A fuzzy genetic algorithm with varying population size to solve an inventory model with credit-linked promotional demand in an imprecise planning horizon

A genetic algorithm (GA) with varying population size is developed where crossover probability is a function of parents’ age-type (young, middle-aged, old, etc.) and is obtained using a fuzzy rule base and possibility theory. It is an improved GA where 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 used to make managerial decision for an inventory model of a newly launched product. It is assumed that lifetime of the product is finite and imprecise (fuzzy) in nature. Here wholesaler/producer offers a delay period of payment to its retailers to capture the market. Due to this facility retailer also offers a fixed credit-period to its customers for some cycles to boost the demand. During these cycles demand of the item increases with time at a decreasing rate depending upon the duration of customers’ credit-period. Models are formulated for both the crisp and fuzzy inventory parameters to maximize the present value of total possible profit from the whole planning horizon under inflation and time value of money. Fuzzy models are transferred to deterministic ones following possibility/necessity measure on fuzzy goal and necessity measure on imprecise constraints. Finally optimal decision is made using above mentioned GA. Performance of the proposed GA on the model with respect to some other GAs are compared.     

An inventory model for a deteriorating item with displayed stock dependent demand under fuzzy inflation and time discounting over a random planning horizon

An inventory model for a deteriorating item (seasonal product) with linearly displayed stock dependent demand is developed in imprecise environment (involving both fuzzy and random parameters) under inflation and time value of money. It is assumed that time horizon, i.e., period of business is random and follows exponential distribution with a known mean. The resultant effect of inflation and time value of money is assumed as fuzzy in nature. The particular case, when resultant effect of inflation and time value is crisp in nature, is also analyzed. A genetic algorithm (GA) is developed with roulette wheel selection, arithmetic crossover, random mutation. For crisp inflation effect, the total expected profit for the planning horizon is maximized using the above GA to derive optimal inventory decision. On the other hand when inflationary effect is fuzzy then the above expected profit is fuzzy in nature too. Since optimization of fuzzy objective is not well defined, the optimistic/pessimistic return of the expected profit is obtained using possibility/necessity measure of fuzzy event. Fuzzy simulation process is proposed to determine this optimistic/pessimistic return. Finally a fuzzy simulation based GA is developed and is used to maximize the above optimistic/pessimistic return to get optimal decision. The models are illustrated with some numerical examples and some sensitivity analyses have been presented.     

Possibility and necessity representations of fuzzy inequality and its application to two warehouse production-inventory problem

In this paper, possibility and necessity representations of fuzzy inequality constraints are presented and then crisp versions of the constraints are derived. Here analogous to chance constraints, real-life necessity and possibility constraints in the context of two warehouse multi-item dynamic production-inventory control system are defined and defuzzified following fuzzy relations. Hence, a realistic two warehouse multi-item production-inventory model with fuzzy constraints has been formulated for a finite period of time and solved for optimal production with the objective of having maximum profit. The rate of production is unknown, assumed to be a function of time and considered as a control variable. Also the present system produces some defective units alongwith the perfect ones and the rate of produced defective units is stochastic in nature. Demand of the good units is stock dependent and known and the defective units are sold at a reduced price. The space required per unit item and available storage space are assumed to be imprecise. The inequality of budget constraints is also imprecise. The space and budget constraints are expressed as necessity and/or possibility types. The model is reduced to an equivalent deterministic model using fuzzy relations 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 pictorial forms.     

Two storage inventory model with fuzzy deterioration over a random planning horizon

An inventory model for a deteriorating item with stock dependent demand is developed under two storage facilities over a random planning horizon, which is assumed to follow exponential distribution with known parameter. For crisp deterioration rate, the expected profit is derived and maximized via genetic algorithm (GA). On the other hand, when deterioration rate is imprecise then optimistic/pessimistic equivalent of fuzzy objective function is obtained using possibility/necessity measure of fuzzy event. Fuzzy simulation process is proposed to maximize the optimistic/pessimistic return and finally fuzzy simulation-based GA is developed to solve the model. The models are illustrated with some numerical data. Sensitivity analyses on expected profit function with respect to distribution parameter λ and confidence levels α₁ and α₂ are also presented.     

