Analysis of critical paths in a project network with fuzzy activity times

This paper proposes an approach to critical path analysis for a project network with activity times being fuzzy numbers, in that the membership function of the fuzzy total duration time is constructed. The basic idea is based on the extension principle and linear programming formulation. A pair of linear programs parameterized by possibility level α is formulated to calculate the lower and upper bounds of the fuzzy total duration time at α. By enumerating different values of α, the membership function of the fuzzy total duration time is constructed, and the fuzzy critical paths are identified at the same time. Moreover, by applying the Yager ranking method, definitions of the most critical path and the relative degree of criticality of paths are developed; and these definitions are theoretically sound and easy to use in practice. Two examples with activity times being fuzzy numbers of L-R and L-L types discussed in previous studies are solved successfully to demonstrate the validity of the proposed approach. Since the total duration time is completely expressed by a membership function rather than by a crisp value, the fuzziness of activity times is conserved completely, and more information is provided for critical path analysis.     

Analysis of maximum total return in the continuous knapsack problem with fuzzy object weights

This paper proposes a parametric programming approach to analyze the fuzzy maximum total return in the continuous knapsack problem with fuzzy objective weights, in that the membership function of the maximum total return is constructed. The idea is based on Zadeh’s extension principle, α-cut representation, and the duality theorem of linear programming. A pair of linear programs parameterized by possibility level α is formulated to calculate the lower and upper bounds of the fuzzy maximum total return at α, through which the membership function of the maximum total return is constructed. To demonstrate the validity of the proposed procedure, an example studied by the previous studies is investigated successfully. Since the fuzzy maximum total return is completely expressed by a membership function rather than by a crisp value reported in previous studies, the fuzziness of object weights is conserved completely, and more information is provided for making decisions in real-world resource allocation applications. The generalization of the proposed approach for other types of knapsack problems is also straightforward.     

Solving fuzzy queueing decision problems via a parametric mixed integer nonlinear programming method

This paper proposes a mathematical programming method to construct the membership functions of the fuzzy objective value of the cost-based queueing decision problem with the cost coefficients and the arrival rate being fuzzy numbers. On the basis of Zadeh’s extension principle, three pairs of mixed integer nonlinear programs (MINLP) parameterized by the possibility level α are formulated to calculate the lower and upper bounds of the minimal expected total cost per unit time at α, through which the membership function of the minimal expected total cost per unit time of the fuzzy objective value is constructed. To provide a suitable optimal service rate for designing queueing systems, the Yager’s ranking index method is adopted. Two numerical examples are solved successfully to demonstrate the validity of the proposed method. Since the objective value is completely expressed by a membership function rather than by a crisp value, it conserves the fuzziness of the input information, thus more information is provided for designing queueing systems. The successful extension of queueing decision models to fuzzy environments permits queueing decision models to have wider applications in practice.     

New fuzzy models for time-cost trade-off problem

The time-cost trade-off problem is a specific type of the project scheduling problem which studies how to modify project activities so as to achieve the trade-off between the completion time and the project cost. In real projects, the trade-off between the project cost and the completion time, and the uncertainty of the environment are both considerable aspects for managers. In this paper, three new fuzzy time-cost trade-off models are proposed, in which credibility theory is applied to describe the uncertainty of activity duration times. A searching method by integrating fuzzy simulation and genetic algorithm is produced to search the quasi-optimal schedules under some decision-making criteria. The purpose of the paper is to reveal how to obtain the optimal balance of the completion time and the project cost in fuzzy environments.     

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.     

A bulk arrival queueing model with fuzzy parameters and varying batch sizes

This paper develops a nonlinear programming approach to derive the membership functions of the steady-state performance measures in bulk arrival queueing systems with varying batch sizes, in that the arrival rate and service rate are fuzzy numbers. The basic idea is based on Zadeh’s extension principle. Two pairs of mixed integer nonlinear programs (MINLP) with binary variables are formulated to calculate the upper and lower bounds of the system performance measure at possibility level α. From different values of α, the membership function of the system performance measure is constructed. For practice use, the defuzzification of performance measures is also provided via Yager ranking index. To demonstrate the validity of the proposed method, a numerical example is solved successfully.     

