Decision making under uncertainty - is sensitivity analysis of any use?

Transcription

1 Decision making under uncertainty - is sensitivity analysis of any use? Stein W. Wallace Molde University College 1

2 Decision making under uncertainty Sensitivity analysis Parametric optimization What-if -analysis Scenario analysis Do these work? 2

3 Mathematical programming, especially linear programming and related network and combinatorial methods, usually form the OR/MS deterministic subfield. It is time to recognize that this categorization is restrictive and does not serve the field well. Those of us who work in the area are, in a sense, blessed and lucky. We have in the linear programming mathematical model and in its solution by the simplex method a readily available analysis that answers important data sensitivity questions and, at the same time, yields critical related economic information. Coupling such an analysis with computationally simple studies provides a rather nondeterministic view of the modelling situation. Thus, those of us who teach and practice mathematical programming have the means of emphasizing and answering concerns about validity, robustness, uncertain data, base case and scenario analysis, and in achieving the truism that modeling is more about gaining insight than in producing numbers. We can and do cut across the dichotomy. 3 (Saul Gass )

4 In all LP models the coefficients of the objective function and the constraints are supplied as input data or as parameters to the model. The optimal solution obtained by the simplex method is based on the values of these coefficients. In practice the values of these coefficients are seldom known with absolute certainty, because many of them are functions of some uncontrollable parameters. For instance, future demands, the cost of raw materials, or the cost of energy resources cannot be predicted with complete accuracy before the problem is solved. Hence the solution of a practical problem is not complete with the mere determination of the optimal solution. Each variation in the values of the data coefficients changes the LP problem, which may in turn affect the optimal solution found earlier. In order to develop an overall strategy to meet the various contingencies, one has to study how the optimal solution will change with changes in the input (data) coefficients. This is known as sensitivity analysis or post-optimality analysis. 4 (Ravindran, Phillips and Solberg 1987)

5 Example You have two lots of land that you can develop Developing the land Building the plant now Building the plant later Lot Lot Plants produce one unit of a good sold at price p, presently unknown 5

6 Nine possible solutions No 9 is to do nothing 6

7 Sensitivity analysis Sample (or construct) possible futures, and solve for each of these possible futures. Compare (combine) solutions to find an overall solution. p < 700 : Do nothing Delayed construction is never used 700 < p < 800 P > 800 No other scenario solutions are possible Flexibility has no value 7

8 Assume p can take on two different values Pr(p=210)=Pr(p=1250)=0.5 Expected value p=730 Deterministic problem: Use p=730 and get Decision 4 with profit =30 Expected value of scenario solutions: Decision 4: * *1250=30 Decision 9: 0 Decision 7: *2* *2*1250= -40 8

9 Decision Invest Income if Income if Expected p=210 p=1250 profit Expected profit for all scenarios Delayed construction used whenever profitable

10 A continuous example max 3x + 2 y + z x y ξ 1 ξ Substitutes with demand summing to one x + y + z 1 Production capacity x, y, z 0 10

11 We can find the optimal solution for all possible demands x y z = = 1 t = t 0 for t [ 0,1] All scenario solutions are infeasible when you try to calculate the expected performance Optimal solution: x = y = 0, z = 1 11

12 What do we see? All scenario solutions have x+y=1, forcing z=0. That is exactly what we do not want. We have solved all possible problems All scenario solutions have the same optimal basis. The optimal solution has a different optimal basis. The space spanned by scenario solutions does not contain the optimal solution 12

13 Connections to options Scenario solutions disregard flexibility. Options (= flexible alternatives) are worthless in a deterministic analysis, unless they are free. We do not have follow-up (recourse) decisions as required. 13

14 The IQ of hindsight After the fact one of the scenario solutions will turn out to be best The stochastic programming solution is hardly ever best Be careful with how decision makers are evaluated 14

15 An example A MIP model has been made to support the development of oil and gas fields in addition to the transportation infrastructure in the North Sea. The model takes more than one hour to run. 50 runs have been made with different levels of demand, prices, and field sizes. 15

16 In 80% of all cases, the gas pipe from A to B has a diameter of 24. In the other cases it is larger. Hence, we can conclude that 24 is a lower bound. In all cases, field C is developed between 2010 and Hence, we can safely assume that we have found a time interval in which the field is developed. Choosing 2015 is clearly a good approximation. 16

