Multi-item integrated supply chain model for deteriorating items with stock dependent demand under fuzzy random and bifuzzy environments

In this paper, we have investigated multi-item integrated production-inventory models of supplier and retailer with a constant rate of deterioration under stock dependent demand. Here we have considered supplier’s production cost as nonlinear function depending on production rate, retailers procurement cost exponentially depends on the credit period and suppliers transportation cost as a non-linear function of the amount of quantity purchased by the retailer. The models are optimized to get the value of the credit periods and total time of the supply chain cycle under the space and budget constraints. The models are also formulated under fuzzy random and bifuzzy environments. The ordering cost, procurement cost, selling price of retailer’s and holding costs, production cost, transportation cost, setup cost of the supplier’s and the total storage area and budget are taken in imprecise environments. To show the validity of the proposed models, few sensitivity analyses are also presented under the different rate of deterioration. The models are also discussed in non deteriorating items as a special case of the deteriorating items. The deterministic optimization models are formulated for minimizing the entire monetary value of the supply chain and solved using genetic algorithm (GA). A case study has been performed to illustrate those models numerically.     

A breakable multi-item multi stage solid transportation problem under budget with Gaussian type-2 fuzzy parameters

This paper is based on two mathematical models for multi-item multi-stage solid transportation problem with budget on total transportation cost in Gaussian type-2 fuzzy environment considering the fixed opening charge and operating cost in distribution center. The first model is about transportation of breakable/damageable items, and the second one considers non breakable/damageable items. The main aspect here is to develop the mathematical formulation of multi stage related solid transportation problem where several items are available for transportation. In order to deal with the Gaussian type-2 fuzziness, two chance-constrained programming models are developed based on generalized credibility measures for the objective function as well as the constraints sets with the help of the CV-based reductions method. Finally the reduced model is turned into its equivalent parametric programming problem. The problem is of high complexity and is difficult to find the optimal solution by any classical method and hence a time and space based meta-heuristic Genetic Algorithm has been proposed. Also the equivalent crisp models are solved using GA and LINGO 13.0 and after comparison, GA results are better. The proposed models and techniques are finally illustrated by providing numerical examples. Some sensitivity analysis and particular cases are presented and discussed. Degrees of efficiency is also evaluated for both the techniques.     

A deteriorating multi-item inventory model with price discount and variable demands via fuzzy logic under resource constraints

An inventory model of deteriorating seasonal products with Maximum Retail Price (MRP) for a wholesaler having showrooms at different places under a single management system is considered under random business periods with fuzzy resource constraints. The wholesaler replenishes the products instantaneously and earns commissions on MRP which vary with the ordered quantities following All Unit Discount (AUD), Incremental Quantity Discount (IQD) or IQD in AUD policy. Demand at showrooms are imprecise and related to selling prices by ‘verbal words’ following fuzzy logic. The wholesaler shares a part of commission with customers. The business periods follows normal distribution and converted to deterministic ones through chance constraint technique. The fuzzy space and budget constraints and fuzzy relations are defuzzified using possibility measures, surprise function and Mumdani fuzzy inference technique. The model is formulated as profit maximization for the wholesaler and solved using a real coded Genetic Algorithm (GA) and illustrated through some numerical examples and some sensitivity analysis. A real-life problem of a developing country is presented, solved using the above mentioned procedures and an appropriate inventory policy is suggested.     

