Shih-Pin Chen

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

Abstract 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.

Parametric programming to the analysis of fuzzy queues

Abstract This paper proposes a general procedure to construct the membership functions of the performance measures in queueing systems when the interarrival time and service time are fuzzy numbers. The basic idea is to reduce a fuzzy queue into a family of crisp queues by applying the 7-cut approach. A pair of parametric programs is formulated to describe that family of crisp queues, via which the membership functions of the performance measures are derived. To demonstrate the validity of the proposed procedure. four fuzzy queues, namely. M / F /1, F / M /1, F / F /1, and FM / FM /1, are exemplified. The discussion of this paper is confined to systems with one and two fuzzy variables; nevertheless, the procedure can be extended to systems with more than two fuzzy variables.

Time–cost trade-off analysis of project networks in fuzzy environments

This paper proposes a novel approach for time-cost trade-off analysis of a project network in fuzzy environments. Different from the results of previous studies, in this paper the membership function of the fuzzy minimum total crash cost is constructed based on Zadehs extension principle and fuzzy solutions are provided. A pair of two-level mathematical programs parameterized by possibility level [alpha] is formulated to calculate the lower and upper bounds of the fuzzy minimum total crash cost at [alpha]. By enumerating different values of [alpha], the membership function of the fuzzy minimum total crash cost is constructed, and the corresponding optimal activity time for each activity is also obtained at the same time. An example of time-cost trade-off problem with several fuzzy parameters is solved successfully to demonstrate the validity of the proposed approach. Since the minimum total crash cost is expressed by a membership function rather than by a crisp value, the fuzziness of parameters is conserved completely, and more information is provided for time-cost trade-off analysis in project management. The proposed approach also can be applied to time-cost trade-off problems with other characteristics.

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.

Parametric nonlinear programming for analyzing fuzzy queues with finite capacity

Abstract This paper proposes a procedure for constructing the membership functions of the performance measures in finite-capacity queueing systems with the arrival rate and service rate being fuzzy numbers. The basic idea is to transform a fuzzy queue with finite capacity to a family of conventional crisp queues with finite capacity by applying the α-cut approach. A pair of parametric nonlinear programs is formulated to describe the family of crisp finite-capacity 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 real life serve as examples. Since the performance measures are expressed by membership functions rather than by crisp values, more information is provided for designing queueing systems in real life. By extending to fuzzy environment, the finite-capacity queueing models would have wider applications.

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

Abstract 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.

Tolerance allocation via simulation embedded sequential quadratic programming

Tolerance allocation is an important problem frequently encountered in the synthesis process, for designers as well as manufacturing engineers. Under the objective of minimizing the manufacturing cost while attaining an acceptable yield, the problem can be formulated as a stochastic program. Owing to the nonlinear nature of the stochastic program, a sequential quadratic programming algorithm is developed to solve the problem. The cumbersome multivariate integration in calculating the yield is approximated by a Monte Carlo simulation and the highly nonlinear yield constraint is supported by some auxiliary constraints. In limited experiments, the proposed method has performed efficiently and robustly. Compared with some previous studies, the designs solved in this paper have smaller manufacturing costs and higher yields, indicating that the proposed method is very promising in solving tolerance allocation problems.

ROBUST TOLERANCE ALLOCATION USING STOCHASTIC PROGRAMMING

Abstract Tolerance allocation in manufacturing is a prominent industrial task for enhancing productivity and reducing manufacturing costs. The classical tolerance allocation problem can be formulated as a stochastic program to determine the assignment of component tolerances such that the manufacturing cost is minimized. However, tolerance design is a prerequisite to the overall quality and cost of a product; robust tolerance design is particularly important and should be considered. In this paper, robustness is considered in formulating the tolerance allocation problem by minimizing the manufacturing costs sensitivity. Moreover, from a practical perspective, the process capability index for each component and the upper bound of the manufacturing cost are also considered. To effectively and efficiently resolve the robust tolerance allocation problem, a sequential quadratic programming algorithm embedded with a Monte Carlo simulation is developed. To demonstrate this design methods robustness, two common...

Simulation response optimization via direct conjugate direction method

This paper modifies Powells conjugate direction method for unconstrained, continuous, local optimization problems to adapt to the stochastic environment in simulation response optimization. The main idea underlying the proposed method is to conduct several replications at each trial point to obtain reliable estimate of the theoretical response. To avoid misjudging the real difference between two points due to the stochastic nature, a t-test of the statistical hypothesis is employed to replace the simple comparison of the mean responses. In an experimental comparison, the proposed method outperforms the Nelder-Mead simlex method, a quasi-Newton method, and several other methods in solving a stochastic Watson function with nine variables, a queueing problem with two variables, and an inventory problem with two variables.

Time value of delays in unreliable production systems with mixed uncertainties of fuzziness and randomness

This paper proposes a parametric programming approach to address the notion of the time value of delays in the presence of mixed (random and fuzzy) uncertainties that result from unreliable systems. To consider different types of delay time values, the system states are appropriately and carefully identified and defined, and a cost-based fuzzy decision model that incorporates several unreliability factors is constructed. Then, the proposed model is transformed into a pair of nonlinear programs parameterized by the possibility level α to identify the lower and upper bounds on the minimal total cost per unit time at α and thus construct the membership function. To provide analytical expressions, a special case with analytical results is also presented. In contrast to existing studies, the results derived from the proposed solution procedure conserve the fuzziness of the input information, representing a significant difference from the crisp results obtained using approaches based on probability theory. The results indicate that the proposed approach can provide more precise information to managers and improve decision-making in practical system design.