A supplier-selection model with classification and joint replenishment of inventory items

Since inventory costs are closely related to suppliers, many models in the literature have selected the suppliers and also allocated orders, simultaneously. Such models usually consider either a single inventory item or multiple inventory items which have independent holding and ordering costs. However, in practice, ordering multiple items from the same supplier leads to a reduction in ordering costs. This paper presents a model in capacity-constrained supplier-selection and order-allocation problem, which considers the joint replenishment of inventory items with a direct grouping approach. In such supplier-selection problems, the following items are considered: a fixed major ordering cost to each supplier, which is independent from the items in the order; a minor ordering cost for each item ordered to each supplier; and the inventory holding and purchasing costs. To solve the developed NP-hard problem, a simulated annealing algorithm was proposed and then compared to a modified genetic algorithm of the literature. The numerical example represented that the number of groups and selected suppliers were reduced when the major ordering cost increased in comparison to other costs. There were also more savings when the number of groups was determined by the model in comparison to predetermined number of groups or no grouping scenarios.     

A goal programming model for joint decision making of inventory lot-size,supplier selection and carrier selection

In this paper, we address a problem in which a storage space constrained buyer procures a single product in multiple periods from multiple suppliers. The production capacity constrained suppliers offer all-unit quantity discounts. The late deliveries and rejections are also incorporated in sourcing. In addition, we consider transportation cost explicitly in decision making which may vary because of freight quantity and distance of shipment between the buyer and a supplier. We propose a multi-objective integer linear programming model for joint decision making of inventory lot-sizing, supplier selection and carrier selection problem. In the multi-objective formulation, net rejected items, net costs and net late delivered items are considered as three objectives that have to be minimized simultaneously over the decision horizon. The intent of the model is to determine the timings, lot-size to be procured, and supplier and carrier to be chosen in each replenishment period. We solve the multi-objective optimization problem using three variants of goal programming (GP) approaches: preemptive GP, non-preemptive GP and weighted max–min fuzzy GP. The solution of these models is compared at different service-level requirements using value path approach.     

Optimal replenishment decision in an EPQ model with defective items under supply chain trade credit policy

The traditional inventory of the economic order quantity model assumes perfect items in an ordered lot and an infinite replenishment rate. However, such conditions are rare in actual production environments. Additionally, most studies of this problem have only considered suppliers offering the wholesaler a grace period. In practice, wholesalers often extend a fixed credit period to downstream customers as well. This study therefore proposes a production model for a lot-size inventory system with finite production rate and defective quality under the condition of two-level trade credit policy and the condition that defective items involve both imperfect quality and scrap items. Thus, optimal wholesaler replenishment decisions can be determined for defective items under two-level trade credit policy in the EPQ framework. Four theorems for determining the optimal cycle time and the results in this study can be deduced as a special case of earlier models. Finally, illustrative examples are provided to verify the theoretical results.     

A supplier selection and order allocation problem with stochastic demands

We consider a system comprising a retailer and a set of candidate suppliers that operates within a finite planning horizon of multiple periods. The retailer replenishes its inventory from the suppliers and satisfies stochastic customer demands. At the beginning of each period, the retailer makes decisions on the replenishment quantity, supplier selection and order allocation among the selected suppliers. An optimisation problem is formulated to minimise the total expected system cost, which includes an outer level stochastic dynamic program for the optimal replenishment quantity and an inner level integer program for supplier selection and order allocation with a given replenishment quantity. For the inner level subproblem, we develop a polynomial algorithm to obtain optimal decisions. For the outer level subproblem, we propose an efficient heuristic for the system with integer-valued inventory, based on the structural properties of the system with real-valued inventory. We investigate the efficiency of the proposed solution approach, as well as the impact of parameters on the optimal replenishment decision with numerical experiments.     

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.     

Optimizing replenishment polices using Genetic Algorithm for single-warehouse multi-retailer system

Coordinating inventory and transportation policies can lead to substantial cost savings and improved service levels especially when the companies relay on third-party logistics providers to transport the products across the supply chain. In this paper, therefore focus has been given on a supply chain system of multi-supplier, single warehouse and multi-retailer with backlogging and transportation capacity. The paper aims to suggest replenishment policies that can minimize system-wide cost by taking advantage of quantity discounts in the transportation cost structures. The problem considered in this paper has been formulated as an integer programming model. The supply chain problem is usually complex and involves massive calculations hence it is difficult to obtain an optimal solution. Therefore, to overcome this issue a Genetic Algorithm (GA) based approach has been suggested to resolve the problem. The computational results demonstrate the robustness and efficacy of the GA in optimizing replenishment policies.     

