Two-echelon supply chain inventory model with controllable lead time and service level constraint

This paper considers a two-echelon supply chain inventory problem consisting of a single-vendor and a single-buyer. In the system under study, a vendor produces a product in a batch production environment and supplies it to a buyer facing a stochastic demand, which is assumed to be normally distributed. Also, buyer’s lead time is controllable which can be shortened at an added cost and all shortages are backordered. A model has been formulated for an integrated vendor–buyer problem to jointly determine the optimal order quantity, lead time and the number of shipments from the vendor to the buyer during a production cycle while minimizing the total expected cost of the vendor–buyer integrated system. It is often difficult to estimate the shortage cost in inventory systems. Therefore, instead of having a shortage cost term in the objective function, a service level constraint (SLC) is included in the model that requires a certain proportion of demands to be met in each cycle. An efficient procedure has been suggested to find the bounds on number of shipments and then, an algorithm is developed to obtain the optimal solution of the proposed model. A numerical example is included to illustrate the algorithmic procedure and the effects of key parameters are studied to analyze the behavior of the model. Finally, the savings of buyer and vendor are investigated from implementation of joint optimization model over the model in which they minimize their own cost independently.     

Periodic review fuzzy inventory model with variable lead time and fuzzy demand

This paper investigates a periodic review fuzzy inventory model with lead time, reorder point, and cycle length as decision variables. The main goal of this study is to minimize the expected total annual cost by simultaneously optimizing cycle length, reorder point, and lead time for the whole system based on fuzzy demand. Two models are considered in this paper: one with normal demand distribution and another with a distribution‐free approach. The model assumes a logarithmic investment function for lost‐sale rate reduction. Furthermore, two separate efficient computational algorithms are explained to obtain the optimal solution. Some numerical examples are given to illustrate the model.     

Computational procedure of optimal inventory model involving controllable backorder rate and variable lead time with defective units

This article considers that the number of defective units in an arrival order is a binominal random variable. We derive a modified mixture inventory model with backorders and lost sales, in which the order quantity and lead time are decision variables. In our studies, we also assume that the backorder rate is dependent on the length of lead time through the amount of shortages and let the backorder rate be a control variable. In addition, we assume that the lead time demand follows a mixture of normal distributions, and then relax the assumption about the form of the mixture of distribution functions of the lead time demand and apply the minimax distribution free procedure to solve the problem. Furthermore, we develop an algorithm procedure to obtain the optimal ordering strategy for each case. Finally, three numerical examples are also given to illustrate the results.     

An integrated inventory problem with transportation in a divergent supply chain under service level constraint

This study investigates an integrated inventory problem with transportation in a single-vendor and multi-buyer divergent supply chain. The vendor manufactures a product and delivers the product to the buyers located in different locations by a fleet of vehicles of identical capacity. The external demands per unit time on the buyers are independent and normally distributed. The lead time components of the buyers, excluding transportation time, can be reduced at an added crash cost. A model has been formulated to minimize the total expected cost of the system associated with the production, inventory, transportation and lead time reduction to find the optimal production, inventory and routing decisions while satisfying the service level constraint of the buyers. We propose a coordinated two-phase iterative approach to solve the model, which has been illustrated through a numerical example.     

A mixed inventory model with variable lead time and random back-order rate

In this paper, we study the determination of the optimal lead time, reorder point and order quantity considering that the back-order probability of a demand made during a stock-out period depends on the interval from the moment in which the order is placed until the next replenishment. Two models are analysed for the specification of the back-order probability: exponential functions and piecewise constant functions. The distribution of the lead time demand is assumed to be Poisson. An algorithm for the determination of the optimal order quantity, reorder point and lead time is given. A numerical example is presented to illustrate the results.     

Mixture inventory model involving variable lead time and controllable backorder rate

This paper allows the backorder rate as a control variable to widen applications of Ouyang et al.'s model [J. Oper. Res. Soc. 47 (1996) 829]. In this study, we assume that the backorder rate is dependent on the length of lead time through the amount of shortages. We discuss two models that are perfect and partial information about the lead time demand distribution, that is, we first assume that the lead time demand follows a normal distribution, and then remove this assumption by only assuming that the first and second moments of the probability distribution of lead time demand are known. For each case, we develop an algorithm to find the optimal ordering strategy. Three numerical examples are given to illustrate solution procedure.     

Inventory control model for a supply chain system with multiple types of items and minimum order size requirements

This paper considers a replenishment problem for a single buyer who orders multiple types of items from two or more heterogeneous suppliers in order to sell to end customers. The buyer periodically orders each type of item from the suppliers according to a select inventory control policy. Processing the order, each supplier enforces the policy that an order from the buyer must meet a predetermined minimum order quantity (MOQ). Therefore, the buyer must decide how much to order from each supplier considering the current inventory level, demand forecast, and MOQ requirement. The buyer's problem is formulated as an integer programming model and an efficient implementation strategy is suggested to apply the model to real problems. Numerical experiments are performed to test the validity of the proposed model as well as the efficiency of the implementation strategy. The experimental results show that this model combined with the implementation method yields a considerable cost reduction compared to the most efficient policy currently available.     