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 note on chance constrained programming with fuzzy coefficients

This paper deals with nonlinear chance constrained programming as well as multiobjective case and goal programming with fuzzy coefficients occurring in not only constraints but also objectives. We also present a fuzzy simulation technique for handling fuzzy objective constraints and fuzzy goal constraints. Finally, a fuzzy simulation based genetic algorithm is employed to solve a numerical example.     

A class of multiobjective linear programming model with fuzzy random coefficients

The aim of this paper is to deal with a multiobjective linear programming problem with fuzzy random coefficients. Some crisp equivalent models are presented and a traditional algorithm based on an interactive fuzzy satisfying method is proposed to obtain the decision maker’s satisfying solution. In addition, the technique of fuzzy random simulation is adopted to handle general fuzzy random objective functions and fuzzy random constraints which are usually hard to be converted into their crisp equivalents. Furthermore, combined with the techniques of fuzzy random simulation, a genetic algorithm using the compromise approach is designed for solving a fuzzy random multiobjective programming problem. Finally, illustrative examples are given in order to show the application of the proposed models and algorithms.     

Solving the knapsack problem with imprecise weight coefficients using genetic algorithms

This paper investigates solving the knapsack problem with imprecise weight coefficients using genetic algorithms. This work is based on the assumption that each weight coefficient is imprecise due to decimal truncation or coefficient rough estimation by the decision-maker. To deal with this kind of imprecise data, fuzzy sets provide a powerful tool to model and solve this problem. We investigate the possibility of using genetic algorithms in solving the fuzzy knapsack problem without defining membership functions for each imprecise weight coefficient. The proposed approach simulates a fuzzy number by distributing it into some partition points. We use genetic algorithms to evolve the values in each partition point so that the final values represent the membership grade of a fuzzy number. The empirical results show that the proposed approach can obtain very good solutions within the given bound of each imprecise weight coefficient than the fuzzy knapsack approach. The fuzzy genetic algorithm concept approach is different, but gives better results than the traditional fuzzy 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.     

A new data envelopment analysis model with fuzzy random inputs and outputs

This paper first presents several formulas for mean chance distributions of triangular fuzzy random variables and their functions, then develops a new class of fuzzy random data envelopment analysis (FRDEA) models with mean chance constraints, in which the inputs and outputs are assumed to be characterized by fuzzy random variables with known possibility and probability distributions. According to the established formulas for the mean chance distributions, we can turn the mean chance constraints into their equivalent stochastic ones. On the other hand, since the objective in the FRDEA model is the expectation about the ratio of the weighted sum of outputs and the weighted sum of inputs for a target decision-making unite (DMU), for general fuzzy random inputs and outputs, we suggest an approximation method to evaluate the objective; and for triangular fuzzy random inputs and outputs, we propose a method to reduce the objective to its equivalent stochastic one. As a consequence, under the assumption that the inputs and the outputs are triangular fuzzy random vectors, the proposed FRDEA model can be reduced to its equivalent stochastic programming one, in which the constraints contain the standard normal distribution function, and the objective is the expectation for a function of the normal random variable. To solve the equivalent stochastic programming model, we design a hybrid algorithm by integrating stochastic simulation and genetic algorithm (GA). Finally, one numerical example is presented to demonstrate the proposed FRDEA modeling idea and the effectiveness of the designed hybrid algorithm.     

Two warehouse inventory models for single vendor multiple retailers with price and stock dependent demand

Here a single vendor multiple retailer inventory model of an item is developed where demand of the item at every retailer is linearly dependent on stock and inversely on some powers of selling price. Item is produced by the vendor and is distributed to the retailers following basic period policy. According to this policy item is replenished to the retailers at a regular time interval (T₁) called basic period (BP) and replenishment quantity is sufficient to last for the period T₁. Due to the scarcity of storage space at market places, every retailer uses a showroom at the market place and a warehouse to store the item, little away from the market place. Item is sold from the showroom and is filled up from the warehouse in a bulk release pattern. Some of the inventory parameters are considered as fuzzy in nature and model is formulated to maximize the average profit from the whole system. Imprecise objective is transformed to equivalent deterministic ones using possibility/necessity measure of fuzzy events with some degree of optimism/pessimism. A genetic algorithm (GA) is developed with roulette wheel selection, arithmetic crossover and random mutation and is used to solve the model. In some complex cases, with the help of above GA, fuzzy simulation process is used to derive the optimal decision. The model is illustrated through numerical examples and some sensitivity analyses are presented.     

