A complement to “A comprehensive note on: An economic order quantity with imperfect quality and quantity discounts”

Chang [1] [H.-C. Chang, A comprehensive note on: an economic order quantity with imperfect quality and quantity discounts, Appl. Math. Model. 35 (10) (2011) 5208-5216] corrects a flaw in Lin’s inventory model [T.Y. Lin, An economic order quantity with imperfect quality and quantity discounts, Appl. Math. Model. 34 (10) (2010) 3158–3165]. Then, he develops an algorithm to find the optimal solution for the corrected Lin’s inventory model and furthermore derives close form expressions to determining the optimal solution to an EOQ inventory model considering items with imperfect quality with different holding costs for good and defective items. In both models there is a discrete variable and he presents some inequalities in order to find the integer value. This paper provides some simple formulas to obtain, in an easy way, the integral value for the discrete variable.     

An economic order quantity with imperfect quality and quantity discounts

Previous studies in the issue of inventory models with imperfect quality assumed the defectives could be sold in a batch by the end of the inspection process and the manufacturing systems were push systems. However, the above assumptions may not be true in the pull system in which buyer is powerful. Therefore, in this paper, we develop a new inventory model for items with imperfect quality and quantity discounts where buyer has exerted power over its supplier. Based on the concept of powerful buyer, there are three considerations included in this new model: (1) the order quantity is manufactured at one setup and is shipped over multiple deliveries, (2) the defectives are screened out by a 100% inspection for each shipment but sold in a batch by the end of inspection at the last shipment of each cycle, and (3) the supplier offers quantity discounts to response the request of the powerful buyer. Further, an algorithm is developed to help the powerful buyer to determine the optimal order policy accurately and quickly. Two numerical examples are available in this paper to illustrate the proposed model and algorithm. Besides, based on the numerical examples, a sensitivity analysis is made to investigate the effects of four important parameters (the inspection rate, the defective rate, the receiving cost, and the ordering cost) on the optimal solution.     

A modeling framework and local search solution methodology for a production-distribution problem with supplier selection and time-aggregated quantity discounts

Supplier selection with quantity discounts has been an active research problem in the literature. In this paper, we focus on a new real-world quantity discounts scheme, where suppliers are selected in the beginning of a strategic planning period (e.g., 5 years). Monthly orders are placed from the selected suppliers, but the quantity discounts are based on the aggregated annual order quantities. We incorporate this type of cost structure in a multi-period, multi-product, multi-echelon supply chain planning problem, and develop a mixed integer linear programming (MIP) model for it. Our model is highly intractable; leading commercial solvers cannot construct high quality feasible solutions for realistic instances even after multiple hours of solution time. We develop an algorithm that constructs an initial feasible solution and a large neighborhood search method that combines two customized iterative algorithms based on MIP-based local search and improves such solution. We report numerical results for a food supply chain application and show the efficiency of using our methodology in getting very high quality primal solutions quickly.     

Heuristics for sourcing from multiple suppliers with alternative quantity discounts

In this paper, we analyze the impact of supplier pricing schemes and supplier capacity limitations on the optimal sourcing policy for a single firm. We consider the situation where the total quantity to be procured for a single period is known by the firm and communicated to the supplier set. In response to this communication, each supplier quotes a price and a capacity limit in terms of a maximum quantity that can be supplied to the buyer. Based on this information, the buyer makes a quantity allocation decision among the suppliers and corresponding to this decision is the choice of a subset of suppliers who will receive an order. Based on industry observations, a variety of supplier pricing schemes from the constituent group of suppliers are analyzed, including linear discounts, incremental units discounts, and all units discounts. Given the complexity of the optimization problem for certain types of pricing schemes, heuristic solution methodologies are developed to identify a quantity allocation decision for the firm. Through an extensive computational comparison, we find that these heuristics generate near-optimal solutions very quickly. Data from a major office products retailer is used to illustrate the resulting sourcing strategies given different pricing schemes and capacity limitations of suppliers in this industry. We find for the case of capacity constrained suppliers, the optimal quantity allocations for two complex pricing schemes (linear discount, and incremental units discount) are such that at most one selected supplier will receive an order quantity that is less than its capacity.     