Supply chain coordination model with controllable lead time and service level constraint

We consider the coordination issue in a decentralized supply chain composed of a vendor and a buyer in this paper. The vendor offers a single product to the buyer who is faced with service level constraint. In addition, lead time can be reduced by added crashing cost. We analyze two supply chain inventory models. The first one is developed under decentralized mode based on Stackelberg model, the other one is developed under centralized mode of the integrated supply chain. The solution procedures are also provided to get the optimal solutions of these two models. Finally, a price discount mechanism is proposed to induce both the vendor and the buyer to accept the centralized model. The feasibility and efficiency of the proposed models are manifested by numerical examples and some managerial implications are highlighted.     

A minimax distribution free procedure for mixed inventory model involving variable lead time with fuzzy demand

In a recent paper, Ouyang and Wu applied the minimax decision approach to solve a continuous review mixed inventory model in which the lead time demand distribution information is unknown but the annual demand is fixed and given. However, in the practical situation, the annual demand probably incurs disturbance due to various uncertainties. In this article, we attempt to modify Ouyang and Wu's model by considering two fuzziness of annual demand (i.e., fuzzy number of annual demand and statistic-fuzzy number of annual demand) and to investigate a computing schema for the continuous review inventory model in the fuzzy sense. We give an algorithm procedure to obtain the optimal ordering strategy for each case.Scope and purposeIn most of the early literature dealing with inventory problems, either using deterministic or probabilistic models, lead time is viewed as a prescribed constant or a stochastic variable. Recently, some researchers (e.g., Liao and Shyu, Ben-Daya and Raouf, and Ouyang and Wu) incorporated the crashing lead time idea to continuous review inventory models, in which the annual demand is given and fixed. However, in the real situation, the annual demand will probably have a little disturbance due to various uncertainties. The purpose of this article is to modify the Ouyang and Wu's model to accommodate this reality, specifically, we apply the fuzzy set concepts to deal with the uncertain annual demand. We first consider a case where the annual demand is treated as the triangular fuzzy number. Then, we employ the statistical method to construct a confidence interval for the annual demand, and through it to establish the corresponding fuzzy number (namely, the statistic-fuzzy number). For each fuzzy case, we investigate a computing schema for the new model and develop an algorithm to find the optimal ordering strategy.     

The distribution free continuous review inventory system with a service level constraint

The stochastic inventory models require the information on the lead time demand. However, the distributional information of the lead time demand is often limited in practice. We relax the assumption that the cumulative distribution function, say F, of the lead time demand is completely known and merely assume that the first two moments of F are known and finite. The distribution free approach for the inventory model consists of finding the most unfavorable distribution for each decision variable and then minimizing over the decision variable. We apply the distribution free approach to the continuous review inventory system with a service level constraint. We develop an iterative procedure to find the optimal order quantity and reorder level.     

On the distribution free continuous-review inventory model with a service level constraint

We consider a continuous-review (Q, r) inventory model with a fill rate service constraint and relax the assumption that the distribution of lead time demand is known. We adopt a distribution free approach: We assume that only the first two moments of the lead time demand distribution are known, and then, optimize the policy parameters against the worst possible distribution. We are able to derive closed-form expressions for the optimal order quantity and reorder point.     

An alternative simple solution algorithm of an inventory model with fixed and variable lead time crash costs under unknown demand distribution

Inventory models with controllable lead time both for known and unknown demand distributions have been proposed in the literature. A model is useless unless it is formulated correctly and feasible. A simple solution procedure of a model also plays an important role in its application. This article highlights an erroneous formulation of an inventory model developed with fixed and variable lead time crash costs under unknown demand distribution, and also demonstrates its infeasibility. To attain feasibility we extend the model to include a constraint. Then, we present an alternative simple solution technique of the modified model and carry out a comparative study on a numerical example to show its potential significance.     

A multiple-vendor single-buyer integrated inventory model with a variable number of vendors

Supply chain management is concerned with the coordination of material and information flows in multi-stage production systems. A closer look at the literature reveals that previous research on the coordination of multi-stage production systems has predominantly focused on the sales side of the supply chain, whereas problems that arise on the supply side have often been neglected. This article closes this gap by studying the coordination of a supplier network in an integrated inventory model. Specifically, we consider a buyer sourcing a product from heterogeneous suppliers and tackle both the supplier selection and lot size decision with the objective to minimise total system costs. First, we provide mathematical formulations for the problem under study, and then suggest a two-stage solution procedure to derive a solution. Numerical studies indicate that our solution procedure reduces the total number of supplier combinations that have to be tested for optimality, and that it may support initiatives which aim on increasing the efficiency of the supply chain as a heuristic planning tool.     

