Yankui Liu

Methods of critical value reduction for type-2 fuzzy variables and their applications

A type-2 fuzzy variable is a map from a fuzzy possibility space to the real number space; it is an appropriate tool for describing type-2 fuzziness. This paper first presents three kinds of critical values (CVs) for a regular fuzzy variable (RFV), and proposes three novel methods of reduction for a type-2 fuzzy variable. Secondly, this paper applies the reduction methods to data envelopment analysis (DEA) models with type-2 fuzzy inputs and outputs, and develops a new class of generalized credibility DEA models. According to the properties of generalized credibility, when the inputs and outputs are mutually independent type-2 triangular fuzzy variables, we can turn the proposed fuzzy DEA model into its equivalent parametric programming problem, in which the parameters can be used to characterize the degree of uncertainty about type-2 fuzziness. For any given parameters, the parametric programming model becomes a linear programming one that can be solved using standard optimization solvers. Finally, one numerical example is provided to illustrate the modeling idea and the efficiency of the proposed DEA model.

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.

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

We develop a new two-stage optimization method for MO-SCND under uncertainty.The proposed model includes uncertain transportation costs and customer demands.An approximation method is adapted to approximate original optimization model.A new MO-BBO algorithm is designed to solve the approximate optimization model.Computational results demonstrate the effectiveness of the MO-BBO algorithm. 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.

Solving fuzzy p-hub center problem by genetic algorithm incorporating local search

The p-hub center problem has extensive applications in various real-world fields such as transportation and telecommunication systems. This paper presents a new risk aversion p-hub center problem with fuzzy travel times, in which value-at-risk (VaR) criterion is adopted in the formulation of objection function. For trapezoidal and normal fuzzy travel times, we first turn the original VaR p-hub center problem into its equivalent parametric mixed-integer programming problem, then develop a hybrid algorithm by incorporating genetic algorithm and local search (GALS) to solve the parametric mixed-integer programming problem. In our designed GALS, the GA is used to perform global search, while LS strategy is applied to each generated individual (or chromosome) of the population. Finally, we conduct two sets of numerical experiments and discuss the experimental results obtained by general-purpose LINGO solver, standard GA and GALS. The computational results show that the GALS achieves the better performance than LINGO solver and standard GA.

Robust optimization of supply chain network design in fuzzy decision system

This paper presents a new robust optimization method for supply chain network design problem by employing variable possibility distributions. Due to the variability of market conditions and demands, there exist some impreciseness and ambiguousness in developing procurement and distribution plans. The proposed optimization method incorporates the uncertainties encountered in the manufacturing industry. The main motivation for building this optimization model is to make tools available for producers to develop robust supply chain network design. The modeling approach selected is a fuzzy value-at-risk (VaR) optimization model, in which the uncertain demands and transportation costs are characterized by variable possibility distributions. The variable possibility distributions are obtained by using the method of possibility critical value reduction to the secondary possibility distributions of uncertain demands and costs. We also discuss the equivalent parametric representation of credibility constraints and VaR objective function. Furthermore, we take the advantage of structural characteristics of the equivalent optimization model to design a parameter-based domain decomposition method. Using the proposed method, the original optimization problem is decomposed to two equivalent mixed-integer parametric programming sub-models so that we can solve the original optimization problem indirectly by solving its sub-models. Finally, we present an application example about a food processing company with four suppliers, five plants, five distribution centers and five customer zones. We formulate our application example as parametric optimization models and conduct our numerical experiments in the cases when the input data (demands and costs) are deterministic, have fixed possibility distributions and have variable possibility distributions. Experimental results show that our parametric optimization method can provide an effective and flexible way for decision makers to design a supply chain network.

An improved hybrid particle swarm optimization algorithm for fuzzy p-hub center problem

The p-hub center problem is useful for the delivery of perishable and time-sensitive system such as express mail service and emergency service. In this paper, we propose a new fuzzy p-hub center problem, in which the travel times are uncertain and characterized by normal fuzzy vectors. The objective of our model is to maximize the credibility of fuzzy travel times not exceeding a predetermined acceptable efficient time point along all paths on a network. Since the proposed hub location problem is too complex to apply conventional optimization algorithms, we adapt an approximation approach (AA) to discretize fuzzy travel times and reformulate the original problem as a mixed-integer programming problem subject to logic constraints. After that, we take advantage of the structural characteristics to develop a parametric decomposition method to divide the approximate p-hub center problem into two mixed-integer programming subproblems. Finally, we design an improved hybrid particle swarm optimization (PSO) algorithm by combining PSO with genetic operators and local search (LS) to update and improve particles for the subproblems. We also evaluate the improved hybrid PSO algorithm against other two solution methods, genetic algorithm (GA) and PSO without LS components. Using a simulated data set of 10 nodes, the computational results show that the improved hybrid PSO algorithm achieves the better performance than GA and PSO without LS in terms of runtime and solution quality.

