یک مدل تولید - موجودی فروشنده - خریدار یکپارچه با توزیع نرمال زمان انجام

By relaxing the long-term assumption of the deterministic lead time, recently three coordinated vendor–buyer models with exponential distribution of lead time in a two-stage supply chain were presented. The vendor produces a product at a finite rate and delivers the lot to the buyer with a number of equal-sized batches (sub-lots) to meet the deterministic demand. The next batch is ordered when the previous one drops to a reorder point. Shortages were allowed and completely backordered. However, in exponential distribution of lead time, the probability of arrival of a batch earlier is higher than the probability of arrival of a batch late or in the mean lead time. But usually, probability of arrival of a batch earlier or late appears to be smaller than the probability of arrival of a batch in the mean lead time. Thus normal distribution of lead time seems to be a better fit to the problem. Hence their models seem unfit to the concerned problem in practice. Based on this notion, we develop a vendor–buyer integrated production–inventory model following normal distribution of lead time but retaining their other assumptions. To make the model more realistic, set up time per set up of a machine, the highest limit on the capacity of the transport vehicle and the transportation cost and time per batch are imposed. Then we derive an optimal solution technique to the model to obtain minimum expected joint total cost that follows development of the solution algorithm. Extensive comparative studies on the results of some numerical problems are carried out to highlight the potential significance of the present method. Sensitivity analysis to the solutions with variations of some parameter values are also carried out.

This paper deals with the development of a vendor–buyer integrated production–inventory model with normal distribution of the delivery lead time of a batch, and its optimal solution technique. It has been agreed by the researchers that the integrated inventory system plays an important role for efficient and effective management of inventories across the entire supply chain. Vendor–buyer coordination is essential for successful implementation of an integrated inventory model (e.g., Goyal and Guptha, 1989, Chandra and Fisher, 1994, Thomas and Griffin, 1996, Sharma et al., 2006 and Tarantilis, 2008). Integrated inventory models have been developed including various concerned factors of the system (see for detail Sajadieh et al., 2009) to enrich the literature. One of the important factors is the delivery lead time of a batch. Although most of the models have been developed based on the deterministic lead time, researchers have enriched the literature with the development of models by taking into account controllable lead time with extra cost (e.g., Ben-Daya and Rouf, 1994, Ouyang et al., 1996, Ouyang and Wu, 1997, Ouyang and Wu, 1998, Moon and Choi, 1989, Lan et al., 1999, Pan and Hsiao, 2001, Pan et al., 2002, Pan and Yang, 2002, Chang, 2005, Pan and Hsiao, 2005, Hoque and Goyal, 2006, Chang et al., 2006, Hoque, 2009 and Ye and Xu, 2010). Inventory policies have also been developed with stochastic lead time (e.g., Yano, 1987, Kumar, 1989, Fujiwara and Sedarage, 1997 and Rossi et al., 2010). Glock (2012) developed an integrated inventory model for the single-vendor single-buyer with stochastic demand and variable lead time under different lead time reduction strategies. He adopted the formulation of the lead time from Hsiao (2008). The lead time of the first batch is formulated by considering production time and setup and transportation time while the lead time for 2,3,…,n batch is only the transportation time considered in the model. However, a little attention has been given for developing a vendor–buyer integrated production–inventory model with stochastic lead time. Recently Sajadieh et al. (2009) developed such a model with exponential distribution of lead time and allowing backordering of shortages. They developed three models: two of them considering total cost of the vendor and the buyer individually and the third for the integrated system, for delivering a lot of a product (produced by the vendor) to the buyer in n number of equal-sized batches to meet the deterministic demand. An order is placed by the buyer when the stock level falls to a certain level, called the reorder point. They identified some environmental causes such as order processing and transportation times of a batch, inspection etc. as factors of leading to lead time uncertainty. Being motivated by this idea they developed the models with exponential distribution of lead time and allowing backordering of shortages. The purpose was to show significant reduction in the minimal total cost of ordering, set up, inventory holding and shortages by the integrated system. In the exponential distribution, the probability of a variable less than of its mean value is always higher than the probability of a variable greater than or equal to its mean value. In case of the exponential distribution of lead time, if a batch arrives early, then its lead time is less than the mean lead time and if a batch arrives late, its lead time is greater than the mean lead time. So, the probability of earlier arrival of a batch is always higher than the probability of arrival of a batch late or in the mean lead time in this distribution, application of which in their models seems to be impractical. To meet the demand without allowing planned shortages, the set up time of a machine plus the processing time of the first batch of a next lot plus its loading, unloading, transfer and inspection time (t) must equal to the time of meeting the demand by the units at the reorder point. This constraint has not been taken into consideration in their models, and hence it may lead to planned early arrivals or late deliveries always. To overcome this deficiency we have taken into account this constraint in developing the model in this paper. Besides, sometimes the processing time of the first batch plus t may deviate because of various factors. Sometimes, other batches may arrive earlier or late because of variations in t. Thus, in practice, the probability of arrival earlier or late of a batch seems to be smaller than the probability of arrival of a batch in the mean lead time, and usually the former probabilities appear to be more or less symmetrical to the latter ones. Thus the lead time seems to follow the normal distribution. For this reason, for a known value of t, here we develop a vendor–buyer integrated production–inventory model following the normal distribution of the lead time but retaining Sajadieh et al. (2009) other assumptions. Also, when a batch arrives late to the buyer, it is kept elsewhere for the same amount of time of delay and creates an extra inventory there. If the late delivery is due to late start of production of a product, then its raw material creates an extra inventory. Sajadieh et al. (2009) did not take into account this extra inventory in developing their models. But this is taken into account in developing the present model here. In addition, we have considered the highest capacity of the transport vehicle used to transport the product because it may not be unlimited. Moreover, the set up time per set up of a machine, the transportation cost and time for transporting a batch to the buyer are also imposed. Thus we have developed the model with the assumption of the normal distribution of the lead time (but retaining their other assumptions), and taking into account the mentioned extra inventory and the constraints along with the set up time and the transportation cost and time per batch. Then a number of properties leading to the minimal total cost (of set up, ordering, inventory holding, transportation and shortage) to the model are established, and hence the optimal solution algorithm is obtained. Extensive comparative studies on the results of numerical problems are carried out to highlight the potential significance of the present method. Sensitivity analysis to the solutions found with variations of some parameter values are also carried out. We organize the paper as follows: In the next section we put forward assumptions and notations. Section 3 deals with development of the model with the normal distribution of the lead time, and its minimal total cost solution technique. In Section 4 we carry out extensive comparative studies on the results of numerical problems. Section 5 concludes by highlighting the paper findings, limitations and future research directions.

