Multiproduct production/inventory control under random demands

Studies the optimal production/inventory control policy for a single machine multiproduct production system. The machine produces to fill the end-product inventory stock and the demand is satisfied from the inventory when available; unsatisfied demand is backlogged until the product becomes available as the result of production. For each product, the demand follows a Poisson process and the unit processing time is known. When the machine switches production from one product to another, it incurs a set-up time and a set-up cost. The relevant costs include the set-up cost, a cost per unit time while the machine is running, and linear costs for inventory and backlogging. This problem is modeled as a semi-Markov decision process using the criterion of minimizing expected total cost with discounting over an infinite horizon. Procedures for computing near-optimal policies and their error bounds are developed. The error bound given by the authors' procedure is shown to be much tighter than the one given by the “norm-based” approach. Computational test results are presented to show the structure of the near-optimal policy and how its accuracy is affected by the system characteristics such as capacity utilization and set-up time     

Solution of an imperfect‐quality EOQ model with backorder under fuzzy lock leadership game approach

This paper considers an economic order quantity (EOQ) inventory model for items with imperfect quality and shortage backordering under several styles of managerial leadership via lock fuzzy game theoretic approach. The decision maker (DM) controls several cost components by playing as Player 1 on the one side and the consumers who may accept/reject those items (unwilling to buy those commodities) stands as Player 2 on the other side. First of all, we develop a profit maximization backlogging EOQ model where the imperfect items are screened out batchwise. Because of the fuzzy flexibility of the model parameters we also develop a fuzzy mathematical model by considering the demand, all cost parameters, and other input parameters of the inventory system as triangular lock fuzzy numbers. Then we develop a 3 × 3 matrix game by applying a five‐stage leadership theory employing several key vectors in the model itself. The problem has been solved for crisp, general fuzzy models of several leadership styles also. Numerical results show that for a cooperative game, inventory profit function reaches its maximum rather than the noncooperative game by the use of proper keys. Finally, comparative study, sensitivity analysis, and graphical illustrations are made to justify the new approach.     

Lot size reorder point inventory model with controllable lead time and set-up cost

This paper deals with the lead time and set-up cost reductions problem on the modified lot size reorder point inventory model in which the production process is imperfect. We consider that the lead time can be shortened at an extra crashing cost, which depends on the length of lead time to be reduced and the ordering lot size. The option of investing in reducing set-up cost is also included. Two commonly used investment cost functional forms, logarithmic and power, are employed for set-up cost reduction. We assume that the stochastic demand during lead time follows a Normal distribution. The objective is simultaneously to optimize the lot size, reorder point, set-up cost and lead time. An algorithm of finding the optimal solution is developed, and two numerical examples are given to illustrate the results.     

Supply chain modeling in uncertain environment with bi-objective approach

Supply chain is viewed as a large-scale system that consists of production and inventory units, organized in a serial structure. Uncertainty is the main attribute in managing the supply chains. Managing a supply chain (SC) is very difficult, since various sources of uncertainty and complex interrelationships among various entities exist in the SC. Uncertainty may result from customer’s demand variability or unreliability in external suppliers. In this paper we develop an inventory model for an assembly supply chain network (each unit has at most one immediate successor, but any number of immediate predecessors) which fuzzy demand for single product in one hand and fuzzy reliability of external suppliers in other hand affect on determination of inventory policy in SCM. External supplier’s reliability has determined using a fuzzy expert system. Also the performance of supply chain is assessed by two criteria including total cost and fill rate. To solve this bi-criteria model, hybridization of multi-objective particle swarm optimization and simulation optimization is considered. Results indicate the efficiency of proposed approach in performance measurement.     

A single-period inventory model with fuzzy demand

The single-period inventory models have wide applications in the real world in assisting the decision maker to determine the optimal quantity to order. Due to lack of historical data, the demand has to be subjectively determined in many cases. In this paper, a single-period inventory model for cases of fuzzy demand is constructed. The costs considered include the procurement cost, shortage cost, and holding cost. For different fuzzy total cost resulted from different order quantity, a method for ranking fuzzy numbers is adopted to find the optimal order quantity in terms of the cost. When the profit gained from selling one item is less (greater) than the loss incurred due to one unsold item, the optimal order quantity lies in the range defined for the left-shape (right-shape) function of the fuzzy demand. If the unit profit is equal to the unit loss, then all quantities with a membership grade 1 are optimal to be ordered. The methodology of this paper can be applied to construct other inventory models with fuzzy demand.     