Entropy maximization model for the trip distribution problem with fuzzy and random parameters

Many trip distribution problems can be modeled as entropy maximization models with quadratic cost constraints. In this paper, the travel costs per unit flow between different zones are assumed to be given fuzzy variables and the trip productions at origins and trip attractions at destinations are assumed to be given random variables. For this case, an entropy maximization model with chance constraint is proposed, and is proved to be convex. In order to solve this model, fuzzy simulation, stochastic simulation and a genetic algorithm are integrated to produce a hybrid intelligent algorithm. Finally, a numerical example is presented to demonstrate the application of the model and the algorithm.     

Mean-variance-skewness model for portfolio selection with fuzzy returns

Numerous empirical studies show that portfolio returns are generally asymmetric, and investors would prefer a portfolio return with larger degree of asymmetry when the mean value and variance are same. In order to measure the asymmetry of fuzzy portfolio return, a concept of skewness is defined as the third central moment in this paper, and its mathematical properties are studied. As an extension of the fuzzy mean-variance model, a mean-variance-skewness model is presented and the corresponding variations are also considered. In order to solve the proposed models, a genetic algorithm integrating fuzzy simulation is designed. Finally, several numerical examples are given to illustrate the modelling idea and the effectiveness of the proposed algorithm.     

Modelling redundancy allocation for a fuzzy random parallel-series system

Due to subjective judgment, imprecise human knowledge and perception in capturing statistical data, the real data of lifetimes in many systems are both random and fuzzy in nature. Based on the fuzzy random variables that are used to characterize the lifetimes, this paper studies the redundancy allocation problems to a fuzzy random parallel-series system.Two fuzzy random redundancy allocation models (FR-RAM) are developed through reliability maximization and cost minimization, respectively. Some properties of the FR-RAM are obtained, in which an analytical formula of reliability with convex lifetimes is derived and the sensitivity of the reliability is discussed. To solve the FR-RAMs, we first address the computation of reliability. A random simulation method based on the derived analytical formula is proposed to compute the reliability with convex lifetimes. As for the reliability with nonconvex lifetimes, the technique of fuzzy random simulation together with the discretization method of fuzzy random variable is employed to compute the reliability, and a convergence theorem of the fuzzy random simulation is proved. Subsequently, we integrate the computation approaches of the reliability and genetic algorithm (GA) to search for the approximately optimal redundancy allocation of the models. Finally, some numerical examples are provided to illustrate the feasibility of the solution algorithm and quantify its effectiveness.     

Fuzzy data envelopment analysis (DEA): Model and ranking method

Data Envelopment Analysis (DEA) is a very effective method to evaluate the relative efficiency of decision-making units (DMUs). Since the data of production processes cannot be precisely measured in some cases, the uncertain theory has played an important role in DEA. This paper attempts to extend the traditional DEA models to a fuzzy framework, thus producing a fuzzy DEA model based on credibility measure. Following is a method of ranking all the DMUs. In order to solve the fuzzy model, we have designed the hybrid algorithm combined with fuzzy simulation and genetic algorithm. When the inputs and outputs are all trapezoidal or triangular fuzzy variables, the model can be transformed to linear programming. Finally, a numerical example is presented to illustrate the fuzzy DEA model and the method of ranking all the DMUs.     