Multi-objective inventory models of deteriorating items with some constraints in a fuzzy environment

In this paper, multi-item inventory models of deteriorating items with stock-dependent demand are developed in a fuzzy environment. Here, the objectives of maximizing the profit and minimizing the wastage cost are fuzzy in nature. Total average cost, warehouse space, inventory costs, purchasing and selling prices are also assumed to be vague and imprecise. The impreciseness in the above objective and constraint goals have been expressed by fuzzy linear membership functions and that in inventory costs and prices by triangular fuzzy numbers (TFN). Models have been solved by the fuzzy non-linear programming (FNLP) method based on Zimmerman [Zimmermann, H.-J., Fuzzy linear programming with several objective functions. Fuzzy Sets and Systems, 1978, 1, 46-55] and Lee and Li [Lee, E. S. and Li, R. J., Fuzzy multiple objective programming and compromise programming with Pareto optima. Fuzzy Sets and Systems, 1993, 53, 275-288]. These are illustrated with numerical examples and results of one model are compared with those obtained by the fuzzy additive goal programming (FAGP) [Tiwari, R. N., Dharmar, S. and Rao, J. R., Fuzzy goal programming: an additive model. Fuzzy Sets and Systems, 1987, 24, 27-34] method.     

Fully fuzzy fixed charge multi-item solid transportation problem

This paper presents fully fuzzy fixed charge multi-item solid transportation problems (FFFCMISTPs), in which direct costs, fixed charges, supplies, demands, conveyance capacities and transported quantities (decision variables) are fuzzy in nature. Objective is to minimize the total fuzzy cost under fuzzy decision variables. In this paper, some approaches are proposed to find the fully fuzzy transported amounts for a fuzzy solid transportation problem (FSTP). Proposed approaches are applicable for both balanced and unbalanced FFFCMISTPs. Another fuzzy fixed charge multi-item solid transportation problem (FFCMISTP) in which transported amounts (decision variables) are not fuzzy is also presented and solved by some other techniques. The models are illustrated with numerical examples and nature of the solutions is discussed.     

Multi-item inventory model of breakable items with stock-dependent demand under stock and time dependent breakability rate

Items made of glass, ceramic, etc. are normally stored in stacks and get damaged during the storage due to the accumulated stress of heaped stock. These items are known as breakable items. Here a multi-item inventory model of breakable items is developed, where demands of the items are stock dependent, breakability rates increase linearly with stock and nonlinearly with time. Due to non-linearity and complexity of the problem, the model is solved numerically and final decisions are made using Genetic Algorithm (GA). In a particular case, model is solved analytically as well as numerically and results are compared. Models are developed with both crisp and uncertain inventory costs. For uncertain inventory costs both fuzzy and stochastic parameters are considered. A chance constrained approach is followed to deal with simultaneous presence of stochastic and fuzzy parameters. Different numerical examples are used to illustrate the problem for different cases.     

A simplified new approach for solving fuzzy transportation problems with generalized trapezoidal fuzzy numbers

In a recent paper, Kaur and Kumar (2012) proposed a new method based on ranking function for solving fuzzy transportation problem (FTP) by assuming that the values of transportation costs are represented by generalized trapezoidal fuzzy numbers. Here it is shown that once the ranking function is chosen, the FTP is converted into crisp one, which is easily solved by the standard transportation algorithms. The main contribution here is the reduction of the computational complexity of the existing method. By solving two application examples, it is shown that it is possible to find a same optimal solution without solving any FTP. Since the proposed approach is based on classical approach it is very easy to understand and to apply on real life transportation problems for the decision makers.     

Necessity constraint in two plant optimal production problem with imprecise parameters

The purpose of this paper is to present and solve a real-life problem of two plants producing the same item under fuzzy-stochastic environment. Here, an item alongwith random defective units is produced at two different plants situated in different locations under a single management. The rates of demand, production and defectiveness at these places are different. Demands of the item are primarily met locally from the respective plants but if a stock-out situation occurs in a plant, immediately some stock, from the other plant if available, is rushed to the stock-out plant. The demands are known but production rates are unknown, functions of time are taken as control variables. The available budget for the management house is imprecise. The holding, shortage and transportation costs are assumed to be imprecise and represented by fuzzy numbers which are transformed to corresponding interval numbers. Following interval mathematics and nearest interval approximation, the objective function is changed to respective multi-objective functions and thus the single-objective fuzzy problem is reduced to a crisp multi-objective decision making (MODM) problem. The MODM problem is then again transformed to a single crisp objective function with the help of weighted sum method. Using fuzzy relations, the imprecise budget constraint expressed in the form of necessity constraint is transformed into an equivalent crisp one. Then, total cost consisting of production, holding, shortage and transportation (from one plant to another) costs is expressed as an optimal control problem and solved using weighted sum method, the Kuhn-Tucker conditions, Pontryagin’s Optimal Control principle and generalized reduced gradient (GRG) technique. The model has been illustrated by numerical data. The optimum results are presented in both tabular and graphical forms.     