A simulation-based decision support system for a multi-echelon inventory problem with service level constraints

In this paper, we present a simulation-based decision support system for solving the multi-echelon constrained inventory problem. The goal is to determine the optimal setting of stocking levels to minimize the total inventory investment costs while satisfying the expected response time targets for each field depot. We derive new decision support algorithms to be applied in different scenarios, including small-sample and large-sample cases. The first case requires that the set of alternative solutions is known at the beginning of the experiment, and the number of evaluated solutions may depend on the simulation budget (i.e., the time available to solve the problem). In the second case, the alternative solutions are generated sequentially during the searching process, and we may terminate the algorithm when the specified sampling budget is exhausted. Empirical studies are conducted to compare the performance of the proposed algorithms with other conventional optimization approaches.     

A single period inventory model with a truncated normally distributed fuzzy random variable demand

In this article, a single period inventory model has been considered in the mixed fuzzy random environment by assuming the annual customer demand to be a fuzzy random variable. Since assuming demand to be normally distributed implies that some amount of demand information is being automatically taken to be negative, the model has been developed for two cases, using the non-truncated and the truncated normal distributions. The problem has been developed to represent scenarios where the aim of the decision-maker is to determine the optimal order quantity such that the expected profit is greater than or equal to a predetermined target. This ‘greater than or equal to’ inequality has been modelled as a fuzzy inequality and a methodology has been developed to this effect. This methodology has been illustrated through a numerical example.     

Integrated vendor--buyer inventory system with sublot sampling inspection policy and controllable lead time

This article investigates the impact of inspection policy and lead time reduction on an integrated vendor--buyer inventory system. We assume that an arriving order contains some defective items. The buyer adopts a sublot sampled inspection policy to inspect selected items. The number of defective items in the sublot sampling is a random variable. The buyer's lead time is assumed reducible by adding crash cost. Two integrated inventory models with backorders and lost sales are derived. We first assume that the lead time demand follows a normal distribution, and then relax the assumption about the lead time demand distribution function and apply the minimax distribution-free procedure to solve the problem. Consequently, the order quantity, reorder point, lead time and the number of shipments per lot from the vendor to the buyer are decision variables. Iterative procedures are developed to obtain the optimal strategy.     

Cost allocation model for optimizing supply chain inventory with controllable lead time

The coordination issue of a decentralized supply chain composed of a vendor and a buyer is considered in this paper. The vendor offers a single product to the buyer and the lead time can be controllable with adding crashing cost. Two supply chain inventory models with controllable lead time under different decision modes are considered, one is proposed under decentralized model based on Stackelberg model, the other is proposed under centralized model of the integrated supply chain. The solution procedures are also suggested to get the optimal solutions of these two models. In addition, an asymmetric Nash bargaining model based on satisfaction level is also developed to get the best cost allocation ratio between the vendor and the buyer by taking their individual rationalities into consideration. The results of numerical example show that shortening lead time reasonably can reduce inventory cost and the cost allocation model based on satisfaction level developed in this paper is effective.     

Single product,finite horizon,periodic review inventory model with working capital requirements and short-term debt

In this paper, we build on a single product, finite horizon, periodic review inventory management setting and include key financial aspects such as working capital constraints, payment delays and multiple sources of financing. We numerically solve for the optimal working capital target and the order-up-to level using an embedded Nelder and Mead optimization, and we perform sensitivity analysis on cash flows and short-term debt levels. Our numerical experiments show that when access to short-term debt is granted, the expected cash flows are indeed fairly insensitive to varying short-term debt premiums. However, when short-term debt becomes prohibitive or when downstream payment delays increase, the required working capital target inflates rapidly.     

Manufacturing setup cost reduction and quality improvement for the distribution free continuous-review inventory model with a service level constraint

Over several decades, the continuous-review inventory model has been widely studied based on various assumptions and restrictions such as those related with quality improvement, service level constraint, and setup cost reduction. We extend Moon and Choi's [1] model by assuming setup cost reduction and quality improvement. A distribution free approach is employed such that only mean and standard deviation need to be known. The total system cost is minimized with respect to decision variables against the worst possible distribution scenario. The benefit of using quality improvement and setup cost reduction in this model is shown. Numerical examples show that this model offers significant improvements over existing models. Finally, sensitivity analysis of the key parameters is also presented.     

Two-echelon multi-product multi-constraint product returns inventory model with permissible delay in payments and variable lead time

Supply chain is not limited to delivering products to the end-costumers since the defective products that are returned back to the producers by the consumers. The producers should be superior knowledge to utilize the return products effectively so as to maintain our natural resources and to provide better service to customers. In this paper, a distributor and a warehouse consisting of a serviceable part and a recoverable part supply chain problem is considered in which there are several products, the distributor has limited space capacity and budget to purchase all products. In this supply chain, the defective products are returned back to the warehouse by the distributor and the warehouse recovered those defective products into perfect products having the same value as the procured products. The lead-time of receiving products from a warehouse to a distributor is a variable which is controllable by adding extra crashing cost. For each product, a fraction of the shortage is backordered and the rest are lost. A mathematical model is employed in this study for optimizing the order quantity, lead time and total number of deliveries with the objective of minimizing system total cost. We show that the model of this problem is a constrained non-linear programme and present a simple Lagrangian multiplier technique to solve it. Numerical and sensitivity analysis are given to show the applicability of the proposed model in real-world product returns inventory problems.