Optimal production inventory policy for defective items with fuzzy time period

In this paper, an optimal production inventory model with fuzzy time period and fuzzy inventory costs for defective items is formulated and solved under fuzzy space constraint. Here, the rate of production is assumed to be a function of time and considered as a control variable. Also the demand is linearly stock dependent. The defective rate is taken as random, the inventory holding cost and production cost are imprecise. The fuzzy parameters are converted to crisp ones using credibility measure theory. The different items have the different imprecise time periods and the minimization of cost for each item leads to a multi-objective optimization problem. The model is under the single management house and desired inventory level and product cost for each item are prescribed. The multi-objective problem is reduced to a single objective problem using Global Criteria Method (GCM) and solved with the help of Fuzzy Riemann Integral (FRI) method, Kuhn–Tucker condition and Generalised Reduced Gradient (GRG) technique. In optimum results including production functions and corresponding optimum costs for the different models are obtained and then are presented in tabular forms.     

A point process approach to inventory models

The aim of the present paper is to make use of the modern theory of point processes to study optimal solutions for single‐item inventory. Demand for goods is assumed to occur according to a compound Poisson process and production occurs continuously and deterministically between times of demand, such that the inventory evolves according to a Markov process in continuous time. The aim is to propose a way of finding optimal production schemes by minimizing a certain expected loss over some finite period. There are holding/production costs depending on the stock level, and random penalty amounts will occur due to excess demand which is assumed backlogged. For simplicity we will not incorporate fixed costs. We give some numerical illustrations. Copyright © 2000 John Wiley & Sons, Ltd.     

Efficient algorithms for the offline variable sized bin-packing problem

We addresses a variant of the classical one dimensional bin-packing problem where several types of bins with unequal sizes and costs are presented. Each bin-type includes limited and/or unlimited identical bins. The goal is to minimize the total cost of bins needed to store a given set of items, each item with some space requirements. Four new heuristics to solve this problem are proposed, developed and compared. The experiments results show that higher quality solutions can be obtained using the proposed algorithms.     

Unbounded knapsack problem with controllable rates: the case of a random demand for items

This paper focuses on a dynamic, continuous-time control generalization of the unbounded knapsack problem. This generalization implies that putting items in a knapsack takes time and has a due date. Specifically, the problem is characterized by a limited production horizon and a number of item types. Given an unbounded number of copies of each type of item, the items can be put into a knapsack at a controllable production rate subject to the available capacity. The demand for items is not known until the end of the production horizon. The objective is to collect items of each type in order to minimize shortage and surplus costs with respect to the demand. We prove that this continuous-time problem can be reduced to a number of discrete-time problems. As a result, solvable cases are found and a polynomial-time algorithm is suggested to approximate the optimal solution with any desired precision.     

Heuristics with guaranteed performance bounds for a manufacturing system with product recovery

We consider a manufacturing system with product recovery. The system manufactures a new product as well as remanufactures the product from old, returned items. The items remanufactured with the returned products are as good as new and satisfy the same demand as the new item. The demand rate for the new item and the return rate for the old item are deterministic and constant. The relevant costs are the holding costs for the new item and the returned item, and the fixed setup costs for both manufacturing and remanufacturing. The objective is to determine the lot sizes and production schedule for manufacturing and remanufacturing so as to minimize the long-run average cost per unit time. We first develop a lower bound among all classes of policies for the problem. We then show that the optimal integer ratio policy for the problem obtains a solution whose cost is at most 1.5% more than the lower bound.     

Lot streaming for quality control in two-stage batch production

We consider a two-stage batch manufacturing process in which the first stage shifts out-of-control at iid exponential times after starting in control. To improve quality, a production batch at Stage 1 is subjected to lot streaming: it is divided into sublots that are processed at Stage 1 and then passed one-by-one to Stage 2 for simultaneous inspection and processing. In any sublot, Stage 1 produces good items before the shift and bad items after. The state of Stage 1 is known as soon as a bad item is encountered in Stage 2, at which time Stage 1 is re-set to the in-control state. We examine both cases of continuous first-stage and continuous second-stage production. For each case we examine both LIFO and FIFO inspection and processing policies at Stage 2. We use nonlinear programming to develop lot streaming policies which minimize the expected number of defective items for LIFO and FIFO policies. We also develop simple approximately optimal policies and compare the output performance of optimal, approximately optimal and equal-lot policies (when applicable) in a numerical example.     

A new dynamic programming procedure for three-staged cutting patterns

Three-staged patterns are often used to solve the 2D cutting stock problem of rectangular items. They can be divided into items in three stages: Vertical cuts divide the plate into segments; then horizontal cuts divide the segments into strips, and finally vertical cuts divide the strips into items. An algorithm for unconstrained three-staged patterns is presented, where a set of rectangular item types are packed into the plate so as to maximize the pattern value, and there is no constraint on the frequencies of each item type. It can be used jointly with the linear programming approach to solve the cutting stock problem. The algorithm solves three large knapsack problems to obtain the optimal pattern: One for the item layout on the widest strip, one for the strip layout on the longest segment, and the third for the segment layout on the plate. The computational results indicate that the algorithm is efficient.     