Constrained fuzzy arithmetic approach to fuzzy transportation problems with fuzzy decision variables

Most of the existing methods for solving fully fuzzy mathematical programs are based on the standard fuzzy arithmetic operations and/or Zadeh's extension principle. These methods may produce questionable results for many real-life applications. Due to this fact, this paper presents a novel method based on the constrained fuzzy arithmetic concept to solve fully fuzzy balanced/unbalanced transportation problems in which all of the parameters (source capacities, demands of destinations, transportation costs etc.) as well as the decision variables (transportation quantities) are considered as fuzzy numbers. In the proposed method, the requisite crisp and/or fuzzy constraints between the base variables of the fuzzy components are provided from the decision maker according to his/her exact or vague judgments. Thereafter, fuzzy arithmetic operations are performed under these requisite constraints by taking into account the additional information while transforming the fuzzy transportation model into crisp equivalent form. Therefore, various fuzzy efficient solutions can be generated by making use of the proposed method according to the decision maker's risk attitude. In order to present the efficiency/applicability of the proposed method, different types of fully fuzzy transportation problems are generated and solved as illustrative examples. A detailed comparative study is also performed with other methods available in the literature. The computational analysis have shown that relatively more precise solutions are obtained from the proposed method for “risk-averse” and “partially risk-averse” decision makers. The proposed method also successfully provided fuzzy acceptable solutions for “risk seekers” with high degree of uncertainty similar to the other existing methods in the literature.     

Multi-objective solid transportation problems with budget constraint in uncertain environment

This paper investigates multi-objective solid transportation problems (MOSTP) under various uncertain environments. The unit transportation penalties/costs are taken as random, fuzzy and hybrid variables respectively, in three different uncertain multi-objective solid transportation models and in each case, the supplies, demands and conveyance capacities are fuzzy. Also, apart from source, demand and capacity constraints, an extra constraint on the total budget at each destination is imposed. Chance-constrained programming technique has been used for the first two models to obtain crisp equivalent forms, whereas expected value model is formulated for the last. We provide an another approach using the interval approximation of fuzzy numbers for the first model to obtain its crisp form and compare numerically two approaches for this model. Fuzzy programming technique and a gradient based optimisation - generalised reduced gradient (GRG) method are applied to beget the optimal solutions. Three numerical examples are provided to illustrate the models and programming.     

A fuzzy bi-criteria transportation problem

In this paper, a fuzzy bi-criteria transportation problem is studied. Here, the model concentrates on two criteria: total delivery time and total profit of transportation. The delivery times on links are fuzzy intervals with increasing linear membership functions, whereas the total delivery time on the network is a fuzzy interval with a decreasing linear membership function. On the other hand, the transporting profits on links are fuzzy intervals with decreasing linear membership functions and the total profit of transportation is a fuzzy number with an increasing linear membership function. Supplies and demands are deterministic numbers. A nonlinear programming model considers the problem using the max–min criterion suggested by Bellman and Zadeh. We show that the problem can be simplified into two bi-level programming problems, which are solved very conveniently. A proposed efficient algorithm based on parametric linear programming solves the bi-level problems. To explain the algorithm two illustrative examples are provided, systematically.     

