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.     

Two storage inventory model in a mixed environment

Multi-item inventory models with two storage facility and bulk release pattern are developed with linearly time dependent demand in a finite time horizon under crisp, stochastic and fuzzy-stochastic environments. Here different inventory parameters—holding costs, ordering costs, purchase costs, etc.—are assumed as probabilistic or fuzzy in nature. In particular cases stochastic and crisp models are derived. Models are formulated as profit maximization principle and three different approaches are proposed for solution. In the first approach, fuzzy extension principle is used to find membership function of the objective function and then it’s Graded Mean Integration Value (GMIV) for different optimistic levels are taken as equivalent stochastic objectives. Then the stochastic model is transformed to a constraint multi-objective programming problem using Stochastic Non-linear Programming (SNLP) technique. The multi-objective problems are transferred to single objective problems using Interactive Fuzzy Satisfising (IFS) technique. Finally, a Region Reducing Genetic Algorithm (RRGA) based on entropy has been developed and implemented to solve the single objective problems. In the second approach, the above GMIV (which is stochastic in nature) is optimized with some degree of probability and using SNLP technique model is transferred to an equivalent single objective crisp problem and solved using RRGA. In the third approach, objective function is optimized with some degree of possibility/necessity and following this approach model is transformed to an equivalent constrained stochastic programming problem. Then it is transformed to an equivalent single objective crisp problem using SNLP technique and solved via RRGA. The models are illustrated with some numerical examples and some sensitivity analyses have been presented.     

A Multi-Objective Production Inventory Model with Backorder for Fuzzy Random Demand Under Flexibility and Reliability

In this paper, an Economic Production Quantity (EPQ) model is developed with flexibility and reliability consideration of production process in an imprecise and uncertain mixed environment. The model has incorporated fuzzy random demand, an imprecise production preparation time and shortage. Here, the setup cost and the reliability of the production process along with the backorder replenishment time and production run period are the decision variables. Due to fuzzy-randomness of the demand, expected average demand is a fuzzy quantity and also imprecise preparation time is represented by fuzzy number. Therefore, both are first transformed to a corresponding interval number and then using the interval arithmetic, the single objective function for expected profit over the time cycle is changed to respective multi-objective functions. Due to highly nonlinearity of the expected profit functions it is optimized using a multi-objective genetic algorithm (MOGA). The associated profit maximization problem is illustrated by numerical examples and also its sensitivity analysis is carried out.     

An interactive method for inventory control with fuzzy lead-time and dynamic demand

Normally, the real-world inventory control problems are imprecisely defined and human interventions are often required to solve these decision-making problems. In this paper, a realistic inventory model with imprecise demand, lead-time and inventory costs have been formulated and an inventory policy is proposed to minimize the cost using man–machine interaction. Here, demand increases with time at a decreasing rate. The imprecise parameters of lead-time, inventory costs and demand are expressed through linear/non-linear membership functions. These are represented by different types of membership functions, linear or quadratic, depending upon the prevailing supply condition and marketing environment. The imprecise parameters are first transformed into corresponding interval numbers and then following the interval mathematics, the objective function for average cost is changed into respective multi-objective functions. These functions are minimized and solved for a Pareto-optimum solution by interactive fuzzy decision-making procedure. This process leads to man–machine interaction for optimum and appropriate decision acceptable to the decision maker’s firm. The model is illustrated numerically and the results are presented in tabular forms.     

Intuitionistic fuzzy optimization technique for Pareto optimal solution of manufacturing inventory models with shortages

This paper discusses a manufacturing inventory model with shortages where carrying cost, shortage cost, setup cost and demand quantity are considered as fuzzy numbers. The fuzzy parameters are transformed into corresponding interval numbers and then the interval objective function has been transformed into a classical multi-objective EPQ (economic production quantity) problem. To minimize the interval objective function, the order relation that represents the decision maker’s preference between interval objective functions has been defined by the right limit, left limit, center and half width of an interval. Finally, the transformed problem has been solved by intuitionistic fuzzy programming technique. The proposed method is illustrated with a numerical example and Pareto optimality test has been applied as well.     