Polyhedral analysis for the two-item uncapacitated lot-sizing problem with one-way substitution

We consider a production planning problem for two items where the high quality item can substitute the demand for the low quality item. Given the number of periods, the demands, the production, inventory holding, setup and substitution costs, the problem is to find a minimum cost production and substitution plan. This problem generalizes the well-known uncapacitated lot-sizing problem. We study the projection of the feasible set onto the space of production and setup variables and derive a family of facet defining inequalities for the associated convex hull. We prove that these inequalities together with the trivial facet defining inequalities describe the convex hull of the projection if the number of periods is two. We present the results of a computational study and discuss the quality of the bounds given by the linear programming relaxation of the model strengthened with these facet defining inequalities for larger number of periods.     

An economic production lot size model in an imperfect production system

The paper investigates an EPL (Economic Production Lotsize) model in an imperfect production system in which the production facility may shift from an ‘in-control’ state to an ‘out-of-control’ state at any random time. The basic assumption of the classical EPL model is that 100% of produced items are perfect quality. This assumption may not be valid for most of the production environments. More specifically, the paper extends the article of Khouja and Mehrez [Khouja, M., Mehrez, A., 1994. An economic production lot size model with imperfect quality and variable production rate. Journal of the Operational Research Society 45, 1405–1417]. Generally, the manufacturing process is ‘in-control’ state at the starting of the production and produced items are of conforming quality. In long-run process, the process shifts from the ‘in-control’ state to the ‘out-of-control’ state after certain time due to higher production rate and production-run-time.The proposed model is formulated assuming that a certain percent of total product is defective (imperfect), in ‘out-of-control’ state. This percentage also varies with production rate and production-run time. The defective items are restored in original quality by reworked at some costs to maintain the quality of products in a competitive market. The production cost per unit item is convex function of production rate. The total costs in this investment model include manufacturing cost, setup cost, holding cost and reworking cost of imperfect quality products. The associated profit maximization problem is illustrated by numerical examples and also its sensitivity analysis is carried out.     

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.     

The multi-item single period problem with an initial stock of convertible units

This paper deals with the situation of a number of end items, each facing uncertain demand in a single period of interest. Besides being able to purchase units of the end items there is also available a stock of units that can be converted into end items but at unit costs that depend on the specific end item. Efficient solution procedures are presented for two situations: (i) where the end item demand distributions are assumed known (illustrated for the case of normally distributed demand) and (ii) a distribution free approach where only the first two moments of the distributions are assumed known. Computational results for a set of problems are presented.     

Joint Replenishment in Multi-Item Inventory Systems

A multi-item inventory system is considered which has the property that, for each single item, a reorder policy using the E.O.Q. formula would be appropriate. Holding costs are linear, and fixed ordering costs are assumed to be composed of a major set-up cost reflecting the pure fact of placing an order, and a sum of minor set-up costs corresponding to the items included in the order. If it is desirable to form a certain number of groups of items where all items of one group share the same order cycle, it is shown that there is always an optimal grouping in which items are arranged in increasing order of their ratio of yearly holding costs and minor set-up costs.A heuristic for forming the groups is given which turns out to be an optimal algorithm for the case that there are no major set-up costs. After an initial sorting of ratios, the worst-case complexity of this procedure is linear in the number of items.     

Pooling of spare parts between multiple users: How to share the benefits?

Companies that maintain capital goods (e.g., airplanes or power plants) often face high costs, both for holding spare parts and due to downtime of their technical systems. These costs can be reduced by pooling common spare parts between multiple companies in the same region, but managers may be unsure about how to share the resulting costs or benefits in a fair way that avoids free riders. To tackle this problem, we study several players, each facing a Poisson demand process for an expensive, low-usage item. They share a stock point that is controlled by a continuous-review base stock policy with full backordering under an optimal base stock level. Costs consist of penalty costs for backorders and holding costs for on-hand stock. We propose to allocate the total costs proportional to players’ demand rates. Our key result is that this cost allocation rule satisfies many appealing properties: it makes all separate participants and subgroups of participants better off, it stimulates growth of the pool, it can be easily implemented in practice, and it induces players to reveal their private information truthfully. To obtain these game theoretical results, we exploit novel structural properties of the cost function in our (S − 1, S) inventory model.     

Multi-item inventory control with full truckloads: A comparison of aggregate and individual order triggering

In this paper, we consider the stochastic joint replenishment problem in an environment where transportation costs are dominant and full truckloads or full container loads are required. One replenishment policy, taking into account capacity restrictions of the total order volume, is the so-called QS policy, where replenishment orders are placed to raise the individual inventory positions of all items to their order-up-to levels, whenever the aggregate inventory position drops below the reorder level. We first provide a method to compute the policy parameters of a QS policy such that item target service levels can be met, under the assumption that demand can be modeled as a compound renewal process. The approximation formulas are based on renewal theory and are tested in a simulation study which reveals good performance. Second, we compare the QS policy with a simple allocation policy where replenishment orders are triggered by the individual inventory positions of the items. At the moment when an individual inventory position drops below its item reorder level, a replenishment order is triggered and the total vehicle capacity is allocated to all items such that the expected elapsed time before the next replenishment order is maximized. In an extensive simulation study it is illustrated that the QS policy outperforms this allocation policy since it results in lower inventory levels for the same service level. Although both policies lead to similar performance if items are identical, it can differ substantially if the item characteristics vary.     