An integrated replenishment model with quantity discounts, reentry and downward substitution for control wafers

This paper considers control wafers replenishment problem in wafer fabrication factories. A dynamic lot-sizing replenishment problem with reentry and downward substitution is examined in a pulling control production environment. The objective is to set the inventory level so as to minimize the total cost of control wafers, where the costs include order cost, purchase cost, setup cost, production cost and holding cost, while maintaining the same level of production throughput. In addition, purchase quantity discounts and precise inventory level are considered in the replenishment model. The control wafers replenishment problem is first constructed as a network, and is then transformed into a mixed integer programming model. Lastly, an efficient heuristic algorithm is proposed for solving large-scale problems. A numerical example is given to illustrate the practicality for empirical investigation. The results demonstrate that the proposed mixed integer programming model and the heuristic algorithm are effective tools for determining the inventory level of control wafers for multi-grades in multi-periods.     

Supplier selection and procurement decisions with uncertain demand,fixed selection costs and quantity discounts

In this paper, we study the supplier selection and procurement decision problem with uncertain demand, quantity discounts and fixed selection costs. In addition, a holding cost is incurred for the excess inventory if the buyer orders more than the realized demand and the shortage must be satisfied by an emergent purchase at a higher price otherwise. The objective is to select the suppliers and to allocate the ordering quantity among them to minimize the total cost (including selecting, procurement, holding and shortage costs, etc.). The problem is modeled as a Mixed Integer Programming (MIP) and is shown to be NP-hard. Some properties of the optimal policy are provided and an optimal algorithm is proposed based on the generalized Bender's decomposition. Numerical experiments are conducted to show the efficiency of the algorithm and to obtain some managerial insights.     

Supplier selection model with contingency planning for supplier failures

We consider the optimal allocation of demand across a set of suppliers given the risk of supplier failures. We assume items sourced are used in multiple facilities and can be purchased from multiple suppliers with different cost and reliability characteristics. Suppliers have production flexibility that allows them to deliver a contingency quantity in case other suppliers fail. Costs considered include supplier fixed costs and variable costs per unit, while failure to deliver to a demand point results in a particular financial loss. The model utilizes the decision tree approach to consider all the possible states of nature when one or more suppliers fail, as well as expand the traditional transportation problem. Unlike other supplier selection models, this model considers contingency planning in the decision process, minimizing the total network costs. This results in a base allocation to one or more of the available suppliers and a state of nature specific delivery contingency plan from the suppliers to each demand point. A numerical example, as well as sensitivity analysis, is presented to illustrate the model and provide insights.     

A solution procedure for mixed-integer nonlinear programming formulation of supply chain planning with quantity discounts under demand uncertainty

Quantity discount policy is decision-making for trade-off prices between suppliers and manufacturers while production is changeable due to demand fluctuations in a real market. In this paper, quantity discount models which consider selection of contract suppliers, production quantity and inventory simultaneously are addressed. The supply chain planning problem with quantity discounts under demand uncertainty is formulated as a mixed-integer nonlinear programming problem (MINLP) with integral terms. We apply an outer-approximation method to solve MINLP problems. In order to improve the efficiency of the proposed method, the problem is reformulated as a stochastic model replacing the integral terms by using a normalisation technique. We present numerical examples to demonstrate the efficiency of the proposed method.     

A fuzzy model for supplier selection in quantity discount environments

Traditionally, supplier selection should simultaneously take into account numerous heterogeneous criteria, and then is a tedious task for the purchasing decision makers. It becomes especially complicated when quantity discounts are considered at the same time. Under such manner, most studies often formulate such a problem as a Multi-Objective Linear Programming (MOLP) problem, and then scale it down to a Mixed Integer Programming (MIP) problem to handle the inherited multi-objectives simultaneously. However, this approach often neglects to consider scaling and subjective weighting issues. In order to ease the problem mentioned above and to obtain a more reasonable compromise solution for allocating order quantities among suppliers with their quantity discount rate offered, the Analytical Hierarchy Process (AHP) and fuzzy compromise programming are introduced in this study. An illustrated example is presented to demonstrate the proposed model and to illuminate two kinds of attitudes for decision makers. The information from the experiments can be utilized further to explain the suppliers’ possible improvement and to help create win–win policies.     

