþt e ðhbþt h b e c e ðcttþ ð47þ Present value of expected ordering cost for the last cycle is Z EOC L ¼ X ðnþt C 3 C 03 edðnþ e NT f ðhþdh ð e T Þ ¼ C 3 ð e ðþt Þ ð e T Þ C0 3 ed ð e ðdttþ Þ ð48þ Present value of expected reduced selling price from the last cycle is X Z ðnþt ESP L ¼ m p 0 e cðnþ e h qðhþ f ðhþdh ¼ m p 0 e cx m p 0 e cx Z t e cn e h qðhþ f ðhþdh ¼ ESP L ESP L2 Z ðnþt e cn e h qðhþ f ðhþdh t ð49þ 23

15 J Ind Eng Int (205) where ESP L ¼ m p 0 e cx and P a ¼ m p 0 h b Z t e cn e h qðhþ f ðhþdh e ðþt e ðhbþt h b e c ð50þ eðcttþ ESP L2 ¼ m p 0 e cx Z ðnþt e cn e h qðhþ f ðhþdh t a ¼ m p 0 h b h b e ðhbþtðhbþt e ðþt e ðþt e ðþt e c e ðcttþ ð5þ Calculations for interest earned and interest paid in last cycle Case ð M t ðn ÞTÞ Subcase (a) ð t MÞ Present value of interest earned in last cycle, IEL ; is IEL ¼ I e m 0 p 0 e cðnþ a bqðtþ ðh tþe t dt ah Pb ¼ I e m 0 p 0 e cðnþ ðh Þe ðh bþ ah Pb e h ðh bþ 2 e bðp aþ ðh Þe ðh bþðh b Þ bðp aþ e ðhbþðhbþh ðh bþðh b Þ 2 e ð52þ Present expected value of interest earned in the last cycle, EIEL ; is EIEL ¼ X Z ðnþt IEL f ðhþdh ah Pb ¼ I e m 0 p 0 Te T et ðh bþ ah Pb ðh bþ 2 e ðþt ðe T Þ bðp aþ Te T et ðh bþðh b Þ bðp aþ ðh bþðh b Þ 2 e ðhbþt h b ðe T e c Þ e ðcttþ ð53þ Subcase (b) ð M t t Þ Present value of interest earned in last cycle, IEL 2 ; is Z M IEL 2 ¼ I e m 0 p 0 e cðnþ a bqðtþ ð M tþe t dt ¼ I e m 0 p 0 e ðctþn e c ah Pb ah Pb M ðh bþ ðh bþ 2 e M bðp aþ ðh bþðh b Þ M bðp aþ ðh bþðh b Þ 2 e ðhbþm ð54þ Present expected value of interest earned in the last cycle, EIEL 2 ; is EIEL 2 ¼ X Z ðnþt IEL 2 f ðhþdh ah Pb ah Pb ¼ I e m 0 p 0 M ðh bþ ðh bþ 2 e M bðp aþ ðh bþðh b Þ M bðp aþ ðh bþðh b Þ 2 e ðhbþm ð et Þe c e ðcttþ ð55þ Present value of interest paid in the last cycle, IPL 2 ; is 23

16 570 J Ind Eng Int (205) IPL 2 ¼ I c p 0 e cðnþ qðtþe t dt M ¼ I c p 0 e cðnþ P a e ðmþ e h h b e ðhbþðhbþh e ðhbþm h b ð56þ Present expected value of interest paid in the last cycle, EIPL 2 ; is EIPL 2 ¼ X Z ðnþt IPL 2 f ðhþdh ¼ I c p 0 P a e M e T h b e ðþt e ðhbþm e T h b e ðhbþt h b e c e ðcttþ ð57þ Subcase (c) ð t t ðn ÞTÞ The expression of the present expected value of interest earned in the last cycle, EIEL 3 ; is same as Eq. (55). Present value of interest paid in the last cycle, IPL 3 ; is IPL 3 ¼ I c p 0 e cðnþ qðtþe t dt M Z t ¼ I c p 0 e cðnþ qðtþe t dt M qðtþe t dt ¼ I c p 0 e cðnþ t P a e h b ðem e t Þ e ðhbþm e ðhbþt h b a e ðhbþt h b h b e ðhbþt e ðhbþhðhbþ e t e h ð58þ Present expected value of interest paid in the last cycle, EIPL 3 ; is 23

17 J Ind Eng Int (205) EIPL 3 ¼ X Z ðnþt IPL 3 f ðhþdh P a ¼ I c p 0 h b ðem e t Þ h b e ðhbþm e ðhbþt ð e T Þ a e ðhbþtðhbþt ð e T Þ h b h b h b ðeðhbþt e ðþt Þ e t ð e T Þ ð eðþt Þ e c e ðcttþ ð59þ EI E L ¼ EIEL EIEL 2 EIEL 3 ð60þ Total expected interest paid for the last cycle,ei P L ; is EI P L ¼ EIPL 2 EIPL 3 ð6þ Case 2 ð t M ðn ÞTÞ Subcase (2a) ð t t Þ The expression of the present expected value of interest earned in the last cycle, EIEL 2 ; is same as Eq. (53). Subcase (2b) ð t t MÞ Present value of interest earned in last cycle, IEL 22 ; is Total expected interest earned for the last cycle,ei E L ; is IEL 22 ¼ I e m 0 p 0 e cðnþ a bqðtþ ðh tþe t dt Z t ¼ I e m 0 p 0 e cðnþ a bqðtþ ðh tþe t dt t ah Pb ¼ I e m 0 p 0 e cðnþ ðh bþ e t e t h bðp aþ t e ðhbþt h ðh bþðh b Þ e ah h t e ðtþ ðh bþ eh baeðhbþðnþt h t ðh bþðh b Þ h b a bqðtþ ðh tþe t dt ðe t Þ h b e ðhbþt e ðhbþðtþ eh h b ð62þ 23

