A multi-item transportation problem with fuzzy tolerance

This paper presents the recently introduced modified subgradient method for optimization and its effectiveness in a fuzzy transportation model. Here a multi-item balanced transportation problem (MIBTP) is formulated where unit transportation costs are imprecise. Also available spaces and budgets at destinations are limited but imprecise. The objective is to find a shipment schedule for the items that minimizes the total cost subjected to imprecise warehouse and budget constraints at destinations. The proposed model is reduced to a multi-objective optimization problem using tolerances, then to a crisp single-objective one using fuzzy non-linear programming (FNLP) technique and Zimmermann's method. The above fuzzy MIBTP is also reduced to another form of deterministic one using modified sub-gradient method (MSM). These two crisp optimization problems are solved by Genetic Algorithm (GA). As an extension, fuzzy multi-item balanced solid transportation problems (STPs) with and without restrictions on some routes and items are formulated and reduced to deterministic ones following FNLP and Zimmermann's methods. These models are also solved by GA. Models are illustrated numerically, optimum results of fuzzy MIBTP from two deductions are compared. Results are also presented for different GA parameters.     

A two-phase approach for multi-objective programming problems with fuzzy coefficients

In this study, a two-phase procedure is introduced to solve multi-objective fuzzy linear programming problems. The procedure provides a practical solution approach, which is an integration of fuzzy parametric programming (FPP) and fuzzy linear programming (FLP), for solving real life multiple objective programming problems with all fuzzy coefficients. The interactive concept of the procedure is performed to reach simultaneous optimal solutions for all objective functions for different grades of precision according to the preferences of the decision-maker (DM). The procedure can be also performed to obtain lexicographic optimal and/or additive solutions if it is needed. In the first phase of the procedure, a family of vector optimization models is constructed by using FPP. Then in the second phase, each model is solved by FLP. The solutions are optimal and each one is an alternative decision plan for the DM.     

Fuzzy fixed charge solid transportation problem and algorithm

This paper mainly investigates the fixed charge solid transportation problem under fuzzy environment, in which the direct costs, the fixed charges, the supplies, the demands and the conveyance capacities are supposed to be fuzzy variables. As a result, several new models, i.e., expected value model, chance-constrained programming model and dependent-chance programming model, are constructed on the basis of credibility theory. After that, the crisp equivalences are also discussed for different models. In order to solve the models, hybrid intelligent algorithm is designed based on the fuzzy simulation technique and tabu search algorithm. Finally, two application results are given to show the applications of the models and algorithm.     

Uncertain goal programming models for bicriteria solid transportation problem

This paper investigates an uncertain bicriteria solid transportation problem. The supplies, demands, conveyance capacities, transportation cost and transportation time are regarded as uncertain variables. According to two types of methods to rank the uncertain variables, expected value goal programming model and chance-constrained goal programming model for the bicriteria solid transportation problem are constructed. It is proved that the expected value goal programming model and chance-constrained goal programming model can be respectively transformed into the corresponding deterministic equivalents by taking advantage of some properties of uncertainty theory. Based on these equivalence relations, the optimal transportation plans of the uncertain goal programming models can be obtained. Several numerical experiments are presented to illustrate the applications of the models.     

Robust and fuzzy goal programming optimization approaches for a novel multi-objective hub location-allocation problem: A supply chain overview

In this paper, a novel multi-objective mathematical model is developed to solve a capacitated single-allocation hub location problem with a supply chain overview. Three mathematical models with various objective functions are developed. The objective functions are to minimize: (a) total transportation and installation costs, (b) weighted sum of service times in the hubs to produce and transfer commodities and the tardiness and earliness times of the flows including raw materials and finished goods, and (c) total greenhouse gas emitted by transportation modes and plants located in the hubs. To come closer to reality, some of the parameters of the proposed mathematical model are regarded as uncertain parameters, and a robust approach is used to solve the given problem. Furthermore, two methods, namely fuzzy multi-objective goal programming (FMOGP) and the Torabi and Hassini's (TH) method are used to solve the multi-objective mathematical model. Finally, the concluding part presents the comparison of the obtained results.     