Note: An application of the EOQ model with nonlinear holding cost to inventory management of perishables

In this note, we consider a variation of the economic order quantity (EOQ) model where cumulative holding cost is a nonlinear function of time. This problem has been studied by Weiss [Weiss, H., 1982. Economic order quantity models with nonlinear holding costs. European Journal of Operational Research 9, 56–60], and we here show how it is an approximation of the optimal order quantity for perishable goods, such as milk, and produce, sold in small to medium size grocery stores where there are delivery surcharges due to infrequent ordering, and managers frequently utilize markdowns to stabilize demand as the product’s expiration date nears. We show how the holding cost curve parameters can be estimated via a regression approach from the product’s usual holding cost (storage plus capital costs), lifetime, and markdown policy. We show in a numerical study that the model provides significant improvement in cost vis-à-vis the classic EOQ model, with a median improvement of 40%. This improvement is more significant for higher daily demand rate, lower holding cost, shorter lifetime, and a markdown policy with steeper discounts.     

An arithmetic-geometric mean inequality approach for determining the optimal production lot size with backlogging and imperfect rework process

Some classical studies on economic production quantity (EPQ) models with imperfect production processes have focused on determining the optimal production lot size. However, these models neglect the fact that the total production-inventory costs can be reduced by reworking imperfect items for a relatively small repair and holding cost. To account for the above phenomenon, we take the out of stock and rework into account and establish an EPQ model with imperfect production processes, failure in repair and complete backlogging. Furthermore, we assume that the holding cost of imperfect items is distinguished from that of perfect ones, as well as, the costs of repair, disposal, and shortage are all included in the proposed model. In addition, without taking complex differential calculus to determine the optimal production lot size and backorder level, we employ an arithmetic-geometric mean inequality method to determine the optimal solutions. Finally, numerical examples and sensitivity analysis are analyzed to illustrate the validity of the proposed model. Some managerial insights are obtained from the numerical examples.     

Dynamic demand lot-sizing rules for incremental quantity discounts

Numerous studies have developed and compared lot-sizing procedures for finite-horizon dynamic demand, material requirements planning (MRP), environments when either no purchase discounts exist or for the case of all-units quantity discounts. This paper examines lot-sizing rules when product price schedules follow incremental quantity discounts. The optimal (non-discount) procedure and some traditional heuristic procedures are modified to incorporate incremental quantity discounts. We further modify two heuristics with a ‘look-ahead enhancement’ that performs very well under experimentation. Numerical tests revealed the overall best-performing heuristic in this study to be a modified ‘least-unit cost’ method with a look-ahead enhancement. That procedure produced an average cost penalty vs optimal of 0.26%.     

The optimal quantity discount that a supplier should offer

Quantity discounts are a useful mechanism for coordination. Here we investigate such discounts from the supplier's perspective, both from a non-cooperative game-theoretical approach and a joint decision model. Taking into account the price elasticity of demand, this analysis aids a sole supplier in establishing an all-unit quantity discount policy in light of the buyer's best reaction. The Stackelberg equilibrium and Pareto optimal solution set are derived for the non-cooperative and joint-decision cases, respectively. Our research indicates that channel efficiency can be improved significantly if the quantity discount decision is made jointly rather than non-cooperatively. Moreover, we extend our model in three directions: (1) the product is transported by a private fleet; (2) the buyer may choose to offer her customers a different percentage discount than that she obtained from the supplier; and (3) the case of heterogeneous buyers. Numerical case studies are employed throughout the paper to illustrate the practical applications of the models presented and the sensitivity to model parameters.     

Optimal purchase of two-itemized drugs for a disease

This paper submits a mathematical model called the two-itemized drugs purchasing decision (TDPD) model to conduct the optimal purchase of two-itemized drugs for healing a given disease. This TDPD model considers not only the annual estimated number of patients suffering from the given disease, the treatment courses, and the unit profit of each drug, but also the strength of the patient's perception on treatment quality, the different purchasing price of each drug with quantity discounts, the ordering cost, the holding cost, and other related costs to determine the optimal order quantity of each drug for achieving the maximum profit. A computerized solving program proposed in this paper applies tabu search technique as the main program and the all-units quantity discounts for economic order quantity as the sub-program to solve such a combinatorial problem, as well as perform sensitivity analyses on the changes by the strength of the patient's perception on treatment quality and the changes of annual estimated number of patients for the given disease. Two drugs, Lipitor and Gemnpid, to heal a disease ‘Hyperlipidaemia’ are considered as a case example in this study. This paper contributes a prototype quantitative method of purchasing decision for a given disease to a decision-maker with profound insights.     