Economic lot sizing in a production system with random demand

An extended economic production quantity model that copes with random demand is developed in this paper. A unique feature of the proposed study is the consideration of transient shortage during the production stage, which has not been explicitly analysed in existing literature. The considered costs include set-up cost for the batch production, inventory carrying cost during the production and depletion stages in one replenishment cycle, and shortage cost when demand cannot be satisfied from the shop floor immediately. Based on renewal reward process, a per-unit-time expected cost model is developed and analysed. Under some mild condition, it can be shown that the approximate cost function is convex. Computational experiments have demonstrated that the average reduction in total cost is significant when the proposed lot sizing policy is compared with those with deterministic demand.     

Incorporating human learning into a fuzzy EOQ inventory model with backorders

Even though publications on fuzzy inventory problems are constantly increasing, modelling the decision maker’s characteristics and their effect on his/her decisions and consequently on the planning outcome has not attracted much attention in the literature. In order to fill this research gap and model reality more accurately, this paper develops a new fuzzy EOQ inventory model with backorders that considers human learning over the planning horizon. The paper is an extension of an existing EOQ inventory model with backorders in which both demand and lead times are fuzzified. Here, the assumption of constant fuzziness is relaxed by incorporating the concept of learning in fuzziness into the model considering that the degree of fuzziness reduces over the planning horizon. The proposed fuzzy EOQ inventory model with backorders and learning in fuzziness has a good performance in efficiency. Finally, it is worth mentioning that learning in fuzziness decreases the total inventory cost.     

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.     

按单装配系统中组件生产和库存分配控制策略研究

针对由两种组件、三类顾客需求组成的按单装配系统, 本文研究了其中的组件生产控制与库存分配问题. 在各类顾客需 求是泊松到达过程, 各种组件加工时间服从指数分布的假设下, 我们运用马尔科夫决策理论建立了无限期折扣总成本模型, 根据Lippman转换得到了相应归一化后的离散最优方程, 在此基础之上分析了生产和库存分配联合最优控制策略的结构性质. 本文证明了最优策略是依赖于系统状态的动态策略. 组件的最优生产策略是动态基库存策略, 其中基库存水平是关于系统中其他组件库存水平的非减函数. 而最优的分配策略是动态的阈值策略, 对于只需一种组件构成的顾客需求, 组件的分配阈值是系统中另一组件库存水平的增函数; 而对于同时需要两种组件组成的顾客需求, 其各组件的分配阈值是另一组件库存水平的减函数. 最后通过数值试验给出了各个参数对联合最优控制策略的影响, 并得到了相应的管理启示.     

Continuous review inventory models with a mixture of backorders and lost sales under fuzzy demand and different decision situations

In this paper, continuous review inventory models in which a fraction of demand is backordered and the remaining fraction is lost during the stock out period are considered under fuzzy demands. In order to find the optimal decision under different situations, two decision methods are proposed. The first one is finding a minimum value of the expected annual total cost, and the second one is maximizing the credibility of an event that the total cost in the planning periods does not exceed a certain budget level. For the first decision method, an approach of ranking fuzzy numbers by their possibilistic mean value is adopted to achieve the optimal solution. For the second one, the technique of fuzzy simulation and differential evolution algorithms are integrated to design hybrid intelligent algorithms to solve the fuzzy models. Subsequently, the two decision models are compared and some advices about inventory cash flow management are given. Further, sensitivity analysis is conducted to give more general situations to illustrate the rationality of the management advices.     

A hybrid fuzzy goal programming approach with different goal priorities to aggregate production planning

In this study a hybrid (including qualitative and quantitative objectives) fuzzy multi objective nonlinear programming (H-FMONLP) model with different goal priorities will be developed for aggregate production planning (APP) problem in a fuzzy environment. Using an interactive decision making process the proposed model tries to minimize total production costs, carrying and back ordering costs and costs of changes in workforce level (quantitative objectives) and maximize total customer satisfaction (qualitative objective) with regarding the inventory level, demand, labor level, machines capacity and warehouse space. A real-world industrial case study demonstrates applicability of proposed model to practical APP decision problems. GENOCOP III (Genetic Algorithm for Numerical Optimization of Constrained Problems) has been used to solve final crisp nonlinear programming problem.     

