A modified Silver–Meal heuristic for dynamic lot sizing under incremental quantity discounts

This note presents a variation of the well-known Silver–Meal heuristic to deal with lot sizing under a combination of a known, but time-varying, demand pattern along with an incremental quantity discount structure. The heuristic is shown to perform very well on a set of experiments presented in a recent paper in this journal. Additional experiments are performed to further explore the time horizon effects on the relative performance of the newly proposed heuristic compared to the two best performing ones from the previous paper.     

Joint price and lot size determination under conditions of permissible delay in payments and quantity discounts for freight cost

This paper deals with the problem of determining the retailer's optimal price and lot size simultaneously under conditions of permissible delay in payments. It is assumed that the ordering cost consists of a fixed set-up cost and a freight cost, where the freight cost has a quantity discount offered due to the economies of scale. The constant price elasticity demand function is adopted, which is a decreasing function of retail price. Investigation of the properties of an optimal solution allows us to develop an algorithm whose validity is illustrated through an example problem.     

A dynamic quantity discount lot size model with resales

This paper deals with a dynamic lot size problem in which the unit purchasing price depends on the quantity of an order and resale of the excess is possible at the end of each period. We assume an all units discount system with a single price break point. Investigation of the properties of an optimal solution allows us to develop a dynamic programming algorithm. A problem example is solved to illustrate the algorithm.     

An integrated model for lot sizing with supplier selection and quantity discounts

Good inventory management is essential for a firm to be cost competitive and to acquire decent profit in the market, and how to achieve an outstanding inventory management has been a popular topic in both the academic field and in real practice for decades. As the production environment getting increasingly complex, various kinds of mathematical models have been developed, such as linear programming, nonlinear programming, mixed integer programming, geometric programming, gradient-based nonlinear programming and dynamic programming, to name a few. However, when the problem becomes NP-hard, heuristics tools may be necessary to solve the problem. In this paper, a mixed integer programming (MIP) model is constructed first to solve the lot-sizing problem with multiple suppliers, multiple periods and quantity discounts. An efficient Genetic Algorithm (GA) is proposed next to tackle the problem when it becomes too complicated. The objectives are to minimize total costs, where the costs include ordering cost, holding cost, purchase cost and transportation cost, under the requirement that no inventory shortage is allowed in the system, and to determine an appropriate inventory level for each planning period. The results demonstrate that the proposed GA model is an effective and accurate tool for determining the replenishment for a manufacturer for multi-periods.     

A comprehensive note on: An economic order quantity with imperfect quality and quantity discounts

Lin [T.Y. Lin, An economic order quantity with imperfect quality and quantity discounts, Appl. Math. Model. 34 (10) (2010) 3158–3165] recently proposed an EOQ model with imperfect quality and quantity discounts, where the lot-splitting shipments policy is adopted. In this note we first rectify the holding cost terms showed in Lin to obtain a new objective function, then resolve the problem and develop an easy to implement algorithm to find the overall optimal solutions for the model. Besides, we present a new model for items with imperfect quality, where lot-splitting shipments and different holding costs for good and defective items are considered. The closed-form formulas for determining the optimal ordering and shipping policies are derived. Also, the results are examined analytically and numerically to gain more insights of the solutions.     

Polynomial cases of the economic lot sizing problem with cost discounts

In this paper we study the economic lot sizing problem with cost discounts. In the economic lot sizing problem a facility faces known demands over a discrete finite horizon. At each period, the ordering cost function and the holding cost function are given and they can be different from period to period. There are no constraints on the quantity ordered in each period and backlogging is not allowed. The objective is to decide when and how much to order so as to minimize the total ordering and holding costs over the finite horizon without any shortages. We study two different cost discount functions. The modified all-unit discount cost function alternates increasing and flat sections, starting with a flat section that indicates a minimum charge for small quantities. While in general the economic lot sizing problem with modified all-unit discount cost function is known to be NP-hard, we assume that the cost functions do not vary from period to period and identify a polynomial case. Then we study the incremental discount cost function which is an increasing piecewise linear function with no flat sections. The efficiency of the solution algorithms follows from properties of the optimal solution. We computationally test the polynomial algorithms against the use of CPLEX.     

Optimal policy for an inventory system with backlogging and all-units discounts: Application to the composite lot size model

This paper examines an inventory model with full backlogging and all-units quantity discounts. The practical scenario of a salesperson offering compensation to a client so as not to lose the sale is considered. The cost of a backorder thus includes both a fixed cost and a further cost which is proportional to the length of time the said backorder exists. A first algorithm is developed to determine the optimal policy while some extensions to this algorithm are obtained that include additional conditions on the model. In particular, the well known composite lot size model, developed by Tersine, is solved, incorporating a new stockout cost and a new all-units discount. Numerical examples are provided to illustrate the application of the algorithms.     

Using the complete squares method to analyze a lot size model when the quantity backordered and the quantity received are both uncertain

Several researchers have recently derived formulae for economic order quantities (EOQs) with some variants without reference to the use of derivatives, neither for first-order necessary conditions nor for second-order sufficient conditions. In addition, this algebraic derivation immediately produces an individual formula for evaluating the minimum expected annual cost. The purpose of this paper is threefold. First, this study extends the previous result to the EOQ formula, taking into account the scenario where the quantity backordered and the quantity received are both uncertain. Second, the complete squares method can readily derive global optimal expressions from a non-convex objective function in an algebraic manner. Third, the explicit identification of some analytic cases can be obtained: it is not as easy to do this using decomposition by projection. A numerical example has been solved to illustrate the solution procedure. Finally, some special cases can be deduced from the EOQ model under study, and concluding remarks are drawn.     