An improved solution to the replenishment policy for the EMQ model with rework and multiple shipments

Recently, Chiu et al. (2012) [1] present an alternative optimization procedure to derive the optimal replenishment lot size for an economic manufacturing quantity (EMQ) model with rework and multiple shipments. This inventory model was proposed by Chiu et al. (2011) [2]. Both papers do not consider the determining of the number of shipments. This paper determines both the optimal replenishment lot size and the optimal number of shipments jointly. The solution of this paper is better than the solutions of Chiu et al.  and .     

Production–shipment policy for EPQ model with quality assurance and an improved delivery schedule

This article is concerned with determining the production–shipment policy for an economic production quantity model with quality assurance and an improved delivery schedule. We extend a recent work by Chiu et al. [Y.-S.P. Chiu, C.-A.K. Lin, H.-H. Chang, and V. Chiu, Mathematical modeling for determining economic batch size and optimal number of deliveries for EPQ model with quality assurance, Math. Comput. Model. Dyn. Sys. 16 (4) (2010), pp. 373–388] by incorporating an alternative delivery plan that aims at lowering the inventory holding cost for both supplier and buyer in such an integrated inventory system. Mathematical modelling along with Hessian matrix equations is used, and as a result the optimal production batch size and optimal number of deliveries are derived. A numerical example is provided to demonstrate the practical use of the results and the significant savings in stock holding costs for both vendor and buyer.     

Optimal economic production quantity policy for imperfect process with imperfect repair and maintenance

This study integrates maintenance and production programs with the economic production quantity (EPQ) model for an imperfect process involving a deteriorating production system with increasing hazard rate: imperfect repair and rework upon failure (out of control state). The imperfect repair performs some restorations and restores the system to an operating state (in-control state), but leaves its failure until perfect preventive maintenance (PM) is performed. There are two types of PM, namely imperfect PM and perfect PM. The probability that perfect PM is performed depends on the number of imperfect maintenance operations performed since the last renewal cycle. Mathematical formulas are obtained for deriving the expected total cost. For the EPQ model, the optimum run time, which minimizes the total cost, is discussed. Various special cases are considered, including the maintenance learning effect. Finally, a numerical example is presented to illustrate the effects of PM, setup, breakdown and holding costs.     

Process mean determination under constant raw material supply

Setting the mean (target value) for a production process is an important decision for a producer when material cost is a significant portion of production cost. Because the process mean determines the process conforming rate, it affects other production decisions, including, in particular, production setup and raw material procurement policies. In this paper, we consider the situation in which the product of interest is assumed to have a lower specification limit, and the items that do not conform to the specification limit are scrapped with no salvage value. The production cost of an item is a linear function of the amount of the raw material used in producing the item, and the supply rate of the raw material is finite and constant. Furthermore, it is assumed that quantity discounts are available in the raw material cost and that the discounts are determined by the supply rate. Two types of discounts are considered in this paper: incremental quantity discounts and all-unit quantity discounts. A two-echelon model is formulated for a single-product production process to incorporate the issues associated with production setup and raw material procurement into the classical process mean problem. Efficient solution algorithms are developed for finding the optimal solutions of the model.     