Fuzzy opportunity cost for EOQ model with quality improvement investment

The cost of capital (i.e. opportunity cost) is one of the key factors that will influence the inventory and investment decisions. Previously, the classical EOQ model has been extended to include an imperfect production process and quality improvement investment, where the opportunity cost rate (interest rate or discounted rate) for evaluating the cost of capital investment is known with certainty. However, in some practical situations, the opportunity cost rate probably incurs disturbance due to the unstable environments. To capture this reality, this paper attempts to combine the statistical technique and fuzzy sets concept to deal with the unstable opportunity cost rate, so as to modify the aforementioned inventory/investment model. We derive the optimal lot size and the optimal process quality level in the fuzzy sense utilizing the logarithmic investment cost function. A numerical example is provided, and the results are compared with those obtained from a crisp opportunity cost rate model.     

Modeling and analysis of single item multi-period procurement lot-sizing problem considering rejections and late deliveries

Integer linear programming approach has been used to solve a multi-period procurement lot-sizing problem for a single product that is procured from a single supplier considering rejections and late deliveries under all-unit quantity discount environment. The intent of proposed model is two fold. First, we aim to establish tradeoffs among cost objectives and determine appropriate lot-size and its timing to minimize total cost over the decision horizon considering quantity discount, economies of scale in transactions and inventory management. Second, the optimization model has been used to analyze the effect of variations in problem parameters such as rejection rate, demand, storage capacity and inventory holding cost for a multi-period procurement lot-sizing problem. This analysis helps the decision maker to figure out opportunities to significantly reduce cost. An illustration is included to demonstrate the effectiveness of the proposed model. The proposed approach provides flexibility to decision maker in multi-period procurement lot-sizing decisions through tradeoff curves and sensitivity analysis.     

Implementing an optimal policy for set-up time reduction in an economic production quantity model

An economic production quantity (EPQ) system consisting of single and multiple items with an optimal policy of set-up time reduction and a fixed increment cost are discussed in the present study. The set-up time reduction ratio as a decision variable under various cases of demand in the EPQ model is considered. It is assumed that the set-up cost is linearly related to the set-up time. The set-up time reduction ratio and the lot size are solved simultaneously to obtain an optimal value of the total annual cost. Numerical examples are presented to demonstrate the accuracy of the proposed method.     

A production inventory model with stock dependent demand incorporating learning and inflationary effect in a random planning horizon: A fuzzy genetic algorithm with varying population size approach

A production inventory model for a newly launched product is developed incorporating inflation and time value of money. It is assumed that demand of the item is displayed stock dependent and lifetime of the product is random in nature and follows exponential distribution with a known mean. Here learning effect on production and setup cost is incorporated. Model is formulated to maximize the expected profit from the whole planning horizon. Following [Last, M. & Eyal, S. (2005). A fuzzy-based lifetime extension of genetic algorithms. Fuzzy Sets and Systems, 149, 131–147], a genetic algorithm (GA) with varying population size is used to solve the model where crossover probability is a function of parent’s age-type (young, middle-aged, old, etc.) and is obtained using a fuzzy rule base and possibility theory. In this GA a subset of better children is included with the parent population for next generation and size of this subset is a percentage of the size of its parent set. This GA is named fuzzy genetic algorithm (FGA) and is used to make decision for above production inventory model in different cases. The model is illustrated with some numerical data. Sensitivity analysis on expected profit function is also presented. Performance of this GA with respect to some other GAs are compared.     

Modelling the effect of demand variations on the performance of a production system

Modelling the effect of demand variations on a production system manufacturing multiple products is discussed. The various system costs involved in the production system, namely set-up cost and inventory cost incurred due to change in demands for the products with respect to products and planning periods are estimated. A statistical modelling is presented for determining the production capacity and inventory level requirement to satisfy the customer to a certain level decided by the management. Two important factors, (i) number of types of products and (ii) multiple planning horizons are considered to identify the costs as well as the production capacity and inventory level requirements. A statistical method, analysis of variance (ANOVA) is used to study the variations in the demands and costs involved. Finally, an example is presented to explain the application and the behaviour of the statistical model.     