A multi-objective wholesaler–retailers inventory-distribution model with controllable lead-time based on probabilistic fuzzy set and triangular fuzzy number

This paper develops a single wholesaler and multi retailers mixture inventory distribution model for a single item involving controllable lead-time with backorder and lost sales. The retailers purchase their items from the wholesaler in lots at some intervals throughout the year to meet the customers’ demand. Not to loose the demands, the retailers offer a price discount to the customers on the stock-out items. Here, it is assumed that the lead-time demands of retailers are uncertain in both stochastic and fuzzy sense, i.e., these are simultaneously random and imprecise. To implement this behavior of the lead-time demands, at first, these demands are assumed to be random, say following a normal distribution. With these random demands, the expected total cost for each retailer is obtained. Now, the mean lead-time demands (which are crisp ones) of the retailers are fuzzified. This fuzzy nature of the lead-time demands implies that the annual average demands of the retailers must be fuzzy numbers, suppose these are triangular fuzzy numbers. Using signed distance technique for defuzzification, the estimate of total costs for each retailer is derived. Therefore, the problem is reduced to optimize the crisp annual costs of wholesaler and retailers separately. The multi-objective model is solved using Global Criteria method. Numerical illustrations have been made with the help of an example taking two retailers into consideration. Mathematical analyses have been made for global pareto-optimal solutions of the multi-objective optimization problem. Sensitivity analyses have been made on backorder ratio and pareto-optimal solutions for wholesaler and different retailers are compared graphically.     

A deteriorating multi-item inventory model with fuzzy costs and resources based on two different defuzzification techniques

Normally inventory models of deteriorating items, such as food products, vegetables, etc. involve imprecise parameters, like imprecise inventory costs, fuzzy storage area, fuzzy budget allocation, etc. In this paper, we aim to provide two defuzzification techniques for two fuzzy inventory models using (i) extension principle and duality theory of non-linear programming and (ii) interval arithmetic. On the basis of Zadeh’s extension principle, two non-linear programs parameterized by the possibility level α are formulated to calculate the lower and upper bounds of the minimum average cost at α-level, through which the membership function of the objective function is constructed. In interval arithmetic technique the interval objective function has been transformed into an equivalent deterministic multi-objective problem defined by the left and right limits of the interval. This formulation corresponds to the possibility level, α = 0.5. Finally, the multi-objective problem is solved by a multi-objective genetic algorithm (MOGA). The model has been illustrated through a numerical example and solved for different values of possibility level, α through extension principle and for α = 0.5 via MOGA. As a particular case, the results have been obtained for the inventory model without deterioration. Results from two methods for α = 0.5 are compared.     

3.1. OR in banking

In this paper, we describe a deterministic multiperiod capacity expansion model in which a single facility serves the demand for many products. Potential applications for the model can be found in the capacity expansion planning of communication systems as well as in the production planning of heavy process industries. The model assumes that each capacity unit simultaneously serves a prespecified (though not necessarily integer) number of demand units of each product. Costs considered include capacity expansion costs, idle capacity holding costs, and capacity shortage costs. All cost functions are assumed to be nondecreasing and concave. Given the demand for each product over the planning horizon, the objective is to find the capacity expansion policy that minimizes the total cost incurred. We develop a dynamic programming algorithm that finds optimal policies. The required computational effort is a polynomial function of the number of products and the number of time periods. When the number of products equals one, the algorithm reduces to the well-known algorithm for the classical dynamic lot size problem.     

Possibility linear programming with trapezoidal fuzzy numbers

Fuzzy linear programming with trapezoidal fuzzy numbers (TrFNs) is considered and a new method is developed to solve it. In this method, TrFNs are used to capture imprecise or uncertain information for the imprecise objective coefficients and/or the imprecise technological coefficients and/or available resources. The auxiliary multi-objective programming is constructed to solve the corresponding possibility linear programming with TrFNs. The auxiliary multi-objective programming involves four objectives: minimizing the left spread, maximizing the right spread, maximizing the left endpoint of the mode and maximizing the middle point of the mode. Three approaches are proposed to solve the constructed auxiliary multi-objective programming, including optimistic approach, pessimistic approach and linear sum approach based on membership function. An investment example and a transportation problem are presented to demonstrate the implementation process of this method. The comparison analysis shows that the fuzzy linear programming with TrFNs developed in this paper generalizes the possibility linear programming with triangular fuzzy numbers.     

A two warehouse inventory model for a deteriorating item with partially/fully backlogged shortage and fuzzy lead time

An optimization inventory policy for a deteriorating item with imprecise lead-time, partially/fully backlogged shortages and price dependent demand is developed under two-warehouse system. For display and storage, the retailer hires one warehouse of finite capacity at market place, treated as own warehouse (OW) and another warehouse of large capacity as it may be required at a distance place from the market, treated as rented warehouse (RW). Holding cost at RW decreases with the increase of distance from the market place. Units are transferred from RW to OW in bulk release pattern and sold from OW. Using the nearest interval approximation method the estimated fuzzy average profit function is defuzzified and transformed to multiple crisp objective functions which are solved by Global Criteria Method. The models are illustrated numerically. Sensitivity of the inventory costs on the location of RW has been depicted graphically. Also loss in profit due to deteriorations for both models have been presented.     