Optimizing material procurement planning problem by two-stage fuzzy programming

This paper presents a new class of two-stage fuzzy material procurement planning (MPP) models with minimum-risk criteria, in which the material demand, the spot market material unit price and the spot market material supply quantity are uncertain and assumed to be fuzzy variables with known possibility distributions. We formulate the two-stage MPP model with the objective of maximizing the credibility of the total material procurement costs less than a given allowable investment level, and the credibility can be regarded as the material procurement risk criteria in a fuzzy environment. Since the fuzzy material demand, the fuzzy spot market material unit price and the fuzzy spot market material supply quantity are usually continuous fuzzy variables with infinite supports, the proposed MPP model belongs to an infinite-dimensional optimization problem that cannot be solved directly. To avoid this difficulty, we apply an approximation approach (AA) to the proposed two-stage fuzzy MPP model, and turn it into an approximating finite-dimensional optimization one. The convergence about the objective function of the approximating two-stage MPP model to that of the original two-stage MPP one is also discussed. Since the exact analytical expression for the objective function in the approximating fuzzy MPP model is unavailable, and the approximating MPP model is a mixed-integer program that is neither linear nor convex, traditional optimization algorithms cannot be used to solve it. Therefore, we develop two heuristic algorithms to solve the approximating MPP model. The first is a particle swarm optimization (PSO) algorithm based on the AA, and the second is a hybrid PSO algorithm which based on the AA and a neural network (NN). Finally, we provide an actual optimization problem about the fuel procurement to compare the effectiveness of the designed algorithms.

Designing fuzzy supply chain network problem by mean-risk optimization method

In many practical supply chain network design (SCND) problems, the critical parameters such as customer demands, transportation costs and resource capacities are quite uncertain. The significance of uncertainty motivates us to develop a new mean-risk fuzzy optimization method for SCND problem, in which the standard semivariance is suggested to gauge the risk resulted from fuzzy uncertainty. To demonstrate the advantages of the proposed optimization method, we define a new concept about the value of fuzzy solution for our SCND problem. When the transportation costs and the demands of customers have continuous possibility distributions, we approximate the continuous fuzzy vector by a sequence of discrete fuzzy vectors. On the basis of the approximation scheme, we obtain an approximating optimization model, which is a nonlinear mixed-integer programming problem. Furthermore, we design a hybrid memetic algorithm (MA) to solve the approximating optimization problem. The designed hybrid MA incorporates the reduced variable neighborhood search to act as the local search procedure. Finally, we conduct some numerical experiments via an application example to demonstrate the effectiveness of the designed hybrid MA.

Fuzzy two-stage material procurement planning problem

Material procurement planning (MPP) deals with the problem that purchasing the right quantity of material from the right supplier at the right time, a purchaser can reduce the material procurement costs via a reasonable MPP model. In order to handle the MPP problem in a fuzzy environment, this paper presents a new class of two-stage fuzzy MPP models, in which the material demand, the spot market material unit price and the spot market material supply quantity are assumed to be fuzzy variables with known possibility distributions. In addition, the procurement decisions are divided into two groups. Some procurement decisions, called first-stage decisions, must be taken before knowing the the particular values taken by the fuzzy variables; while some other decisions, called second-stage decisions, can be taken after the realizations of the fuzzy variables are known. The objective of the proposed fuzzy MPP model is to minimize the expected material procurement costs over the two stages. On other hand, since the fuzzy material demand, the fuzzy spot market material unit price and the fuzzy spot market material supply quantity are usually continuous fuzzy variables with infinite supports, the proposed MPP model belongs to an infinite-dimensional optimization problem whose objective function cannot be computed exactly. To avoid this difficulty, we suggest an approximation approach (AA) to evaluating the objective function, and turn the original MPP model into an approximating finite-dimensional one. To show the credibility of the AA, the convergence about the objective function of the approximating MPP model to that of the original MPP one is discussed. Since the exact analytical expression for the objective function in the approximating fuzzy MPP model is unavailable, and the approximating MPP model is a mixed-integer program that is neither linear nor convex, the traditional optimization algorithms cannot be used to solve it. Therefore, we design an AA-based particle swarm optimization to solve the approximating two-stage fuzzy MPP model. Finally, we apply the two-stage MPP model to an actual fuel procurement problem, and demonstrate the effectiveness of the designed algorithm via numerical experiments.

Distributionally robust fuzzy project portfolio optimization problem with interactive returns

Graphical abstractDisplay Omitted HighlightsA novel credibilistic optimization model is developed for project portfolio selection problem.The uncertain returns and work times are characterized by parametric possibility distributions.The original credibilistic project portfolio selection model is turned into its equivalent parametric programming models.Computational results demonstrate the advantages of the proposed new optimization method. Effective project selection and staff assignment strategies directly impact organizational profitability. Based on critical value optimization criterion, this paper discusses how uncertainty and interaction impact the project portfolio return and staff allocation. Since the exact possibility distributions of uncertain parameters in practical project portfolio problems are often unavailable, we adopt variable parametric possibility distributions to characterize uncertain model parameters. Furthermore, this paper develops a novel parametric credibilistic optimization method for project portfolio selection problem. According to the structural characteristics of variable parametric possibility distributions, we derive the equivalent analytical expressions of credibility constraints, and turn the original credibilistic project portfolio model into its equivalent nonlinear mixed-integer programming models. To show the advantages of the proposed parametric credibilistic optimization method, some numerical experiments are conducted by setting various values of distribution parameters. The computational results support our arguments by comparing with the optimization method under fixed possibility distributions.