Fractional Goal Programming for Fuzzy Solid Transportation Problem with Interval Cost

In this paper, we study a solid transportation problem with interval cost using fractional goal programming approach (FGP). In real life applications of the FGP problem with multiple objectives, it is difficult for the decision-maker(s) to determine the goal value of each objective precisely as the goal values are imprecise, vague, or uncertain. Therefore, a fuzzy goal programming model is developed for this purpose. The proposed model presents an application of fuzzy goal programming to the solid transportation problem. Also, we use a special type of non-linear (hyperbolic) membership functions to solve multi-objective transportation problem. It gives an optimal compromise solution. The proposed model is illustrated by using an example.     

A hybrid intelligent algorithm for portfolio selection problem with fuzzy returns

Portfolio selection theory with fuzzy returns has been well developed and widely applied. Within the framework of credibility theory, several fuzzy portfolio selection models have been proposed such as mean–variance model, entropy optimization model, chance constrained programming model and so on. In order to solve these nonlinear optimization models, a hybrid intelligent algorithm is designed by integrating simulated annealing algorithm, neural network and fuzzy simulation techniques, where the neural network is used to approximate the expected value and variance for fuzzy returns and the fuzzy simulation is used to generate the training data for neural network. Since these models are used to be solved by genetic algorithm, some comparisons between the hybrid intelligent algorithm and genetic algorithm are given in terms of numerical examples, which imply that the hybrid intelligent algorithm is robust and more effective. In particular, it reduces the running time significantly for large size problems.     

An Algorithm for Solving Optimization Problems with One Linear Objective Function and Finitely Many Constraints of Fuzzy Relation Inequalities

An optimization model with one linear objective function and fuzzy relation equation constraints was presented by Fang and Li (1999) as well as an efficient solution procedure was designed by them for solving such a problem. A more general case of the problem, an optimization model with one linear objective function and finitely many constraints of fuzzy relation inequalities, is investigated in this paper. A new approach for solving this problem is proposed based on a necessary condition of optimality given in the paper. Compared with the known methods, the proposed algorithm shrinks the searching region and hence obtains an optimal solution fast. For some special cases, the proposed algorithm reaches an optimal solution very fast since there is only one minimum solution in the shrunk searching region. At the end of the paper, two numerical examples are given to illustrate this difference between the proposed algorithm and the known ones.     

Two-storage inventory model with lot-size dependent fuzzy lead-time under possibility constraints via genetic algorithm

Multi-item inventory models with stock dependent demand and two storage facilities are developed in a fuzzy environment where processing time of each unit is fuzzy and the processing time of a lot is correlated with its size. These are order-quantity reorder-point models with back-ordering if required. Here possibility and crisp constraints on investment and capacity of the small storehouse respectively are considered. The models are formulated as fuzzy chance constrained programming problem and is solved via generalized reduced gradient (GRG) technique when crisp equivalent of the constraints are available. A genetic algorithm (GA) is developed based on fuzzy simulation and entropy where region of search space gradually decreases to a small neighborhood of the optima and it is used to solve the models whenever the equivalent crisp form of the constraint is not available. The models are illustrated with some numerical examples and some sensitivity analyses have been done. For some particular cases results observed via GRG and GA are compared.     

Generalized minimal cost flow problem in fuzzy nature: An application in bus network planning problem

The aim of minimal cost flow problem (MCFP) in fuzzy nature, which is denoted with FMCFP, is to find the least cost of the shipment of a commodity through a capacitated network in order to satisfy imprecise concepts in supply or demand of network nodes and capacity or cost of network links. Fuzzy supply–demand may arise in real problems, where incomplete statistical data or simulation results are used. Also, variation in the cost or capacity of links is commonly happening. In the present paper, after defining a total order on LR type fuzzy numbers, three models are studied; MCFP with fuzzy costs, MCFP with fuzzy supply–demand and a combination of two cases. For the first model, scaling negative cycle cancelling algorithm, which is a polynomial time algorithm, is proposed. For the second model, “nominal flow” is introduced which provides an efficient scheme for finding fuzzy flow. For the third model, we present an exact and some heuristic methods. Numerical examples are illustrated to demonstrate the efficiency of the proposed schemes. Finally, an application of this viewpoint in bus network planning problem is provided.