Two storage inventory problem with dynamic demand and interval valued lead-time over finite time horizon under inflation and time-value of money

A finite time horizon inventory problem for a deteriorating item having two separate warehouses, one is a own warehouse (OW) of finite dimension and other a rented warehouse (RW), is developed with interval-valued lead-time under inflation and time value of money. Due to different preserving facilities and storage environment, inventory holding cost is considered to be different in different warehouses. The demand rate of item is increasing with time at a decreasing rate. Shortages are allowed in each cycle and backlogged them partially. Shortages may or may not be allowed in the last cycle and under this circumstance, there may be three different types of model. Here it is assumed that the replenishment cycle lengths are of equal length and the stocks of RW are transported to OW in continuous release pattern. For each model, different scenarios are depicted depending upon the re-order point for the next lot. Representing the lead-time by an interval number and using the interval arithmetic, the single objective function for profit is changed to corresponding multi-objective functions. These functions are maximized and solved by Fast and Elitist Multi-objective Genetic Algorithm (FEMGA). The models are illustrated numerically and the results are presented in tabular form.     

Posynomial geometric programming with interval exponents and coefficients

Geometric programming provides a powerful tool for solving nonlinear problems where nonlinear relations can be well presented by an exponential or power function. In the real world, many applications of geometric programming are engineering design problems in which some of the problem parameters are estimates of actual values. This paper develops a solution method when the exponents in the objective function, the cost and the constraint coefficients, and the right-hand sides are imprecise and represented as interval data. Since the parameters of the problem are imprecise, the objective value should be imprecise as well. A pair of two-level mathematical programs is formulated to obtain the upper bound and lower bound of the objective values. Based on the duality theorem and by applying a variable separation technique, the pair of two-level mathematical programs is transformed into a pair of ordinary one-level geometric programs. Solving the pair of geometric programs produces the interval of the objective value. The ability of calculating the bounds of the objective value developed in this paper might help lead to more realistic modeling efforts in engineering optimization areas.     

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.     

An interval approach based on expectation optimization for fuzzy random bilevel linear programming problems

This paper considers a class of bilevel linear programming problems in which the coefficients of both objective functions are fuzzy random variables. The main idea of this paper is to introduce the Pareto optimal solution in a multi-objective bilevel programming problem as a solution for a fuzzy random bilevel programming problem. To this end, a stochastic interval bilevel linear programming problem is first introduced in terms of α-cuts of fuzzy random variables. On the basis of an order relation of interval numbers and the expectation optimization model, the stochastic interval bilevel linear programming problem can be transformed into a multi-objective bilevel programming problem which is solved by means of weighted linear combination technique. In order to compare different optimal solutions depending on different cuts, two criterions are given to provide the preferable optimal solutions for the upper and lower level decision makers respectively. Finally, a production planning problem is given to demonstrate the feasibility of the proposed approach.     

Multi-item inventory models with price dependent demand under flexibility and reliability consideration and imprecise space constraint: A geometric programming approach

In this paper, multi-item economic production quantity (EPQ) models with selling price dependent demand, infinite production rate, stock dependent unit production and holding costs are considered. Flexibility and reliability consideration are introduced in the production process. The models are developed under two fuzzy environments–one with fuzzy goal and fuzzy restrictions on storage area and the other with unit cost as fuzzy and possibility–necessity restrictions on storage space. The objective goal and constraint goal are defined by membership functions and the presence of fuzzy parameters in the objective function is dealt with fuzzy possibility/necessity measures. The models are formed as maximization problems. The first one—the fuzzy goal programming problem is solved using Fuzzy Additive Goal Programming (FAGP) and Modified Geometric Programming (MGP) methods. The second model with fuzzy possibility/necessity measures is solved by Geometric Programming (GP) method. The models are illustrated through numerical examples. The sensitivity analyses of the profit function due to different measures of possibility and necessity are performed and presented graphically.     