17 Subfields D and E compete for the same production capacity at a platform. In all cases subfield D is chosen. Hence, we can safely disregard subfield E in the analyses to follow. Does anything change if we are able to make these statements for all scenarios and not just the 50 chosen? 17

18 The conclusions based on sensitivity analysis may be good or correct The arguments leading to the conclusions are false. 18

19 Option values All decision have embedded option values, positive or negative A decision opens and closes doors The higher variance, the higher option value If we underestimate the randomness, we underestimate the option values Scenario analysis gives no value to options 19

20 Organizational issues The float of information in an organization randomness is often lost between departments Will result in reduced randomness and hence reduced value of flexibility. 20

21 Board rooms Leaders want clear advise and explanations Results in randomness being suppressed and hence flexibility being undervalued 21

22 Discount rates Many companies use very high discount rates to express that only very profitable projects will be accepted Investments in flexibility normally have costs early and potential income late Results in reduced option values and incorrect investments in accepted projects 22

23 Subjective issues Over confidence in own estimates reduces option values Hindsight learning - we have problems seeing that other scenarios could have happened bad learning 23

24 Flexibility is often important, but... Disregarded by methods Lost in organizations Undervalued by individuals 24

25 Regret and robust solutions Expected value solutions may have a large variance. We may be risk averse. Maybe we regret not having made another decision? We especially hate large regrets. 25

26 Limits on the regret We may choose to maximize expected profit with a bound on the largest regret Maximize the expected profit minus a function of the variance Examples To follow Markowitz mean-variance model 26

27 Regret for the nine possible solutions If we limit regret to maximum 500, we will choose Decision 4, and the expected profit will drop from 195 to

28 Alternative Buy insurance (an option) against high prices High prices happen with 50% probability Pay 10 (plus a fee) to get 20 if prices are high The expected profit for Decision 3 is now 195 minus the insurance fee, and the maximal regret is down to

29 Time related entities Period = time step in model Stage = point in time where it makes sense to make a new decision New information is needed to make new decisions New information is only interesting if new decisions can be made 29

30 How many stages? Enough to capture all the important consequences of the first stage solution Enough to capture a time span similar to the real decision problem 30

31 Correct use of sensitivity analysis A priori analysis, under uncertainty, of decisions that eventually will be made under certainty. Analysis of changes in deterministic parameters. 31

32 Mathematical programming, especially linear programming and related network and combinatorial methods, usually form the OR/MS deterministic subfield. It is time to recognize that this categorization is restrictive and does not serve the field well. Those of us who work in the area are, in a sense, blessed and lucky. We have in the linear programming mathematical model and in its solution by the simplex method a readily available analysis that answers important data sensitivity questions and, at the same time, yields critical related economic information. Coupling such an analysis with computationally simple studies provides a rather nondeterministic view of the modelling situation. Thus, those of us who teach and practice mathematical programming have the means of emphasizing and answering concerns about validity, robustness, uncertain data, base case and scenario analysis, and in achieving the truism that modeling is more about gaining insight than in producing numbers. We can and do cut across the dichotomy. (Saul Gass )

33 In all LP models the coefficients of the objective function and the constraints are supplied as input data or as parameters to the model. The optimal solution obtained by the simplex method is based on the values of these coefficients. In practice the values of these coefficients are seldom known with absolute certainty, because many of them are functions of some uncontrollable parameters. For instance, future demands, the cost of raw materials, or the cost of energy resources cannot be predicted with complete accuracy before the problem is solved. Hence the solution of a practical problem is not complete with the mere determination of the optimal solution. Each variation in the values of the data coefficients changes the LP problem, which may in turn affect the optimal solution found earlier. In order to develop an overall strategy to meet the various contingencies, one has to study how the optimal solution will change with changes in the input (data) coefficients. This is known as sensitivity analysis or post-optimality analysis. (Ravindran, Phillips and Solberg )

34 The sad conclusion Sensitivity analysis, and its relatives, are not theoretically valid methods for analyzing decision making under uncertainty. They can lead to arbitrarily bad conclusions. Methods like stochastic programming and (real) option theory are needed. 34