Production Engineering and Management under Fuzziness

In project or production management, an activity network is classically defined by a set of tasks (activities) and a set of precedence constraints expressing which tasks cannot start before others are completed. When there are no resource constraints, we can display the network as a directed acyclic graph. With such a network the goal is to find critical activities, and to determine optimal starting times of activities, so as to minimize the makespan. The first step is to determine the earliest ending time of the project. This problem was posed in the fifties, in the framework of project management, by Malcolm et al. [32] and the basic underlying graph-theoretic approach, called Project Evaluation and Review Technique, is now popularized under the acronym PERT. The determination of critical activities is carried out via the so-called critical path method (Kelley [29]). The usual assumption in scheduling is that the duration of each task is precisely known, so that solving the PERT problem is rather simple. However, in project management, the durations of tasks are seldom precisely known in advance, at the time when the plan of the project is designed. Detailed specifications of the methods and resources involved for the realization of activities are often not available when the tentative plan is made up. This difficulty has been noticed very early by the authors that introduced the PERT approach. They proposed to model the duration of tasks by probability distributions, and tried to estimate the mean value and standard deviation of earliest starting times of activities. Since then, there has been an extensive literature on probabilistic PERT (see Adlakha and Kulkarni [1] and Elmaghraby [18] for a bibliography and recent views). Even if the task duration times are independent random variables, it is admitted that the problem of finding the distribution of the ending time of a project is intractable, due to the dependencies induced by the topology of the network [25]. Another difficulty, not always pointed out, is the possible lack of statistical data validating the choice of activity duration distributions. Even if statistical data are available, they may be partially inadequate because each project takes place in a specific environment, and is not the exact replica of past projects.

The basic paradigm for simulation of production and logistic systems is the probabilistic approach to describing the real world uncertainty. However, in many cases we do not have the information that would be precise enough to build corresponding probabilistic models or there are some human factors preventing doing so. In such situations, the mathematical tools of fuzzy sets theory may be successfully used. It seems that a simple and natural way to do this is the replacement of probability densities with appropriate fuzzy intervals and the use of fuzzy arithmetic to build adequate fuzzy models. Since the models of production and logistic systems are usually used for the optimization of simulated processes, the problem of fuzzy optimization arises. The problems of simulation and optimization in the fuzzy setting can be solved with use of fuzzy and interval arithmetic, but there are some inherent problems in formulation of basic mathematical operations on fuzzy and interval objects. The more important of them (especially for the fuzzy optimization) is the interval and fuzzy objects comparison. The paper presents a new method for crisp and fuzzy interval comparison based on the probabilistic approach. The use of this method in the synthesis with α-cut representation of fuzzy values and usuall interval arithmetic rules, makes it possible to develope an affective approach to fuzzy simulation and optimization. This approach is illustrated by the examples of fuzzy simulation of linear production line and logistic system and by the example of fuzzy solution of optimal goods distribution problem. The results obtained with use of the proposed approach are compared with those obtained using Monte-Carlo method.

In traditional investment planning investment decisions are usually taken to be now-or-never, which the firm can either enter into right now or abandon forever. The decision on to close/not close a production plant has been understood to be a similar now-or-never decision for two reasons: (i) to close a plant is a hard decision and senior management can make it only when the facts are irrefutable; (ii) there is no future evaluation of what-if scenarios after the plant is closed. However, it is often possible to postpone,modify or split up a complex decision in strategic components, which can generate important learning effects and therefore essentially reduce uncertainty. If we close a plant we lose all alternative development paths which could be possible under changing conditions; on the other hand, senior management may have a difficult time with shareholders if they continue operating a production plant in conditions which cut into its profitability as their actions are evaluated and judged every quarter. In these cases we can utilize the idea of real options. The new rule, derived from option pricing theory, is that we should only close the plant now if the net present value of this action is high enough to compensate for giving up the value of the option to wait. Because the value of the option to wait vanishes right after we irreversibly decide to close the plant, this loss in value is actually the opportunity cost of our decision. In this work we will use fuzzy real option models for the problem of closing/not closing a production plant in the forest products industry sector.