Inventory Positioning in Multiple Product Supply Chains

In this paper we address the problem of inventory positioning, i.e., the determination of the supply chain node where inventory should be held, to minimize holding costs given a pre-specified order fill rate. A single-echelon inventory system with multiple products models the problem. The value of inventory is assumed to be an increasing function of the amount of processing performed at upstream nodes, while achieved fill-rates are dependent on the distance or time between the inventory storage and customer locations. We propose a novel analytical approach to solve the problem for the case of normally distributed demand that is based on iterative calculations of inventory holding costs at the various potential inventory locations.     

Modeling fuzzy multi-period production planning and sourcing problem with credibility service levels

A great deal of research has been done on production planning and sourcing problems, most of which concern deterministic or stochastic demand and cost situations and single period systems. In this paper, we consider a new class of multi-period production planning and sourcing problem with credibility service levels, in which a manufacturer has a number of plants and subcontractors and has to meet the product demand according to the credibility service levels set by its customers. In the proposed problem, demands and costs are uncertain and assumed to be fuzzy variables with known possibility distributions. The objective of the problem is to minimize the total expected cost, including the expected value of the sum of the inventory holding and production cost in the planning horizon. Because the proposed problem is too complex to apply conventional optimization algorithms, we suggest an approximation approach (AA) to evaluate the objective function. After that, two algorithms are designed to solve the proposed production planning problem. The first is a PSO algorithm combining the AA, and the second is a hybrid PSO algorithm integrating the AA, neural network (NN) and PSO. Finally, one numerical example is provided to compare the effectiveness of the proposed two algorithms.     

Time and inventory dependent optimal maintenance policies for single machine workstations: An MDP approach

In this work the problem of obtaining an optimal maintenance policy for a single-machine, single-product workstation that deteriorates over time is addressed, using Markov Decision Process (MDP) models. Two models are proposed. The decision criteria for the first model is based on the cost of performing maintenance, the cost of repairing a failed machine and the cost of holding inventory while the machine is not available for production. For the second model the cost of holding inventory is replaced by the cost of not satisfying the demand. The processing time of jobs, inter-arrival times of jobs or units of demand, and the failure times are assumed to be random. The results show that in order to make better maintenance decisions the interaction between the inventory (whether in process or final), and the number of shifts that the machine has been working without restoration, has to be taken into account. If this interaction is considered, the long-run operational costs are reduced significantly. Moreover, structural properties of the optimal policies of the models are obtained after imposing conditions on the parameters of the models and on the distribution of the lifetime of a recently restored machine.     

Possibility and necessity constraints and their defuzzification—A multi-item production-inventory scenario via optimal control theory

In this paper, analogous to chance constraints, real-life necessity and possibility constraints in the context of a multi-item dynamic production-inventory control system are defined and defuzzified following fuzzy relations. Hence, a realistic multi-item production-inventory model with shortages and fuzzy constraints has been formulated and solved for optimal production with the objective of having minimum cost. Here, the rate of production is assumed to be a function of time and considered as a control variable. Also the present system produces some defective units along with the perfect ones and the rate of produced defective units is constant. Here demand of the good units is time dependent and known and the defective units are of no use. The space required per unit item, available storage space and investment capital are assumed to be imprecise. The space and budget constraints are of necessity and/or possibility types. The model is formulated as an optimal control problem and solved for optimum production function using Pontryagin’s optimal control policy, the Kuhn–Tucker conditions and generalized reduced gradient (GRG) technique. The model is illustrated numerically and values of demand, optimal production function and stock level are presented in both tabular and graphical forms. The sensitivity of the cost functional due to the changes in confidence level of imprecise constraints is also presented.     

A novel method for solving linear programming problems with symmetric trapezoidal fuzzy numbers

Linear programming (LP) is a widely used optimization method for solving real-life problems because of its efficiency. Although precise data are fundamentally indispensable in conventional LP problems, the observed values of the data in real-life problems are often imprecise. Fuzzy sets theory has been extensively used to represent imprecise data in LP by formalizing the inaccuracies inherent in human decision-making. The fuzzy LP (FLP) models in the literature generally either incorporate the imprecisions related to the coefficients of the objective function, the values of the right-hand-side, and/or the elements of the coefficient matrix. We propose a new method for solving FLP problems in which the coefficients of the objective function and the values of the right-hand-side are represented by symmetric trapezoidal fuzzy numbers while the elements of the coefficient matrix are represented by real numbers. We convert the FLP problem into an equivalent crisp LP problem and solve the crisp problem with the standard primal simplex method. We show that the method proposed in this study is simpler and computationally more efficient than two competing FLP methods commonly used in the literature.