Configuring a supply network in the presence of volume discounts

This paper considers a configuration problem for a manufacturer’s supply network in the presence of volume discounts. An integrated mixed-integer nonlinear programming (MINLP) model is developed to simultaneously determine the selection of suppliers and production levels under an uncertain demand environment. The suppliers provide discounts in terms of the total value of purchased items instead of the quantity of items. The objective of the model is to maximize the manufacturer’s expected profit, subject to both manufacturer and supplier capacities. An external function is developed to deal with integral parts of the model and is integrated with the general algebraic modeling system (GAMS) and its solvers to solve the complex problem. Two numerical examples are used to illustrate the effectiveness of the proposed model and solution approach. Management insights from the sensitivity analysis and comparisons of volume discount are also discussed.     

Inventory replenishment model using fuzzy multiple objective programming: A case study of a high-tech company in Taiwan

The progress in high technology has led to the wide use of thin film transistor-liquid crystal display (TFT-LCD). The evolution of the manufacturing technology of TFT-LCD keeps increasing the size of TFT-LCD since a larger TFT-LCD allows a larger display application and an improved productivity. However, as the size of TFT-LCD increases, the size of TFT-array substrates and color filter substrates has to increase simultaneously. This leads to a more complicated inventory problem of large-sized substrates. Therefore, this paper considers a color filter replenishment problem in TFT-LCD manufacturing with the consideration of storage space, yield rate, quantity discounts and multiple suppliers. We first formulate the color filter replenishment problem as a fuzzy multiple objective programming, and then a fuzzy multiple objective programming with assigned weights for objectives based on experts’ opinions is proposed. An example with four cases is given to illustrate the practicality for empirical investigation. The results demonstrate that both methods are effective tools for inventory management of color filters for multi-periods. In addition, the methods can be applied or modified for managing inventory in general.     

Kombinatorische Auktionen in der betrieblichen BeschaffungEine Analyse grundlegender Entwurfsprobleme – An Analysis of Design Problems in Combinatorial Procurement Auctions

Traditional auction mechanisms support price negotiations on a single item. The Internet allows for the exchange of much more complex offers in real-time. This is one of the reasons for much research on multidimensional auction mechanisms allowing negotiations on multiple items, multiple units, or multiple attributes of an item, as they can be regularly found in procurement. Combinatorial auctions, for example, enable suppliers to submit bids on bundles of items. A number of laboratory experiments has shown high allocative efficiency in markets with economies of scope. For suppliers it is easier to express cost savings due to bundling (e. g., decreased transportation or production costs). This can lead to significant savings in total cost of the procurement manager. Procurement negotiations exhibit a number of particularities:

It is often necessary to consider qualitative attributes or volume discounts in bundle bids. These complex bid types have not been sufficiently analyzed.

The winner determination problem requires the consideration of a number of additional business constraints, such as limits on the spend on a particular supplier or the number of suppliers.

Iterative combinatorial auctions have a number of advantages in practical applications, but they also lead to new problems in the determination of ask prices.

In this paper, we will discuss fundamental problems in the design of combinatorial auctions and the particularities of procurement applications. Reprint of an article from WIRTSCHAFTSINFORMATIK 47(2)2005:126–134.     

Dynamic pricing,promotion and replenishment policies for a deteriorating item under permissible delay in payments

We study the problem of dynamic pricing, promotion and replenishment for a deteriorating item subject to the supplier's trade credit and retailer's promotional effort. In this paper we adopt a price- and time-dependent demand function to model the finite time horizon inventory for deteriorating items. The objective of this paper is to determine the optimal retail price, the promotional effort and the replenishment quantity so that the net profit is maximized. We discuss the properties and develop an algorithm for solving the problem described. The numerical analyses show that an appropriate promotion policy can benefit the retailer and that the promotion policy is important, especially for deteriorating items. Furthermore dynamic decision-making is shown to be superior to fixed decision-making in terms of profit maximization. Some special cases, such as with no credit period and for non-deteriorating items, are discussed as is the influence of the time-varying demand, the rate of deterioration and the credit period on the retailer behavior.     