Optimal replenishment policy for imperfect quality EMQ model with rework and backlogging

This paper derives the optimal replenishment policy for imperfect quality economic manufacturing quantity (EMQ) model with rework and backlogging. The classic EMQ model assumes that all items produced are of perfect quality. However, in real‐life manufacturing settings, generation of imperfect quality items is almost inevitable. In this study, a random defective rate is assumed. All items produced are inspected and the defective items are classified as scrap and repairable. A rework process is involved in each production run when regular manufacturing process ends, and a rate of failure in repair is also assumed. Unit disposal cost and unit repairing and holding costs are included in our mathematical modelling and analysis. The renewal reward theorem is employed in this study to cope with the variable cycle length. The optimal replenishment policy in terms of lot‐size and backlogging level that minimizes expected overall costs for the proposed imperfect quality EMQ model is derived. Special cases of the model are identified and discussed. Numerical example is provided to demonstrate its practical usage. Copyright © 2006 John Wiley & Sons, Ltd.     

A two-stage supply chain coordination mechanism considering price sensitive demand and quantity discounts

This paper explores the coordination between a supplier and a buyer within a decentralized supply chain, through the use of quantity discounts in a game theoretic model. Within this model, the players face inventory and pricing decisions. We propose both cooperative and non-cooperative approaches considering that the product traded experiences a price sensitive demand. In the first case, we study the dynamics of the game from the supplier's side as the leader in the negotiation obtaining a Stackelberg equilibrium, and then show how the payoff of this player could still improve from this point. In the second case, a cooperative model is formulated, where decisions are taken simultaneously, emulating a centralized firm, showing the benefits of the cooperation between the players. We further formulate a pricing game, where the buyer is allowed to set different prices to the final customer as a reaction to the supplier's discount decisions. For the latter we investigate the difference between feasibility of implementing a retail discount given a current coordination mechanism and without it. Finally the implications of transportation costs are analyzed in the quantity discount schedule. Our findings are illustrated with a numerical example showing the difference in the players’ payoff in each case and the optimal strategies, comparing in each case our results with existing work.     

Integrated analytical hierarch process and mathematical programming to supplier selection problem with quantity discount

In this article an integration of analytical hierarchy process and non-linear integer and multi-objective programming under some constraints such as quantity discounts, capacity, and budget is applied to determine the best suppliers and to place the optimal order quantities among them. This integration-based multi-criteria decision making methodology takes into account both qualitative and quantitative factors in supplier selection. While the analytical hierarchy process matches item characteristics with supplier characteristics, non-linear integer programming model analytically determines the best suppliers and the optimal order quantities among the determined suppliers. The objectives of the mathematical models constructed are maximizing the total value of purchase (TVP), minimizing the total cost of purchase (TCP) or maximizing TVP and minimizing TCP simultaneously. In addition, several “what if” scenarios are facilitated and the quality of the resulting models is evaluated on real-life data.     

An EOQ model with backorders and all-units discounts

In this paper we study an inventory model with backorders where the purchase unit price depends on the ordered quantity. This situation appears in practice when a salesperson offers a fixed compensation to a client for not losing the sale and there are quantity discounts. The optimal policy is obtained through a sequential optimization procedure in two stages that relies on a quadratic function (first stage) and on the objective function of the classical EOQ model (second stage). An algorithm is developed for the model and some extensions are commented.     

Random fuzzy EOQ model with imperfect quality items

This paper investigates an economic order quantity (EOQ) problem with imperfect quality items, where the percentage of imperfect quality items in each lot is characterized as a random fuzzy variable while the setup cost per lot, the holding cost of each unit item per day, and the inspection cost of each unit item are characterized as fuzzy variables, respectively. In order to maximize the expected long-run average profit, a random fuzzy EOQ model is constructed. Since it is almost impossible to find an analytic method to solve the proposed model, a particle swarm optimization (PSO) algorithm based on the random fuzzy simulation is designed. Finally, the effectiveness of the designed algorithm is illustrated by a numerical example.     

Robust economic order quantity models

In this paper, we study the classical economic order quantity (EOQ) model under significant. In particular, the problem under consideration is the economic order quantity model with the input data of the demand rate, the order cost, and the holding cost rate being uncertain. A robustness approach based on scenario characterization of the input data is adopted to describe the uncertainties. The aim of the approach is to find an inventory policy that performs well under all realizable input data scenarios. An efficient linear time algorithm is devised to find the robust decisions. Analytical results are obtained for the case where input data are defined in continuous intervals. Comparisons on performances between the robust decisions and the stochastic optimization decisions are conducted. The results demonstrate the advantages of robustness approach.