Fuzzy profit measures for a fuzzy economic order quantity model

Changing economic conditions make the selling price and demand quantity more and more uncertain in the market. The conventional inventory models determine the selling price and order quantity for a retailer’s maximal profit with exactly known parameters. This paper develops a solution method to derive the fuzzy profit of the inventory model when the demand quantity and unit cost are fuzzy numbers. Since the parameters contained in the inventory model are fuzzy, the profit value calculated from the model should be fuzzy as well. Based on the extension principle, the fuzzy inventory problem is transformed into a pair of two-level mathematical programs to derive the upper bound and lower bound of the fuzzy profit at possibility level α. According to the duality theorem of geometric programming, the pair of two-level mathematical programs is transformed into a pair of conventional geometric programs to solve. By enumerating different α values, the upper bound and lower bound of the fuzzy profit are collected to approximate the membership function. Since the profit of the inventory problem is expressed by the membership function rather than by a crisp value, more information is provided for making decisions.     

Optimization models and a GA-based algorithm for stochastic time-cost trade-off problem

In real-life projects, both the trade-off between the project cost and the project completion time, and the uncertainty of the environment are considerable aspects for decision-makers. However, the research on the time-cost trade-off problem seldom concerns stochastic environments. Besides, optimizing the expected value of the objective is the exclusive decision-making criterion in the existing models for the stochastic time-cost trade-off problem. In this paper, two newly developed alternative stochastic time-cost trade-off models are proposed, in which the philosophies of chance-constrained programming and dependent-chance programming are adopted for decision-making. In addition, a hybrid intelligent algorithm integrating stochastic simulations and genetic algorithm is designed to search the quasi-optimal schedules under different decision-making criteria. The goal of the paper is to reveal how to obtain the optimal balance of the project completion time and the project cost in stochastic environments.     

Parametric nonlinear programming approach to fuzzy queues with bulk service

This paper proposes a procedure to construct the membership functions of the performance measures in bulk service queuing systems with the arrival rate and service rate are fuzzy numbers. The basic idea is to transform a fuzzy queue with bulk service to a family of conventional crisp queues with bulk service by applying the α-cut approach. On the basis of α-cut representation and the extension principle, a pair of parametric nonlinear programs is formulated to describe that family of crisp bulk service queues, via which the membership functions of the performance measures are derived. To demonstrate the validity of the proposed procedure, two fuzzy queues often encountered in transportation management are exemplified. Since the performance measures are expressed by membership functions rather than by crisp values, they completely conserve the fuzziness of input information when some data of bulk-service queuing systems are ambiguous. Thus the proposed approach for vague systems can represent the system more accurately, and more information is provided for designing queuing systems in real life. By extending to fuzzy environment, the bulk service queuing models would have wider applications.     

A new path-based cutting plane approach for the discrete time-cost tradeoff problem

Determining discrete time-cost tradeoffs in project networks allows for the control of the processing time of an activity via the amount of non-renewable resources allocated to it. Larger resource allocations with associated higher costs reduce activities’ durations. Given a set of execution modes (time-cost pairs) for each activity, the discrete time-cost tradeoff problem (DTCTP) involves selecting a mode for each activity so that either: (i) the project completion time is minimized, given a budget, or (ii) the total project cost is minimized, given a deadline, or (iii) the complete and efficient project cost curve is constructed over all feasible project durations. The DTCTP is a problem with great applicability prospects but at the same time a strongly N P{\mathcal N}\,P-hard optimization problem; solving it exactly has been a real challenge. Known optimal solution methodologies are limited to networks with no more than 50 activities and only lower bounds can be computed for larger, realistically sized, project instances. In this paper, we study a path-based approach to the DTCTP, in which a new path-based formulation in activity-on-node project networks is presented. This formulation is subsequently solved using an exact cutting plane algorithm enhanced with speed-up techniques. Extensive computational results reported for almost 5,000 benchmark test problems demonstrate the effectiveness of the proposed algorithm in solving to optimality for the first time some of the hardest and largest instances in the literature. The promising results suggest that the algorithms may be embedded into project management software and, hence, become a useful tool for practitioners in the future.     