连锁门店选址与配送中心选择联合决策的f-MIGP模型

建立了连锁门店选址和配送中心选择联合决策问题的模糊多目标混合整数规划模型．针对该模型的特殊结构。提出一种适用的求解策略：首先确定每个模糊目标的隶属度函数;然后将模糊多目标混合整数规划模型转化为等价的清晰多目标混合整数规划模型,通过最大最小算子求出目标值;最后借助于两阶段算法,求出问题的最优解．通过应用算例进一步说明了该模型的有效性和可行性．     

The multi-level warehouse layout problem with uncertain information: uncertainty theory method

This paper investigates the multi-level warehouse layout problem with indeterminate factors, in which the monthly demands and horizontal transportation distances are described by uncertain variables. We first consider the distribution function of the total cost for transportation. Second, two uncertain models, namely, the chance-constrained programming model and the chance-maximum programming model, are developed to lay out the multi-level warehouse under uncertainty. Some properties of the models are discussed to solve the models. The properties point out that the optimal solution to the chance-constrained programming model is equivalent to a corresponding deterministic model. Additionally, we also discuss the relation between the chance-constrained programming model and the chance-maximum programming model, which leads to an effective approach to search for the optimal solution to the chance-maximum programming model. Finally, a numerical experiment is illustrated to show the ideas of the proposed models.     

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 two-phase possibilistic linear programming methodology for multi-objective supplier evaluation and order allocation problems

In this study, an integration of the analytic hierarchy process (AHP) and a multi-objective possibilistic linear programming (MOPLP) technique is developed to account for all tangible, intangible, quantitative, and qualitative factors which are used to evaluate and select suppliers and to define the optimum order quantities assigned to each. A multi-objective linear programming technique is first employed to solve the problem. To model the uncertainties encountered in the integrated supplier evaluation and order allocation methodology, fuzzy theory is adopted. Hence, possibilistic linear programming (PLP) is proposed for solving the problem, as it is believed to be the best approach for absorbing the imprecise nature of the real world. In the supplier evaluation phase, environmental criteria are also considered.     

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.     

Fuzzy multi-objective chance-constrained programming model for hazardous materials transportation

Hazardous materials transportation is an important and hot issue of public safety. Based on the shortest path model, this paper presents a fuzzy multi-objective programming model that minimizes the transportation risk to life, travel time and fuel consumption. First, we present the risk model, travel time model and fuel consumption model. Furthermore, we formulate a chance-constrained programming model within the framework of credibility theory, in which the lengths of arcs in the transportation network are assumed to be fuzzy variables. A hybrid intelligent algorithm integrating fuzzy simulation and genetic algorithm is designed for finding a satisfactory solution. Finally, some numerical examples are given to demonstrate the efficiency of the proposed model and algorithm.     

Dependent-chance programming with fuzzy decisions

Dependent-chance programming (DCP) is a new type of stochastic programming and has been extended to the area of fuzzy programming. This paper provides a spectrum of DCP and dependent-chance multiobjective programming (DCMOP) as well as dependent-chance goal programming (DCGP) models with fuzzy rather than crisp decisions. The terms of uncertain environment, event, chance function, and induced constraints are discussed in the case of fuzzy decisions. A technique of fuzzy simulation is also designed for computing chance functions. Finally, we present a fuzzy simulation-based genetic algorithm for solving these models and illustrate its effectiveness by some numerical examples     

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.     

Multi-objective biogeography-based optimization for supply chain network design under uncertainty

This paper proposes a new two-stage optimization method for multi-objective supply chain network design (MO-SCND) problem with uncertain transportation costs and uncertain customer demands. On the basis of risk-neutral and risk-averse criteria, we develop two objectives for our SCND problem. We introduce two solution concepts for the proposed MO-SCND problem, and use them to define the multi-objective value of fuzzy solution (MOVFS). The value of the MOVFS measures the importance of uncertainties included in the model, and helps us to understand the necessity of solving the two-stage multi-objective optimization model. When the uncertain transportation costs and customer demands have joined continuous possibility distributions, we employ an approximation approach (AA) to compute the values of two objective functions. Using the AA, the original optimization problem becomes an approximating mixed-integer multi-objective programming model. To solve the hard approximating optimization problem, we design an improved multi-objective biogeography-based optimization (MO-BBO) algorithm integrated with LINGO software. We also compare the improved MO-BBO algorithm with the multi-objective genetic algorithm (MO-GA). Finally, a realistic dairy company example is provided to demonstrate that the improved MO-BBO algorithm achieves the better performance than MO-GA in terms of solution quality.     

Dual models for possibilistic regression analysis

Upper and lower regression models (dual possibilistic models) are proposed for data analysis with crisp inputs and interval or fuzzy outputs. Based on the given data, the dual possibilistic models can be derived from upper and lower directions, respectively, where the inclusion relationship between these two models holds. Thus, the inherent uncertainty existing in the given phenomenon can be approximated by the dual models. As a core part of possibilistic regression, firstly possibilistic regression for crisp inputs and interval outputs is considered where the basic dual linear models based on linear programming, dual nonlinear models based on linear programming and dual nonlinear models based on quadratic programming are systematically addressed, and similarities between dual possibilistic regression models and rough sets are analyzed in depth. Then, as a natural extension, dual possibilistic regression models for crisp inputs and fuzzy outputs are addressed.     