A production–recycling–inventory system with imprecise holding costs

This paper presents an optimal control recycling production inventory system in fuzzy environment. The used items are bought back and then either put on recycling or disposal. Recycled products can be used for the new products which are sold again. Here, the rate of production, recycling and disposal are assumed to be function of time and considered as control variables. The demand inversely depends on the selling price. Again selling price is serviceable stock dependent. The holding costs (for serviceable and non-serviceable items) are fuzzy variables. At first we define the expected values of fuzzy variable, then the system is transferred to the fuzzy expected value model. In this paper, an optimal control approach is proposed to optimize the production, recycling and disposal strategy with respect so that expected value of total profit is maximum. The optimum results are presented both in tabular form and graphically.     

A multi-item newsvendor problem with preseason production and capacitated reactive production

Given items with short life cycles or seasonal demands, one can potentially improve profits by producing during the selling season, especially when its production capacity is substantial. We develop a two-stage, multi-item model incorporating reactive production that employs a firm’s internal capacity. Production occurs in an uncapacitated preseason stage and a capacitated reactive stage. Demands occur in the reactive stage. Reactive capacities are pre-allocated to each item in the preseason stage and cannot be changed during the reactive stage. Reactive production occurs during the selling season with full knowledge of demands. The objective is expected profit maximization. Unsatisfied demand is lost. The revenue, salvage value, and production and lost sales costs are proportional. Assuming no fixed costs, we present a simple algorithm for computing optimal policies. For a model with fixed costs for allocating preseason stage production and reactive stage capacity to product families, we characterize optimal policies and develop optimal and heuristic algorithms.     

A queueing model of delayed product differentiation

We consider a production system in which a supplier produces semi-finished items on a make-to-stock basis for a manufacturer that will customize the items on a make-to-order basis. The proportion of total processing time undertaken by the supplier determines how suitable the semi-finished items will be to meet customer demand. The manufacturer wishes to determine the optimal point of differentiation (the proportion of processing completed by the supplier) and its optimal semi-finished goods buffer size. We use matrix geometric methods to evaluate various performance measures for this system, and then, with enumeration techniques, obtain optimal solutions. We find that delayed product differentiation is attractive when the manufacturer can balance the costs of customer order fulfillment delay with the costs associated with unsuitable items.     

A stocking policy for spare part provisioning under age based preventive replacement

A new policy, called stocking policy for ease of reference, has been advanced for joint optimization of age replacement and spare provisioning. It combines age replacement policy with continuous review (s, S) type inventory policy, where s is the stock reorder level and S is the maximum stock level. The policy is generally applicable to any operating situation having either a single item or a number of identical items. A simulation model has been developed to determine the optimal values of the decision variables by minimizing the total cost of replacement and inventory. The behaviour of the stocking policy has been studied for a number of case problems specifically constructed by 5-factor second order rotatory design and the effects of different cost elements and item failure characteristics have been highlighted. For all case problems, optimal (s, S) policies to-support the Barlow-Proschan age policy have also been determined. Simulation results clearly indicate that the optimal stocking policy is, in general, more cost-effective than the Barlow-Proschan policy.     

Base stock policies for production/inventory problems with uncertain capacity levels

In this paper we consider a single item, stochastic demand production/inventory problem where the maximum amount that can be produced (or ordered) in any given period is assumed to be uncertain. Inventory levels are reviewed periodically. The system operates under a stationary modified base stock policy. The intent of our paper is to present a procedure for computing the optimal base stocl level of this policy under expected average cost per period criterion. This procedure would provide guidance as to the appropriate amount of capacity to store in the form of inventory in the face of stochastic demand and uncertain capacity. In achieving this goal, our main contribution is to establish the analogy between the class of base stock production/inventory policies that operate under demand/capacity uncertainty, and the G/G/1 queues and their associated random walks. We also present example derivations for some important capacity distributions.     

A STOCHASTIC FEEDBACK MODEL FOR OPTIMAL MANAGEMENT OF RENEWABLE RESOURCES

Analytical expressions for optimal harvest of a renewable resource stock which is subject to a stochastic process are found. These expressions give the optimal harvest as an explicit feedback control law. All relations in the model, including the stochastic process, may be arbitrary functions of the state variable (stock). The objective function, however, is at most a quadratic function in the control variable (yield). A quadratic objective function includes the cases of downward sloping demand and increasing marginal costs which are the most common sources for nonlinearities in the economic part of the model. When it is assumed that there is a moratorium on harvest for stock sizes below a certain level (biological barrier), it is shown that the barrier requirements influence the optimal harvest paths throughout.