Measuring performances of multiple-channel queueing systems with imprecise data: a membership function approach

This paper proposes a mixed integer nonlinear programming (MINLP) approach to measure the system performances of multiple-channel queueing models with imprecise data. The main idea is to transform a multiple-channel queue with imprecise data to a family of conventional crisp multiple-channel queues by applying the α-cut approach in fuzzy theory. On the basis of α-cut representation and the extension principle, two pairs of parametric MINLP are formulated to describe the family of crisp multiple-channel queues, via which the membership functions of the performance measures are derived. To demonstrate the validity of the proposed procedure, a real-world case of multiple-channel fuzzy queue is investigated successfully. Since the performance measures are expressed by membership functions rather than by crisp values, the fuzziness of input information is completed conserved. Thus, the results obtained from the proposed approach can represent the system more accurately, and more information is provided for system design in practice.     

Fuzzy variables

The purpose of this study is to explore a possible axiomatic framework from which a rigorous theory of fuzziness may be constructed. The approach we propose is analogous to the sample space concept of probability theory. A fuzzy variable is a mapping from an abstract space (called the pattern space) onto the real line. The membership function is obtained as the extension of a special type of capacity (called a scale) from the pattern space to the real line via the fuzzy variable. In essence we are proposing an entirely new definition of a fuzzy set on the line as a mapping to the line rather than on the line. The current definition of a transformation of a fuzzy set is obtained as a derived result of our model. In addition, we derive the membership function of sums and products of fuzzy sets and present an example which reinforces the credibility of our approach.     

不确定条件下工程项目的多目标模糊均衡优化

在工程项目多目标优化问题研究基础上,研究不确定环境下工程项目多目标均衡优化问题.利用模糊数表示费用变化率和质量变化率,考虑模糊集的不同可能性水平,建立工程项目多目标模糊均衡优化模型,给出模型的求解方法和步骤,得到不同可能性水平下多目标优化问题的最优折衷解变化范围.优化方法使决策者能够根据决策风险的大小进行最优目标值的确定.     

Some comments on fuzzy variables

Nahmias introduced the concept of a fuzzy variable as a possible axiomatic framework from which a rigorous theory of fuzziness may be constructed. In this paper we attempt to shed more light on fuzzy variables in analogy with random variables. In particular, we study the problem: if X₁, X₂,…,X_n are mutually unrelated fuzzy variables with common membership function μ and α₁, α₂,…,α_n are real numbers satisfying α_i ? o for every i and Σ_i=1ⁿα_i=1, when does does Z = Σ_{i = 1}ⁿα_iX_i have the same membership function μ?     

A redundant repairable system with imperfect coverage and fuzzy parameters

This paper proposes a procedure to construct the membership functions of the system characteristics of a redundant repairable system with two primary units and one standby in which the coverage factor is the same for an operating unit failure as that for a standby unit failure. Times to failure and times to repair of the operating and standby units are assumed to follow fuzzified exponential distributions. The α-cut approach is used to extract from the fuzzy repairable system a family of conventional crisp intervals for the desired system characteristics, determined with a set of parametric nonlinear programs using their membership functions. A numerical example is solved successfully to illustrate the practicality of the proposed approach. Because the system characteristics are governed by the membership functions, more information is provided for use by management, and because the redundant system is extended to the fuzzy environment, general repairable systems are represented more accurately and the analytic results are more useful for designers and practitioners.     