再制造/制造系统集成物流网络模糊机会约束规划模型

在再制造/制造(R/M)系统集成物流网络中,回收产品的数量具有不确定性.根据这一特点,将各消费区域废旧产品的回收数量看成是模糊参数,提出了该集成物流网络的模糊机会约束规划模型.通过把模型中模糊机会约束清晰化,将模型转化为确定性的混合整数规划模型.利用实例数据,针对不同的置信水平对模型进行分析,其结果为该集成物流网络的设计提供了依据.     

Uncertain location–allocation decisions for a bi-objective two-stage supply chain network design problem with environmental impacts

In the cases that the historical data of an uncertain event is not available, belief degree-based uncertainty theory is a useful tool to reflect such uncertainty. This study focuses on uncertain bi-objective supply chain network design problem with cost and environmental impacts under uncertainty. As such network may be designed for the first time in a geographical region, this problem is modelled by the concepts of belief degree-based uncertainty theory. This article is almost the first study on belief degree-based uncertain supply chain network design problem with environmental impacts. Two approaches such as expected value model and chance-constrained model are applied to convert the proposed uncertain problem to its crisp form. The obtained crisp forms are solved by some multi-objective optimization approaches of the literature such as TH, Niroomand, MMNV. A deep computational study with several test problems are performed to study the performance of the crisp models and the solution approaches. According to the results, the obtained crisp formulations are highly sensitive to the changes in the value of the cost parameters. On the other hand, Niroomand and MMNV solution approaches perform better than other solution approaches from the solution quality point of view.     

Multi-objective reliability-redundancy allocation problem using particle swarm optimization

This paper considers the multi-objective reliability redundancy allocation problem of a series system where the reliability of the system and the corresponding designing cost are considered as two different objectives. Due to non-stochastic uncertain and conflicting factors it is difficult to reduce the cost of the system and improve the reliability of the system simultaneously. In such situations, the decision making is difficult, and the presence of multi-objectives gives rise to multi-objective optimization problem (MOOP), which leads to Pareto optimal solutions instead of a single optimal solution. However in order to make the model more flexible and adaptable to human decision process, the optimization model can be expressed as fuzzy nonlinear programming problems with fuzzy numbers. Thus in a fuzzy environment, a fuzzy multi-objective optimization problem (FMOOP) is formulated from the original crisp optimization problem. In order to solve the resultant problem, a crisp optimization problem is reformulated from FMOOP by taking into account the preference of decision maker regarding cost and reliability goals and then particle swarm optimization is applied to solve the resulting fuzzified MOOP under a number of constraints. The approach has been demonstrated through the case study of a pharmaceutical plant situated in the northern part of India.     

A goal programming model for joint decision making of inventory lot-size,supplier selection and carrier selection

In this paper, we address a problem in which a storage space constrained buyer procures a single product in multiple periods from multiple suppliers. The production capacity constrained suppliers offer all-unit quantity discounts. The late deliveries and rejections are also incorporated in sourcing. In addition, we consider transportation cost explicitly in decision making which may vary because of freight quantity and distance of shipment between the buyer and a supplier. We propose a multi-objective integer linear programming model for joint decision making of inventory lot-sizing, supplier selection and carrier selection problem. In the multi-objective formulation, net rejected items, net costs and net late delivered items are considered as three objectives that have to be minimized simultaneously over the decision horizon. The intent of the model is to determine the timings, lot-size to be procured, and supplier and carrier to be chosen in each replenishment period. We solve the multi-objective optimization problem using three variants of goal programming (GP) approaches: preemptive GP, non-preemptive GP and weighted max–min fuzzy GP. The solution of these models is compared at different service-level requirements using value path approach.     

Modelling linear fractional transportation problem in belief degree—based uncertain environment

A linear fractional transportation problem in uncertain environment is studied in this paper where the uncertain parameters of the problem are of belief degree—based uncertainty. For the first time, this type of uncertainty is considered for the linear fractional transportation problem. Belief degree—based uncertainty is useful for the cases that no historical information of an uncertain event is available. Zigzag type uncertainty distribution is used to show the belief degree—based uncertainty of the parameters of the problem. As solution methodology, the uncertain linear fractional transportation problem is converted to a crisp form using three approaches of expected value model, expected value and chance-constrained model, and chance-constrained model, separately. An extensive computational study on a real illustrative example shows the efficiency of the proposed formulation and the conversion approaches. The sensitivity analysis over the example illustrates the high dependency of the objective function value to the changes of the confidence level values of the chance constraints in the expected value and chance-constrained programming approach and the chance-constrained programming approach.