Overview of the Operations Research Modeling Approach:DERIVING SOLUTIONS FROM THE MODEL

DERIVING SOLUTIONS FROM THE MODEL

After a mathematical model is formulated for the problem under consideration, the next phase in an OR study is to develop a procedure (usually a computer-based procedure) for deriving solutions to the problem from this model. You might think that this must be the major part of the study, but actually it is not in most cases. Sometimes, in fact, it is a relatively simple step, in which one of the standard algorithms (systematic solution procedures) of OR is applied on a computer by using one of a number of readily available software packages. For experienced OR practitioners, finding a solution is the fun part, whereas the real work comes in the preceding and following steps, including the postoptimality analysis discussed later in this section.

Since much of this book is devoted to the subject of how to obtain solutions for various important types of mathematical models, little needs to be said about it here. However, we do need to discuss the nature of such solutions.

A common theme in OR is the search for an optimal, or best, solution. Indeed, many procedures have been developed, and are presented in this book, for finding such solutions for certain kinds of problems. However, it needs to be recognized that these solutions are optimal only with respect to the model being used. Since the model necessarily is an ideal- ized rather than an exact representation of the real problem, there cannot be any utopian guarantee that the optimal solution for the model will prove to be the best possible solution that could have been implemented for the real problem. There just are too many imponder- ables and uncertainties associated with real problems. However, if the model is well for- mulated and tested, the resulting solution should tend to be a good approximation to an ideal course of action for the real problem. Therefore, rather than be deluded into demand- ing the impossible, you should make the test of the practical success of an OR study hinge on whether it provides a better guide for action than can be obtained by other means.

The late Herbert Simon (an eminent management scientist and a Nobel Laureate in economics) pointed out that satisficing is much more prevalent than optimizing in actual practice. In coining the term satisficing as a combination of the words satisfactory and optimizing, Simon was describing the tendency of managers to seek a solution that is “good enough” for the problem at hand. Rather than trying to develop an overall mea- sure of performance to optimally reconcile conflicts between various desirable objec- tives (including well-established criteria for judging the performance of different segments of the organization), a more pragmatic approach may be used. Goals may be set to establish minimum satisfactory levels of performance in various areas, based per- haps on past levels of performance or on what the competition is achieving. If a solution is found that enables all these goals to be met, it is likely to be adopted without further ado. Such is the nature of satisficing.

The distinction between optimizing and satisficing reflects the difference between the- ory and the realities frequently faced in trying to implement that theory in practice. In the words of one of England’s pioneering OR leaders, Samuel Eilon, “Optimizing is the sci- ence of the ultimate; satisficing is the art of the feasible.”1

OR teams attempt to bring as much of the “science of the ultimate” as possible to the decision-making process. However, the successful team does so in full recognition of the overriding need of the decision maker to obtain a satisfactory guide for action in a reason- able period of time. Therefore, the goal of an OR study should be to conduct the study in an optimal manner, regardless of whether this involves finding an optimal solution for the model. Thus, in addition to pursuing the science of the ultimate, the team should also con- sider the cost of the study and the disadvantages of delaying its completion, and then attempt to maximize the net benefits resulting from the study. In recognition of this concept, OR teams occasionally use only heuristic procedures (i.e., intuitively designed procedures that do not guarantee an optimal solution) to find a good suboptimal solution. This is most often the case when the time or cost required to find an optimal solution for an adequate model of the problem would be very large. In recent years, great progress has been made in developing efficient and effective metaheuristics that provide both a general structure and strategy guidelines for designing a specific heuristic procedure to fit a particular kind of problem. The use of metaheuristics (the subject of Chap. 14) is continuing to grow.

The discussion thus far has implied that an OR study seeks to find only one solution, which may or may not be required to be optimal. In fact, this usually is not the case. An

1S. Eilon, “Goals and Constraints in Decision-making,” Operational Research Quarterly, 23: 3–15, 1972. Address given at the 1971 annual conference of the Canadian Operational Research Society.

optimal solution for the original model may be far from ideal for the real problem, so addi- tional analysis is needed. Therefore, postoptimality analysis (analysis done after finding an optimal solution) is a very important part of most OR studies. This analysis also is sometimes referred to as what-if analysis because it involves addressing some questions about what would happen to the optimal solution if different assumptions are made about future conditions. These questions often are raised by the managers who will be making the ultimate decisions rather than by the OR team.

The advent of powerful spreadsheet software now has frequently given spreadsheets a central role in conducting postoptimality analysis. One of the great strengths of a spread- sheet is the ease with which it can be used interactively by anyone, including managers, to see what happens to the optimal solution (according to the current version of the model) when changes are made to the model. This process of experimenting with changes in the model also can be very helpful in providing understanding of the behavior of the model and increasing confidence in its validity.

In part, postoptimality analysis involves conducting sensitivity analysis to determine which parameters of the model are most critical (the “sensitive parameters”) in determin- ing the solution. A common definition of sensitive parameter (used throughout this book) is the following.

For a mathematical model with specified values for all its parameters, the model’s sensitive parameters are the parameters whose value cannot be changed without changing the optimal solution.

Identifying the sensitive parameters is important, because this identifies the parameters whose value must be assigned with special care to avoid distorting the output of the model.

The value assigned to a parameter commonly is just an estimate of some quantity (e.g., unit profit) whose exact value will become known only after the solution has been implemented. Therefore, after the sensitive parameters are identified, special attention is given to estimating each one more closely, or at least its range of likely values. One then seeks a solution that remains a particularly good one for all the various combinations of likely values of the sensitive parameters.

If the solution is implemented on an ongoing basis, any later change in the value of a sensitive parameter immediately signals a need to change the solution.

In some cases, certain parameters of the model represent policy decisions (e.g., resource allocations). If so, there frequently is some flexibility in the values assigned to these parameters. Perhaps some can be increased by decreasing others. Postoptimality analysis includes the investigation of such trade-offs.

In conjunction with the study phase discussed in Sec. 2.4 (testing the model), postopti- mality analysis also involves obtaining a sequence of solutions that comprises a series of improving approximations to the ideal course of action. Thus, the apparent weaknesses in the initial solution are used to suggest improvements in the model, its input data, and per- haps the solution procedure. A new solution is then obtained, and the cycle is repeated. This process continues until the improvements in the succeeding solutions become too small to warrant continuation. Even then, a number of alternative solutions (perhaps solutions that are optimal for one of several plausible versions of the model and its input data) may be pre- sented to management for the final selection. As suggested in Sec. 2.1, this presentation of alternative solutions would normally be done whenever the final choice among these alter- natives should be based on considerations that are best left to the judgment of management.

Example. Consider again the Rijkswaterstaat OR study of national water management policy for the Netherlands, introduced at the end of Sec. 2.2. This study did not conclude by recommending just a single solution. Instead, a number of attractive alternatives were identified, analyzed, and compared. The final choice was left to the Dutch political process, culminating with approval by Parliament. Sensitivity analysis played a major role in this study. For example, certain parameters of the models represented environmental standards. Sensitivity analysis included assessing the impact on water management problems if the values of these parameters were changed from the current environmental standards to other reasonable values. Sensitivity analysis also was used to assess the impact of changing the assumptions of the models, e.g., the assumption on the effect of future international treaties on the amount of pollution entering the Netherlands. A variety of scenarios (e.g., an extremely dry year or an extremely wet year) also were analyzed, with appropriate probabilities assigned. (Also see Selected References A11 and A13 for other examples where quickly deriving the appropriate kinds of solutions were a key part of an award-winning OR application.)