H∞ estimation for fuzzy membership function optimization

Given a fuzzy logic system, how can we determine the membership functions that will result in the best performance? If we constrain the membership functions to a specific shape (e.g., triangles or trapezoids) then each membership function can be parameterized by a few variables and the membership optimization problem can be reduced to a parameter optimization problem. The parameter optimization problem can then be formulated as a nonlinear filtering problem. In this paper we solve the nonlinear filtering problem using H_∞ state estimation theory. However, the membership functions that result from this approach are not (in general) sum normal. That is, the membership function values do not add up to one at each point in the domain. We therefore modify the H_∞ filter with the addition of state constraints so that the resulting membership functions are sum normal. Sum normality may be desirable not only for its intuitive appeal but also for computational reasons in the real time implementation of fuzzy logic systems. The methods proposed in this paper are illustrated on a fuzzy automotive cruise controller and compared to Kalman filtering based optimization.     

Project scheduling problem with mixed uncertainty of randomness and fuzziness

Project scheduling problem is to determine the schedule of allocating resources so as to balance the total cost and the completion time. This paper considers project scheduling problem with mixed uncertainty of randomness and fuzziness, where activity duration times are assumed to be random fuzzy variables. Three types of random fuzzy models as expected cost minimization model, (α, β)-cost minimization model and chance maximization model are built to meet different management requirements. Random fuzzy simulations for some uncertain functions are given and embedded into genetic algorithm to design a hybrid intelligent algorithm. Finally, some numerical experiments are given for the sake of illustration of the effectiveness of the algorithm.     

An alternative formulation for the fuzzy assignment problem

The existing assignment problems for assigning n jobs to n individuals are limited to the considerations of cost or profit measured as crisp. However, in many real applications, costs are not deterministic numbers. This paper develops a procedure based on Data Envelopment Analysis method to solve the assignment problems with fuzzy costs or fuzzy profits for each possible assignment. It aims to obtain the points with maximum membership values for the fuzzy parameters while maximizing the profit or minimizing the assignment cost. In this method, a discrete approach is presented to rank the fuzzy numbers first. Then, corresponding to each fuzzy number, we introduce a crisp number using the efficiency concept. A numerical example is used to illustrate the usefulness of this new method.     

Determination of the threshold value β of variable precision rough set by fuzzy algorithms

In the past, the choices of β values to be applied to find the β-reducts in VPRS for an information system are somewhat arbitrary. In this study, a systematic method which bridges the fuzzy set methodology and probabilistic approach of RS to solve the threshold value β determination problem in variable precision rough sets (VPRS) is proposed. Different from the existing probabilistic methods, the proposed method relies on the fuzzy membership degrees of each attribute of the objects to calculate β. The proposed method gives the membership degrees and fuzzy aggregation operators the probabilistic interpretations. Based on the probabilistic interpretations, the threshold value β of VPRS is directly derived from fuzzy membership degree by Implication Relations and Fuzzy Algorithms, in which the membership degrees are obtained by the standard Fuzzy C-means method. The argument is that errors of system classification would occur in the fuzzy-clustering phase prior to information classification, therefore the threshold value β should be constrained by the probability of belongingness of an object to the fuzzy clusters, i.e., through the values of membership functions. A few examples are given in the paper to demonstrate the differences with other β-determining methods.     

Bifurcations of fuzzy nonlinear dynamical systems

We present some recent developments of the fuzzy generalized cell mapping method (FGCM) in this paper. The topological property of the FGCM and its finite convergence of membership distribution vector are discussed. Powerful algorithms of digraphs are adopted for the analysis of topological properties of the FGCM systems. Bifurcations of fuzzy nonlinear dynamical systems are studied by using the FGCM method. A backward algorithm is introduced to study the unstable equilibrium solutions and their bifurcation. We have found that near the deterministic bifurcation point, the fuzzy system undergoes a complex transition as the control parameter varies. In this transition region, the steady state membership distribution is dependent on the initial condition. If we use the measure and topology of the α-cut (α = 1) of the steady state membership function of the persistent group representing the stable fuzzy equilibrium solution to characterize the fuzzy bifurcation, assuming the uniform initial condition within the persistent group, the bifurcation of the fuzzy dynamical system is then completed within an interval of the control parameter, rather than at a point as is the case of deterministic systems.