Suitable application situations of different postponement approaches: Standardization vs. modularization

Although postponement benefits manufacturers by increasing flexibility and reducing inventory and product complexity, this strategy may not be suitable for all situations faced by manufacturers. This paper builds a cost model to examine the value of postponement for a firm with two products made in N$N$ stages and compares two different postponement approaches (that is, standardization and modularization) in terms of their costs in the presence of demand uncertainty. The considered trade-offs include processing costs, inventory costs, and the cost of product/process redesign. The analytical results provide two optimal decisions, the static decision and the dynamic decision, and suggest how firms choose a suitable postponement decision according their business environments. In addition, case studies with numerical examples are provided to illustrate the effectiveness of the model. The main contributions of this paper are: (1) the explanation of the effect of environment uncertainty on the decision to use a postponement strategy, (2) the definitions of some environmental circumstances that make postponement economical, and (3) the illustration of the cost structures under different postponement strategies.     

Two-echelon fuzzy stochastic supply chain for the manufacturer–buyer integrated production–inventory system

This paper deals with two-echelon integrated procurement production model for the manufacturer and the buyer integrated inventory system. The manufacturer procures raw material from outside suppliers (not a part of supply chain) then proceed to convert it as finished product, and finally delivers to the buyer, who faces imprecise and uncertain, called fuzzy random demand of customers. The manufacturer and the buyer work under joint channel, in which a centralized decision maker makes all decisions to optimize the joint total relevant cost (JTRC) of entire supply chain. In this account, in one production cycle of the manufacturer we determine an optimal multi-ordering policy for the buyer. To be part of this, we first derive the JTRC in stochastic framework, and then extend it in fuzzy stochastic environment. In order to scalarize the fuzzy stochastic JTRC, we use an evaluation method wherein randomness is estimated by probabilistic expectation and fuzziness is estimated by possibilistic mean based on possibility evaluation measure. To derive the optimal policies for both parties, an algorithm is proposed. A numerical illustration addresses the situations of paddy procurement, conversion to rice and fulfillment of uncertain demand of rice. Furthermore, sensitivity of parameters is examined to illustrate the model and algorithm.     

The Cournot game under a fuzzy decision environment

The conventional precision-based decision analysis methodology is invalid for business decision analysis when precise assessment data seldom exist. This paper considers the Cournot game with fuzzy demand and fuzzy costs that are assumed to be triangular fuzzy numbers. Our model utilizes the weighted center of gravity (WCoG) method to defuzzify the fuzzy profit function into a crisp value. We derive the equilibrium Cournot quantity of each firm by simultaneously solving the first-order condition of each firm. Our model explicitly derives the necessary condition to avoid an unreasonable outcome of negative equilibrium quantities and lack of flexibility for modification of the ranking method. In addition, we examine the standard deviation of the fuzzy profit resulting from the fuzziness of each firm’s cost and market demand functions. We conduct sensitivity analysis to investigate the effect of parameter perturbations on firms’ outcomes. The results indicate that the center of parameter plays an important role in sensitivity analysis and dominates over variations in equilibrium quantity due to a perturbation of fuzzy parameters.     

A mathematical model on EPQ for stochastic demand in an imperfect production system

In the paper, we develop an EPQ (economic production quantity) inventory model to determine the optimal buffer inventory for stochastic demand in the market during preventive maintenance or repair of a manufacturing facility with an EPQ (economic production quantity) model in an imperfect production system. Preventive maintenance, an essential element of the just-in-time structure, may cause shortage which is reduced by buffer inventory. The products are sold with the free minimal repair warranty (FRW) policy. The production system may undergo “out-of-control” state from “in-control” state, after a certain time that follows a probability density function. The defective (non-conforming) items in “in-control” or “out-of-control” state are reworked at a cost just after the regular production time. Finally, an expected cost function regarding the inventory cost, unit production cost, preventive maintenance cost and shortage cost is minimized analytically. We develop another case where the buffer inventory as well as the production rate are decision variables and the expected unit cost considering the above cost functions is optimized also. The numerical examples are provided to illustrate the behaviour and application of the model. Sensitivity analysis of the model with respect to key parameters of the system is carried out.