An interval programming approach for developing economic order quantity model with imprecise exponents and coefficients

This paper develops an economic order quantity (EOQ) model with uncertain data. For modelling the uncertainty in real-world data, the exponents and coefficients in demand and cost functions are considered as interval data and then, the related model is designed. The proposed model maximises the profit and determines the price, marketing cost and lot sizing with the interval data. Since the model parameters are imprecise, the objective value is imprecise, too. So, the upper and lower bounds are specially formulated for the problem and then, the model is transferred to a geometric program. The resulted geometric program is solved by using the duality approach and the lower and upper bounds are found out for the objective function and variables. Two numerical examples and sensitivity analysis are further used to illustrate the performance of the proposed model.     

带时变生产成本的易变质经济批量模型的最优策略分析

考虑了具有时变生产成本的易变质产品经济批量模型．有限计划期内,单位生产成本、生产率以及需求率假定为时间的连续函数,生产固定成本则具有遗忘效应现象．当不允许缺货时,建立了以总成本最小为目标的混合整数优化模型并证明了此问题最优解的相关性质．对于此问题的特殊情形,将成本函数中的离散型变量松弛为连续型变量,通过分析其最优解的存在性及唯一性,求解了此最优解,将其作为初始值设计了求取一般情形最优解的有效算法．最后通过算例验证了理论结果的有效性．     

Profit maximization with quantity discount: An application of geometric programming

Profit maximization is an important issue to the firms that pursue the largest economic profit possible. Traditionally, profit maximization problem is solved by differentiating with respect to input prices. The total differentiation of the first-order conditions might give complicated equations difficult to handle. Different from traditional studies, this paper considers input quantity discount and employs geometric programming technique to derive the objective value for the profit-maximization problem. The geometric programming approach not only gives the global optimum solution but also provides the information that is able to discover the relationship between profit maximization and returns to scale in the solution process. No differentiation is required. Moreover, geometric programming can provide a computationally attractive view of sensitivity analysis for the changes in parameters. Examples are given to illustrate the idea proposed in this paper.     

An elaborative unit cost structure-based fuzzy economic production quantity model

The purpose of this paper is to investigate and propose a fuzzy extended economic production quantity model based on an elaboratively modeled unit cost structure. This unit cost structure consists of the various lot-size correlative components such as on-line setups, off-line setups, initial production defectives, direct material, labor, and depreciation in addition to lot-size non-correlative items. Thus, the unit cost is correlatively modeled to the production quantity. Therefore, the modeling or the annual total cost function developed consists of not only annual inventory and setup costs but also production cost. Moreover, via the concept of fuzzy blurred optimal argument and the vertex method of the α-cut fuzzy arithmetic (or fuzzy interval analysis), two solution approaches are proposed: (1) a fuzzy EPQ and (2) a compromised crisp EPQ in the fuzzy sense. An optimization procedure, which can simultaneously determine the α-cut-vertex combination of fuzzy parameters and the optimizing decision variable value, is also proposed. The sensitivity model for the fuzzy total cost and thus EPQ to the various cost factors is provided. Finally, a numerical example with the original data collected from a firm demonstrates the usefulness of the new model.     

Shortening cycle times in multi-product,capacitated production environments through quality level improvements and setup reduction

This paper addresses the issue of investing in reduced setup times and defect rates for a manufacturer of several products operating in a JIT environment. Production cycle times can be shortened by investing in setup time and defect rate reductions, respectively. The objective is to determine optimal levels of setup time and defect rate reductions along with the corresponding optimal levels of investments respectively, and the optimal production cycle time for each product. The problem is constrained by demand requirements, process improvement budget limitations, and manufacturing and warehousing capacity constraints. We consider the cases of product-specific quality improvements and joint-product quality improvements. A general nonlinear optimization models of these problems are formulated. A convex geometric programming approximation of these models is developed respectively, in order to solve them. The approximation can be made to any desired degree of accuracy. Our empirical findings provide insights into a number of managerial issues surrounding investment decisions in product-specific quality improvements and setup reductions due to a product redesign as well as in joint-product improvements due to a process redesign.     

A note on an economic order quantity model and Taguchi’s cost of poor quality

Tsou [Appl. Math. Model. 31 (2) (2007) 283–291] previously proposed an economic order quantity model with Taguchi’s cost of poor quality. In this note, we correct an error appearing in their numerical results and provide some simple formulas that can be applied to numerical studies with the specific objective of calculating the defective rate and the total profit of his model.     

