DECISION MAKING UNDER UNCERTAINTY: IS SENSITIVITY ANALYSIS OF ANY USE? STEIN W. WALLACE Department of Industrial Economics and Technology Management, Norwegian University of Science and Technology, Trondheim, Norway [email protected] (Received April 1997; revision received January 1998, June 1998; accepted July 1998) Sensitivity analysis, combined with parametric optimization, is often presented as a way of checking if the solution of a deterministic linear program is reliable—even if some of the parameters are not fully known but are instead replaced by a best guess, often a sample mean. It is customary to claim that if the region over which a certain basis is optimal is large, one is fairly safe by using the solution of the linear program. If not, the parametric analysis will provide us with alternative solutions that can be tested. This way, sensitivity analysis is used to facilitate decision making under uncertainty by means of a deterministic tool, namely parametric linear programming. We show in this note that this basic idea of stability has little do with optimality of an optimization problem where the parameters are uncertain. 1. WHY SENSITIVITY ANALYSIS optimal. If uncertainty rests in the right-hand side, the anal- Most, if not all, decisions are made under uncertainty; there ysis shows over which area the given basis is primal feasi- is hardly any disagreement about that. Aspects such as price ble, and therefore over which area the same primal variables and demand, quality of raw materials, and reliability of ma- remain nonnegative. In the ÿrst case, if the area is large, it chines and operators can hardly ever be viewed as determin- is customary to say that the expected value solution (that istic entities. In addition, the model itself will almost always is, the solution corresponding to the expected value prob- be an approximation of the real problem and hence repre- lem) is stable, implying that it is reasonably safe to use it. sents uncertainty with respect to the quality of the resulting In the second case, it is the optimal basis that is stable (and optimal solution. also the dual variables), but at least we know which vari- However, it is important to remember that although all ables (machines, processes, transportation modes) to focus decisions can be viewed as being made under uncertainty, on. Again, in standard texts, the terminology is usually dif- this does not imply that uncertainty is an important aspect ferent because the term expected value solution is not used. of all problems. If, for example, the same decision is the An advanced user would also consider the reduced costs at unique optimum for absolutely all possible values of the the boundaries to make sure the function is not too steep. uncertain parameters, although the objective function value Correspondingly, if the area of optimality is small, there is may be very dependent, the true optimal decision can be a fear that the given solution (or basis) is not very safe to found simply by solving one single problem, normally the use. In that case, we can use the parametric analysis to look one where all parameters are set at their most likely value. In for alternative solutions. such a case it is fair to claim that uncertainty is unimportant In a newly issued book on sensitivity analysis and para- for making decisions. metric programming (Gal and Greenberg 1997), we read An obvious di/culty with this approach is the need to in the foreward: “Mathematical programming, especially know that it works without actually checking that it does. A linear programming and related network and combinatorial common approach is therefore to solve the expected value methods, usually form the OR=MS deterministic subÿeld. problem—that is, the problem with all uncertain parameters It is time to recognize that this categorization is restric- replaced by their expected values—and then perform sen- tive and does not serve the ÿeld well. Those of us who sitivity analysis. This approach is taught in most textbooks work in the area are, in a sense, blessed and lucky. We on mathematical programming and operations research, al- have in the linear programming mathematical model and though it is not common to interpret the linear program that in its solution by the simplex method a readily available is solved as an expected value problem. If uncertainty is in analysis that answers important data sensitivity questions the objective function, the sensitivity analysis will tell over and, at the same time, yields critical related economic in- which area of the parameter set the given (primal) solution is formation. Coupling such an analysis with computationally Subject classiÿcations: Decision analysis, risk: sensitivity analysis, what-if analysis. Programming, linear, parametric: random parameters. Programming stochastic: can sensitivity analyses be used? Area of review: OR CHRONICLE. Operations Research, ? 2000 INFORMS 0030-364X/00/4801-0020 $05.00 Vol. 48, No. 1, January–February 2000, pp. 020–025 20 1526-5463 electronic ISSN WALLACE /21 simple studies provides a rather nondeterministic view of the mean to minimize costs in a setting where costs occur twice, modeling situation. Thus, those of us who teach and practice and the second occurrence is random? In fact, it means noth- mathematical programming have the means of emphasizing ing unless we specify it more clearly. Very often it seems and answering concerns about validity, robustness, uncer- that the underlying assumption (implicit and hidden as it tain data, base case and scenario analysis, and in achiev- may be) is to minimize the sum of the immediate costs aris- ing the truism that modeling is more about gaining insight ing from the decision, and the expected future costs. We than in producing numbers. We can and do cut across the will proceed here as if that was the case. A few aspects of dichotomy.” We see that the usefulness of sensitivity analy- this issue will be discussed in the next section. sis for understanding uncertainty is shared by the authors in Many will argue that we have already gone too far, as they their reference to robustness, uncertain data, and base case are not willing to operate with probability distributions at and scenario analysis. all. In fact, many approaches claim that one of their beneÿts As an example from a more typical text book, sensitiv- is that distributions are not needed. And, of course, prob- ity analysis is introduced the following way in Ravindran et ability distributions are not discussed in textbooks on sen- al. (1987, §2.11): “In all LP models the coe/cients of the sitivity analysis and parametric (linear) programming. The objective function and the constraints are supplied as input arguments made in this paper do not depend on either the data or as parameters to the model. The optimal solution willingness or the ability to estimate probability distribu- obtained by the simplex method is based on the values of tions. Furthermore, we do not imply that users are willing or these coe/cients. In practice the values of these coe/cients able to solve the resulting complex optimization problem. are seldom known with absolute certainty, because many of However, we do assume that users would accept that proba- them are functions of some uncontrollable parameters. For bility distributions would make it possible to ÿnd better so- instance, future demands, the cost of raw materials, or the lutions if the distributions were available and all calculations cost of energy resources cannot be predicted with complete came for free. In other words, we do not disregard practi- accuracy before the problem is solved. Hence the solution of cal arguments against probability distributions, but we do a practical problem is not complete with the mere determi- assume that terms like “expected future costs” make sense, nation of the optimal solution. Each variation in the values even though they possibly cannot be calculated from a prac- of the data coe/cients changes the LP problem, which may tical point of view. in turn aIect the optimal solution found earlier. In order to Let us add that any approach that implicitly or explicitly develop an overall strategy to meet the various contingen- accepts that we do in fact have a problem with at least two cies, one has to study how the optimal solution will change stages (simply meaning that some decisions are made before with changes in the input (data) coe/cients. This is known all relevant parameters are known with certainty) must make as sensitivity analysis or post-optimally analysis.” some kind of assumption about the relationship between the What type of decision problems are we talking about? immediate costs and the future costs. There is no free lunch First, we see from the very wording that uncertainty is an here. element of the decision problem. Second, it seems clear that So, again we conclude that although the expected value some decisions must be made before all the actual parame- solution is not optimal (in hindsight) for all parameter sets, ter values become known, and ÿnally, that in some way it it may still minimize expected costs and hence be optimal. is costly if the decision ÿts badly to the realized parame- But, of course, there may be other solutions that have a ter set. The problem therefore has at least two stages, that better expected performances. is, at least two points in time where costs (proÿts) are in- How would one go about ÿnding alternative solutions? curred, and these points are separated by another point in Again, if we refer to textbooks (for example, Ravindran time where uncertainty is revealed. Typical examples would 1987) as well as practice, there are a few basic approaches.