Assuming exponential distribution of the lead time of delivering a batch from the vendor to the buyer, Sajadieh et al. (2009) developed three vendor–buyer models—two of them considering the total cost of the vendor and the buyer independently, and the third one for the integrated system. The lot is transferred with n number of equal-sized batches and the next batch is ordered by the buyer when the stock level drops to the reorder point. They showed significant total cost reduction by the integrated system. Here we point out that in the exponential distribution of the lead time, the probability of earlier arrival of a batch is always higher than the probability of arrival of a batch late or in the mean lead time. Since the lead time of a batch usually follows approximately the normal distribution, here we develop an integrated vendor–buyer model following this distribution, and present a minimal total cost solution technique to the model. Besides, they did not take into account some extra inventory concerned in their models, and also a constraint to avoid planned early arrival or planned shortages to the buyer in meeting the demand. Although transportation of a batch incurs a cost per batch and the transport vehicle should have highest transport capacity, these have not been considered in their models. In our model we have taken into account the mentioned extra inventory and the constraints along with the transportation cost and time per batch. Also, we have considered the set up time of a machine and the loading, unloading, inspection and transportation time of a batch. For a known batch size Q, the total cost function is found to be convex in n (number of equal-sized batches in a lot), and hence a formula (6) for determining the minimal (real) n is obtained. Substituting for this n in the total cost function, it is expressed in terms of Q only. Although this total cost function in Q is found to be semi-convex theoretically, it has been found convex in solutions of many numerical problems. So, we carry out a simple search over Q to attain its minimal value. For known n, semi-convex nature of the total cost function in Q is also pointed out, and hence the minimal total cost is calculated at an integral Q from (5) for each of the rounding up and rounding down values of the minimal n. Minimum of these total costs along with the associated integer values of Q and n leads to the final solution. Following this solution technique a step by step solution algorithm is presented. A comparative study of solutions of two numerical problems obtained by the method developed in this paper with the corresponding ones found by Sajadieh et al. (2009) is carried out. The current method in this paper is found to reduce the minimal total cost significantly in two cases. Extensive comparative studies on the results of numerical problems are carried out to highlight sensitivities of solutions for changes to various parameter values. For a particular production rate, generally the batch sizes and the number of batches are found almost the same, but the minimal total cost increases (except for P=7000 and 8000 when σ increases from 0.014 to 0.017) almost uniformly as the standard deviation of the lead time (σ) increases (keeping all the remaining parameter values the same). For a particular value of σ, generally the lot sizes and the associated minimal total cost are found to be in increasing trend, while the number of batches are in non-increasing trend as the production rate increases (keeping all the remaining parameter values the same). However, the minimal total cost increases with a decreasing growth rate. For a particular production rate, the increase in the minimal total cost is found mostly due to the increase in the shortage cost. For a particular σ, the general trend of increase in the minimal total cost is found to occur mainly due to the increase in the inventory cost of the buyer. For a particular production rate, generally the lot sizes are found almost the same but the minimal total cost increases as each of π′,hb,hv increases independently keeping all other parameter values the same. As the production rate increases for a particular value of each of π′,hb,hv, although the lot size and the associated minimal total cost are found to increase generally, the latter one decreases when production rate increases from 7000 to 8000. The same numerical problem is solved both for T=0 and T=100. The rate of increase in the minimal total cost up to P=6000 is found to be more for T=0 than the corresponding rate of decrease in the minimal total cost for T=100. Thereafter, the rate of decrease in the minimal total cost is found to be the same in both cases. Although buyer's storage capacity is important in deciding minimal batch sizes, this has not been considered in developing the model in this paper. Imposing limit on the capacity of the buyer's storage an appropriate minimal batch size may be obtained. In our integrated inventory model the lot is transferred with equal sized batches to synchronize the production flow. However, synchronization of the integrated inventory supply chain by transferring the lot with equal and/or unequal sized batches might be more fruitful, and hence this research may be extended in this direction. Moreover, it can be extended to the case of multiple buyers. So, future research might be carried out in those directions.