An economic production quantity model with a positive resetup point under random demand

In this paper, an extended economic production quantity (EPQ) model is investigated, where demand follows a random process. This study is motivated by an industrial case for precision machine assembly in the machinery industry. Both a positive resetup point s and a fixed lot size Q are implemented in this production control policy. To cope with random demand, a resetup point, i.e., the lowest inventory level to start the production, is adapted to minimize stock shortage during the replenishment cycle. The considered cost includes setup cost, inventory carrying cost, and shortage cost, where shortage may occur at the production stage and/or at the end of one replenishment cycle. Under some mild conditions, the expected cost per unit time can be shown to be convex with respect to decision parameters s and Q. Further computational study has demonstrated that the proposed model outperforms the classical EPQ when demand is random. In particular, a positive resetup point contributes to a significant portion of this cost savings when compared with that in the classical lot sizing policy.     

A phase-type geometric process repair model with spare device procurement and repairman’s multiple vacations

This paper analyzes a phase-type geometric process repair model with spare device procurement lead time and repairman’s multiple vacations. The repairman may mean here the human beings who are used to repair the failed device. When the device functions smoothly, the repairman leaves the system for a vacation, the duration of which is an exponentially distributed random variable. In vacation period, the repairman can perform other secondary jobs to make some extra profits for the system. The lifetimes and the repair times of the device are governed by phase-type distributions (PH distributions), and the condition of device following repair is not “as good as new”. After a prefixed number of repairs, the device is replaced by a new and identical one. The spare device for replacement is available only by an order and the procurement lead time for delivering the spare device also follows a PH distribution. Under these assumptions, the vector-valued Markov process governing the system is constructed, and several important performance measures are studied in transient and stationary regimes. Furthermore, employing the standard results in renewal reward process, the explicit expression of the long-run average profit rate for the system is derived. Meanwhile, the optimal maintenance policy is also numerically determined.     

An economic production quantity inventory model involving fuzzy demand rate and fuzzy deterioration rate

Generally, in deriving the solution of economic production quantity (EPQ) inventory model, we consider the demand rate and deterioration rate as constant quantity. But in case of real life problems, the demand rate and deterioration rate are not actually constant but slightly disturbed from their original crisp value. The motivation of this paper is to consider a more realistic EPQ inventory model with finite production rate, fuzzy demand rate and fuzzy deterioration rate. The effect of the loss in production quantity due to faulty/old machine have also been taken into consideration. The methodology to obtain the optimum value of the fuzzy total cost is derived and a numerical example is used to illustrate the computation procedure. A sensitivity analysis is also carried out to get the sensitiveness of the tolarance of different input parameters.     

An economic order quantity model for deteriorating items with partially permissible delay in payments linked to order quantity

To attract more sales suppliers frequently offer a permissible delay in payments if the retailer orders more than or equal to a predetermined quantity W. In this paper, we generalize [Goyal, S.K., 1985. EOQ under conditions of permissible delay in payments. Journal of the Operational Research Society 36, 335–338] economic order quantity (EOQ) model with permissible delay in payment to reflect the following real-world situations: (1) the retailer’s selling price per unit is significantly higher than unit purchase price, (2) the interest rate charged by a bank is not necessarily higher than the retailer’s investment return rate, (3) many items such as fruits and vegetables deteriorate continuously, and (4) the supplier may offer a partial permissible delay in payments even if the order quantity is less than W. We then establish the proper mathematical model, and derive several theoretical results to determine the optimal solution under various situations and use two approaches to solve this complex inventory problem. Finally, a numerical example is given to illustrate the theoretical results.     

A note on article “A tolerance approach to the fuzzy goal programming problems with unbalanced triangular membership function”

Kim and Whang use a tolerance approach for solving fuzzy goal programming problems with unbalanced membership functions [J.S. Kim, K. Whang, A tolerance approach to the fuzzy goal programming problems with unbalanced triangular membership function, European Journal of Operational Research 107 (1998) 614–624]. In this note it is shown that some results in that article are incorrect. The necessary corrections are proposed.     

An economic order quantity model with partial backordering and a special sale price

A constant unit purchase cost is one of the main assumptions in the classic economic order quantity model. In practice, suppliers sometimes offer special sale prices to stimulate sales or decrease inventories of certain items. In this paper we develop an EOQ model with a special sale price and partial backordering. We prove the convexity of the cost-reduction function if a special order is placed at the special sale price. A solution method is proposed and numerical examples are presented.     