Multi-item lot size determination and scheduling under capacity constraints

This paper deals with the planning of a production group, which has to produce several products. For each product there is a delivery plan covering several periods. Moreover, there are capacity constraints. Such a situation requires integrated optimization of lot sizes and lot scheduling. Since exact solution of the problem is in general not feasible, we will present a non exact approach which gives quite good results in some practical cases and might be a good starting point in other cases.     

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.     

Incorporating quantity discounts and their inventory impacts into the centralized purchasing decision

Multi-site organizations must balance conflicting forces to determine the appropriate degree of purchasing centralization for their respective supplies. The ability to garner quantity discounts represents one of the primary reasons that organizations centralize procurement. This paper provides methodologies to calculate optimal order quantities and compute total purchasing and inventory costs when products have quantity discount pricing. Procedures for both all-units and incremental quantity discount schedules are provided for four different strategic purchasing configurations (scenarios): complete decentralization, centralized pricing with decentralized purchasing, centralized purchasing with local distribution, and centralized purchasing and warehousing. For ordering decisions under local distribution, procedures to determine optimal order quantities and costs are presented in a precise form that could be easily implemented into spreadsheets by practicing managers. For the more complicated multi-echelon scenarios, we introduce a single-cycle policy with a tailored aggregation refinement step that performs very well under experimentation when compared to a conservative bound.     

Augmenting the lot sizing order quantity when demand is probabilistic

In this paper we consider a single item, discrete time, lot sizing situation where demand is random and its parameters (e.g., mean and standard deviation) can change with time. For the appealing criterion of minimizing expected total relevant costs per unit time until the moment of the next replenishment we develop two heuristic ways of selecting an appropriate augmentation quantity beyond the expected total demand through to the planned (deterministic) time of the next replenishment. The results of a set of numerical experiments show that augmentation is important, particularly when orders occur frequently (i.e., the fixed cost of a replenishment is low relative to the costs of carrying one period of demand in stock) and the coefficient of variability of demand is relatively low, but also under other specified circumstances. The heuristic procedures are also shown to perform very favourably against a hindsight, baseline (s, S) policy, especially for larger levels of non-stationarity.     

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.     

Stability of the constant cost dynamic lot size model

A mathematical analysis of the dynamic lot size model with constant cost parameters is provided. First, stability regions for so called generalized and optimal solutions are found, which show how the cost input may vary, leaving the solution valid. Based on these results a BAsic dialog program has been designed to display the optimal solution and the stability regions to the decision maker. Secondly, an estimation of the initial optimal solution is given for the case, when the cost inputs leave the stability region.     

A note on the economic lot size of the integrated vendor–buyer inventory system derived without derivatives

Numerous researches on the integrated production inventory models use differential calculus to solve the multi-variable problems. This study simplifies the solution procedure using a simple algebraic method to solve the multi-variable problems. As a result, students who are unfamiliar with calculus may be able to understand the solution procedure with ease. This paper refers to the approach by Grubbström and Erdem [R.W. Grubbström, A. Erdem, The EOQ with backlogging derived without derivatives, International Journal of Production Economics 59 (1999) 529–530] and extends the model by Yang and Wee [P.C. Yang, H.M. Wee, The economic lot size of the integrated vendor–buyer system derived without derivatives. Optimal Control Applications and Methods 23 (2002) 163–169] to derive an algebraic method to solve the three decision variables of the proposed model.     

A new dynamic programming algorithm for the single item capacitated dynamic lot size model

We develop a new dynamic programming method for the single item capacitated dynamic lot size model with non-negative demands and no backlogging. This approach builds the Optimal value function in piecewise linear segments. It works very well on the test problems, requiring less than 0.3 seconds to solve problems with 48 periods on a VAX 8600. Problems with the time horizon up to 768 periods are solved. Empirically, the computing effort increases only at a quadratic rate relative to the number of periods in the time horizon.This research was supported in part by NSF grants DDM-8814075 and DMC-8504786.     

Error bound for the dynamic lot size model with backlogging

This paper finds a tight bound on the error introduced in the dynamic lot size model with backlogging when it is truncated by planning for the interval of time for which data are actually known, rather than for the natural planning interval.Research supported by NSF Grant ECS-8310213 to Yale University.     

A comparison of average and discounted cost models for the dynamic lot size inventory problem

It is a common practice in the inventory literature to use average cost models as approximations to the theoretically correct discounted cost models. An average cost model minimizes the average undiscounted cost per period, while a discounted cost model minimizes the total discounted cost over the problem horizon. This paper attempts to answer an important question: How good are the results (the total discounted costs) for the average cost models compared to those for the discounted cost models? This question has been conclusively answered for the simplest inventory model where the demand rate and other parameters are assumed to remain constant in time. This paper addresses this issue for the first time for the case where demand rates are allowed to be nonstationary in time.A discounted cost model has been developed in the paper to carry out this comparison. It is shown that a simple dynamic programming algorithm can be used to find optimal order policies for the discounted cost model.The effect of the varying interest rates and other parameters on the relative performance of the average cost model has been studied by developing an insightful analysis and also by doing a computational study. The results show that, while the average cost model can cost as much as about 26% more than the discounted cost model in extreme cases, this increase is not significant for the parameter values in the range of the common interest.     

A dynamic programming algorithm for dynamic lot size models with piecewise linear costs

We propose a new algorithm for dynamic lot size models (LSM) in which production and inventory cost functions are only assumed to be piecewise linear. In particular, there are no assumptions of convexity, concavity or monotonicity. Arbitrary capacities on both production and inventory may occur, and backlogging is allowed. Thus the algorithm addresses most variants of the LSM appearing in the literature. Computational experience shows it to be very effective on NP-hard versions of the problem. For example, 48 period capacitated problems with production costs defined by eight linear segments are solvable in less than 2.5 minutes of Vax 8600 cpu time.