Solving the multi-buyer joint replenishment problem with the RAND method

The multi-buyer joint replenishment problem (MJRP) is the multi-item inventory problem of coordinating the replenishment of a group of items that are jointly ordered from a single supplier. Joint replenishment of a group of items reduces cost by decreasing the number of times that the major ordering cost is charged. The objective of MJRP is to develop inventory policies minimizing the total costs over the planning horizon. The single buyer joint replenishment problem may be solved by techniques using the collectively described as the best available heuristics (e.g. Simulated Annealing, Genetic Procedures, Tabu Search, and others), collectively discussed as the RAND method. In this paper, we propose a new efficient RAND method to solve MJRP.     

A heuristic to solve the dynamic lot sizing problem with supplier selection and quantity discounts

In the present study, the single-item dynamic lot sizing problem with supplier selection is investigated. The problem is broken down into two different cases. In the first case, quantity discounts are not taken into account; in the second case, incremental and all-unit quantity discounts are considered. Due to the complexity of the problems, a new heuristic is developed, which is based on the Fordyce–Webster Algorithm (Fordyce and Webster, 1984). In order to solve the problem where multiple suppliers are considered, a third dimension is added to the matrices used in the Fordyce–Webster Algorithm. The solutions gained using the proposed algorithm are similar to those of Parsa, Khiav, Mazdeh, and Mehrani (2013) in terms of accuracy and computational time. However, the implementation of matrices makes this method easy to explain in comparison with other heuristics developed for similar problems.     

Trade credit financing in the vendor–buyer inventory system with ordering cost reduction,transportation cost and backorder price discount when the received quantity is uncertain

Previous ordering cost reduction vendor–buyer inventory models with backorder price discount usually assumed that the buyer must pay to the vendor for the ordered items as soon as the items are received, the received quantity is same as the ordered quantity and the transportation cost is independent of the shipment lot-size. In practice, however, the vendor is willing to offer the buyer a certain credit period without interest to promote market competition as well as the buyer's quantity received may not match with the ordered quantity due to unavailability of the raw material, worker's strike, human errors in counting, transcribing, etc. Furthermore, the discounts are offered for the transportation cost of large ordered quantities. This paper derives a single-vendor single-buyer supply chain model for the ordering cost reduction inventory system with backorder price discount, taking into consideration the effect of transportation cost discount and the condition of permissible delay in payments include the case where the buyer's received quantity does not necessarily match the quantity requisitioned. We take the transportation cost as a function of the shipment lot-size and it is taken to be in an all-unit-discount cost format. Thus we incorporate transportation cost explicitly into the model and develop optimal solution procedures for solving the proposed inventory problem. Numerical example and sensitivity analyses are given to demonstrate the applications and performance of the proposed methodology.     

The stochastic lot-sizing problem with quantity discounts

This paper addresses the stochastic lot-sizing problem with quantity discounts. In particular, we examine the uncapacitated finite-period economic lot-sizing problem in which the parameters in each period are random and discrete. When an order is placed, a fixed cost is incurred and an all-unit quantity discount is awarded based on the quantity ordered. The lead time is zero and the order is delivered immediately. First we study the case with overstocks by which the excess inventory incurs a holding cost. The objective in this case is to minimize the expected total cost including ordering and holding costs. The stochastic dynamics is modeled with a scenario tree. We characterize properties of the optimal policy and propose a polynomial time algorithm with complexity O(n³) for single discount level, where n is the number of nodes in the scenario tree. We extend the results to cases allowing stockout and multi-discount levels. Numerical experiments are conducted to evaluate the performance of the algorithm and to gain the management insights.     

Joint replenishment with imperfect items and price discount

A joint replenishment problem is presented to determine the ordering policy for multiple items having a certain percentage of defective units. The purpose of this paper is to study the impact of the percentage of defective units on the ordering policy. Two different scenarios are presented for joint replenishment problem: (1) without price discount and (2) with price discount. For each scenario, the total expected cost per unit time is derived and algorithms are presented to determine the family cycle length and the integer number of intervals that the replenishment quantity of each item will last. Numerical examples are presented and the results are discussed.