A generalized JM model with applications to imperfect debugging in software reliability

For their nice mathematical properties, state space models have been widely used, especially for forecasting. Over the last decades, the study of tracking software reliability by statistical models has attracted scientists’ attention. However, most of models focus on perfect debugging although practically imperfect debugging arises everywhere. In this paper, a non-Gaussian state space model is modified to predict software failure time with imperfect debugging. In fact, this model is very flexible so that we can modify the system equation in this model to satisfy the various situations. Besides, this model is suitable for tracking software reliability, and applied to two well known datasets on software failures.     

An economic order quantity model with a known price increase and partial backordering

A constant unit purchase cost is one of the main assumptions in the classic Economic Order Quantity model. In practice, suppliers sometimes face a known price increase. In this paper, we develop EOQ models with a known price increase and partial backordering under two different assumptions about when the increase will occur. We prove the concavity of the extra profit functions for both scenarios if a special order is placed just before the price increases. A solution method is proposed and numerical examples are presented.     

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.     

A note on “quality improvement and setup reduction in the joint economic lot size model”

In a recent paper, Affisco et al. [J.F. Affisco, M.J. Paknejad, F. Nasri, Quality improvement and setup reduction in the joint economic lot size model, European Journal of Operations Research 142 (2002) 497–508] propose a quality-adjusted joint economic lot size model that considers investments in quality improvement and setup cost reduction. In particular, they consider a single-vendor, single-buyer, deterministic demand economic lot-sizing problem, and they investigate the potential impact of economic investments in the vendor’s quality improvement and setup cost reduction efforts on the system-wide costs. However, the particular form of the investment function that they use to represent the cost of investments in quality improvement does not represent actual practice in many industries. Hence, in this note, we develop modified models for quality improvement and simultaneous quality improvement and setup cost reduction using a modified form of the investment function. Our fundamental results and conclusions are substantially different than those in Affisco et al. (2002).     

A displayed inventory model with L–R fuzzy number

In this study, we formulate a multi-item displayed inventory model under shelf-space constraint in fuzzy environment. Here demand rate of an item is considered as a function of the displayed inventory level. The problem is formulated to maximize average profit. In real life situation, the goals and inventory parameters are may not precise. Such type of uncertainty may be characterized by fuzzy numbers. Here, the constraint goal and the inventory cost parameters are assumed to be triangular shaped fuzzy numbers with different types of left and right membership functions. The fuzzy numbers are then approximated to a nearest interval number. Using arithmetic of interval numbers, the problem is described as a multi-objective inventory problem. The problem is then solved by fuzzy geometric programming approach. Finally a numerical example is given to illustrate the problem.     

Improving reliability of a shared supplier with competition and spillovers

Supplier reliability is a key determinant of a manufacturer’s competitiveness. It reflects a supplier’s capability of order fulfillment, which can be measured by the percentage of order quantity delivered in a given time window. A perfectly reliable supplier delivers an amount equal to the order placed by its customer, while an unreliable supplier may deliver an amount less than the amount ordered. Therefore, when suppliers are unreliable, manufacturers often have incentives to help suppliers improve delivery reliability. Suppliers, however, often work with multiple manufacturers and the benefit of enhanced reliability may spill over to competing manufacturers. In this study, we explore how potential spillover influences manufacturers’ incentives to improve supplier’s reliability. We consider two manufacturers that compete with imperfectly substitutable products on Type I service level (i.e., in-stock probability). The manufacturers share a common supplier who, due to variations in production quality or yield, is unreliable. Manufacturers may exert efforts to improve the supplier’s reliability in the sense that the delivered quantity is stochastically larger after improvement. We develop a two-stage model that encompasses supplier improvement, uncertain supply and random demand in a competitive setting. In this complex model, we characterize the manufacturers’ equilibrium in-stock probability. Moreover, we characterize sufficient conditions for the existence of the equilibrium of the manufacturers’ improvement efforts. Finally, we numerically test the impact of market characteristics on the manufacturers’ equilibrium improvement efforts. We find that a manufacturer’s equilibrium improvement effort usually declines in market competition, market uncertainty or spillover effect, although its expected equilibrium profit typically increases in spillover effect.     

A note on the equilibria of an economic model with local competition “à la Cournot”

Guirao and Rubio (2010) [12] introduce an economic model, which generalizes the classical duopoly of Cournot type, where the competitors are located around a circle or a line and each firm competes “à la Cournot” with its right and left neighboring. For the case of having three and four players, we describe completely the bifurcations of equilibria in terms of the production costs of each firm and we study the stability of them. Moreover, for the case of four players we provide some information on the two-periodic orbits of the system.