18 572 J Ind Eng Int (205) Present expected value of interest earned in the last cycle, EIEL 22 ; is EIEL 22 ¼ X Z ðnþt IEL 22 f ðhþdh ah Pb ¼ I e m 0 p 0 ð e T Þt e t ðe t Þ Te T ðe T Þð ðh bþ Þ bðp aþ ð e T Þt e ðhbþt ðh bþðh b Þ e ðhbþt Te T ðe T Þð h b Þ ah e t Te T ðe T Þðt ðh bþ Þ e ðþt ð Þ ba e ðhbþtðhbþt Te T ðe T Þðt ðh bþðh b Þ h b Þ e ðþt e c ðh b Þð Þ e ðcttþ ð63þ Subcase (2c) ð M t ðn ÞTÞ Present value of interest earned in last cycle, IEL 23 ; is Z M IEL 23 ¼ I e m 0 p 0 e cðnþ a bqðtþ ð M tþe t dt Z t ¼ I e m 0 p 0 e cðnþ a bqðtþ ð M tþe t dt Z ðnþt a bqðtþ ð M tþe t dt t ah Pb ¼ I e m 0 p 0 e cðnþ M Pb ðh bþ ðh bþ et ðm t Þ ah ðh bþ 2 ð Pb em Þ ðh bþ 2 ð et Þ bðp aþ ðm t Þe ðhbþt M ðh bþðh b Þ ba ðh bþðh b Þ eðhbþtðhbþt bðp aþ ðm t Þ ðh bþðh b Þ 2 e ðhbþm e ðhbþt ba ðh bþðh b Þ 2 eðhbþt e ðhbþt ð64þ 23

19 J Ind Eng Int (205) Present expected value of interest earned in last cycle, EIEL 23 ; is Total expected interest paid for the last cycle, EI 2 P L ; is EIEL 23 ¼ X Z ðnþt IEL 23 f ðhþdh ah Pb ¼ I e m 0 p 0 M Pb ðh bþ ðh bþ et ðm t Þ ah ðh bþ 2 ð Pb em Þ ðh bþ 2 ð et Þ bðp aþ ðm t Þe ðhbþt M ðh bþðh b Þ ba ðh bþðh b Þ eðhbþtðhbþt bðp aþ ðm t Þ ðh bþðh b Þ 2 e ðhbþm e ðhbþt ba ðh bþðh b Þ 2 eðhbþt e c ð e T Þ e ðcttþ e ðhbþt ð65þ Present value of interest paid in the last cycle, IPL 23 ; is IPL 23 ¼ I c p 0 e cðnþ qðtþe t dt M ¼ I c p 0 e cðnþ a h b h b ðe ðhbnþtðhbþm e ðhbþhðhbþðnþt e h e ðmþ ð66þ Present expected value of interest paid in the last cycle, EIPL 23 ; is EIPL 23 ¼ X Z ðnþt IPL 23 f ðhþdh a ¼ I c p 0 e ðhbþtðhbþm ð e T Þ h b h b e ðþt e ðhbþt h b e ðþt ðe MT e M Þ e c e ðcttþ ð67þ Total expected interest earned for the last cycle, EI 2 E L ;,is EI 2 E L ¼ EIEL 2 EIEL 22 EIEL 23 ð68þ EI 2 P L ¼ EIPL 23 eferences ð69þ Alamri AA, Balhi ZT (2007) The effects of learning and forgetting on the optimal production lot size for deteriorating items with time varying demand and deterioration rates. Int J Prod Econ Chang CT, Ouyang LY, Teng JT (2003) An EOQ model for deteriorating items under supplier credits lined to ordering quantity. Appl Math Model Chiu HN, Chen HM (2005) An optimal algorithm for solving the dynamic lot-sizing model with learning and forgetting in setups and production. Int J Prod Econ Chung KH, Kim YH (989) Economic analysis of inventory systems a rejoinder. Eng Econ Chung KJ, Liao JJ (2004) Lot-sizing decisions under trade credit depending on the ordering quantity. Comput Oper es Das D, oy A, Kar S (200a) Improving production policy for a deteriorating item under permissible delay in payments with stoc-dependent demand rate. Comput Math Appl Das D, oy A, Kar S (200b) A production-inventory model for a deteriorating item incorporating learning effect using genetic algorithm. Adv Oper es. doi0.55/200/46042 Das D, Kar MB, oy A, Kar S (202) Two-warehouse production model for deteriorating inventory items with stoc-dependent demand under inflation over a random planning horizon. Cent Eur J Oper es 20(2) Datta TK, Pal AK (992) A note on a replenishment policy for an inventory model with linear trend in demand and shortages. J Oper es Soc Giri BC, Pal SP, Chaudhuri KS (996) An inventory model for deteriorating items with stoc-dependent demand rate. Eur J Oper es

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