Multi-item partial backlogging inventory models over random planninghorizon in random fuzzy environment

In this paper, some multi-item inventory models for deteriorating items are developed in a random planning horizon under inflation and time value money with space and budget constraints. The proposed models allow stock dependent consumption rate and partially backlogged shortages. Here the time horizon is a random variable with exponential distribution. The inventory parameters other than planning horizon are deterministic in one model and in the other, the deterioration and net value of the money are fuzzy, available budget and space are fuzzy and random fuzzy respectively. Fuzzy and random fuzzy constraints have been defuzzified using possibility and possibility–probability chance constraint techniques. The fuzzy objective function also has been defuzzified using possibility chance constraint against a goal. Both deterministic optimization problems are formulated for maximization of profit and solved using genetic algorithm (GA) and fuzzy simulation based genetic algorithm (FAGA). The models are illustrated with some numerical data. Results for different achievement levels are obtained and sensitivity analysis on expected profit function is also presented.Scope and purposeThe traditional inventory model considers the ideal case in which depletion of inventory is caused by a constant demand rate. However for more sale, inventory should be maintained at a higher level. Of course, this would result in higher holding or procurement cost, etc. Also, in many real situations, during a shortage period, the longer the waiting time is, the smaller the backlogging rate would be. For instance, for fashionable commodities and high-tech products with short product life cycle, the willingness for a customer to wait for backlogging diminishes with the length of the waiting time. Most of the classical inventory models did not take into account the effects of inflation and time value of money. But at present, the economic situation of most of the countries has been much deteriorated due to large scale inflation and consequent sharp decline in the purchasing power of money. So, it has not been possible to ignore the effects of inflation and time value of money any further. The purpose of this article is to maximize the expected profit of two inventory control systems in the random planning horizon.     

A hybrid fuzzy regression model and its application in hydrology engineering

The goal of this paper is to handle the large variation issues in fuzzy data by constructing a variable spread multivariate adaptive regression splines (MARS) fuzzy regression model with crisp parameters estimation and fuzzy error terms. It deals with imprecise measurement of response variable and crisp measurement of explanatory variables. The proposed method is a two-phase procedure which applies the MARS technique at phase one and an optimization problem at phase two to estimate the center and fuzziness of the response variable. The proposed method, therefore, handles two problems simultaneously: the problem of large variation issue and the problem of variation spreads in fuzzy observations. A realistic application of the proposed method is also presented, by which the suspended load is modeled using discharge in a hydrology engineering problem. Empirical results demonstrate that the proposed approach is more efficient and more realistic than some well-known least-squares fuzzy regression models.     

Intuitionistic fuzzy optimization technique for solving multi-objective reliability optimization problems in interval environment

In designing phase of systems, design parameters such as component reliabilities and cost are normally under uncertainties. This paper presents a methodology for solving the multi-objective reliability optimization model in which parameters are considered as imprecise in terms of triangular interval data. The uncertain multi-objective optimization model is converted into deterministic multi-objective model including left, center and right interval functions. A conflicting nature between the objectives is resolved with the help of intuitionistic fuzzy programming technique by considering linear as well as the nonlinear degree of membership and non-membership functions. The resultants max–min problem has been solved with particle swarm optimization (PSO) and compared their results with genetic algorithm (GA). Finally, a numerical instance is presented to show the performance of the proposed approach.     

Bi-objective optimization of the reliability-redundancy allocation problem for series-parallel system

The main objective of this paper is to solve the bi-objective reliability redundancy allocation problem for series-parallel system where reliability of the system and the corresponding designing cost are considered as two different objectives. In their formulation, reliability of each component is considered as a triangular fuzzy number. In order to solve the problem, developed fuzzy model is converted to a crisp model by using expected values of fuzzy numbers and taking into account the preference of decision maker regarding cost and reliability goals. Finally the obtained crisp optimization problem has been solved with particle swarm optimization (PSO) and compared their results with genetic algorithm (GA). Examples are shown to illustrate the method. Finally statistical simulation has been performed for supremacy the approach.     

A gamma type-2 defuzzification method for solving a solid transportation problem considering carbon emission

This paper intends to develop a multi-objective solid transportation problem considering carbon emission, where the parameters are of gamma type-2 fuzzy in nature. This paper proposed the defuzzification process for gamma type-2 fuzzy variable using critical value (CV ) and nearest interval approximation method. A chance constraint programming problem is generated using the CV based reduction method to convert the fuzzy problem to its equivalent crisp form. Applying the \(\alpha \)-cut based interval approximation method, a deterministic problem is developed. Some real life data are used to minimize the cost and carbon emission. LINGO standard optimization solver has been used to solve the multi-objective problem using weighted sum method and intuitionistic fuzzy programming technique. The Genetic Algorithm (GA) and Particle Swarm Optimization (PSO) algorithm are implemented to generate efficient optimal solution by converting the multi-objective problem to a single objective problem using penalty cost for carbon emission. After solving the problem, analysis on some particular cases has been presented. The sensitivity analysis has been shown to different credibility levels of cost, emission, source, demand, conveyance to find total cost, emission and transported amount in each level. A comparison study on the performance of three algorithms (LINGO, GA and PSO) is presented. At the end, some graphs have been plotted which shows the effect of emission with different emission parameters.     

Maximum entropy principle with imprecise side-conditions III: Crisp continuous solutions

In this paper we consider the maximum entropy principle with imprecise side-conditions, where the imprecise side-conditions are modeled as fuzzy sets. In two previous papers our solution produced: (1) fuzzy discrete probability distributions and fuzzy probability density functions; and (2) crisp discrete probability distributions. In this paper we consider only continuous probability density functions and we have the constraint that the solution must be crisp (non-fuzzy).     

A generalized approach to fuzzy optimization

A new approach to fuzzy optimization based on the generalization of Bellman-Zadeh's (BZ) concept is proposed in this article. It consists of a parametric generalization of intersection of fuzzy sets and a generalized defuzzification method. This approach allows the solving of a fuzzy mathematical programming (FMP) problem without transformation to a crisp one. It takes into account all possible fuzzy decisions and allows the degree of conjunction of criteria and constraints to vary. BZ method can be considered a special case of the approach proposed here. A simple algorithm for noniterative solving FMP problem is proposed whereas well-known Zimmermann's approach uses numerical methods. an illustrative example is presented. © 1994 John Wiley & Sons, Inc.     

Maximum entropy principle with imprecise side-conditions II: crisp discrete solutions

In this paper we consider the maximum entropy principle with imprecise side-conditions, where the imprecise side-conditions are modeled as fuzzy sets. In a previous paper our solution produced fuzzy discrete probability distributions and fuzzy probability density functions. In this paper we consider only discrete probability distributions and we have the constraint that the solution must be crisp (non-fuzzy).     

Demand and cost forecast error sensitivity analyses in aggregate production planning by possibilistic linear programming models

A production management system contains many imprecise natures. The conventional deterministic and/or stochastic model in a computer integrated production management system (CIPMS) may not capture the imprecise natures well. This study examines how the imprecise natures in the CIPMS affect the planning results. Possibilistic linear programming models are also proposed for the aggregate production planning problem with imprecise natures. The proposed model can adequately describe the imprecise natures in a production system and, in doing so, the CIPMS can adapt to a variety of non-crisp properties in an actual system. For comparison, the classic aggregate production planning problem given by Holt, Modigliani, and Simon (HMS) is solved using the proposed possibilistic model and the crisp model of Hanssmann and Hess (HH). Perturbing the cost coefficients and the demand allows one to simulate the imprecise natures of a real world and evaluate the effect of the imprecise natures to production plans by both the possibilistic and the crisp HH approaches. Experimental results indicate that the possibilistic model does provide better plans that can tolerate a higher spectrum of imprecise properties than those obtained by the crisp HH model.