Random Variables - Stochastic Programming

Stochastic programming concerns mathematical programming models where some parameters are uncertain. Rather than deterministic numbers, these parameters are described as random variables. For stochastic programming, some variables are to be set by a decision maker, these are the decision variables, while some model parameters are determined by chance, these are the random variables.

There are a variety of situations one might consider in this context. One differentiation is based on when the decision maker must make decisions relative to the time when the random variables are realized. There are several possibilities.

No Recourse: The decision maker must choose values for the decision variables before any of the random variables are realized.

Single Stage Recourse: Some of the decision variables must be set before the random variables are realized, while others may wait until after they are realized.

Multistage Recourse: Decisions and random variables are determined in stages. The first set of decision variables must be fixed before all realizations. Then the first set of random variables are realized and the second set of decision variables must be fixed. The process continues through a series of decisions and realizations. Typically the stages represent intervals of time.

Wait and See: The decision maker makes no decisions until all random variables are realized.

Of course, the decision maker prefers the last case. Then the optimum decisions can be made with no uncertainty using a deterministic optimization procedure. The first case is the most difficult because the all decisions must be made before the model is completely known. It is not obvious how decisions should be made in this case, but we investigate some alternatives below using the Algorithm feature of the add-in.

We consider on this page the first and last cases since for these the Function feature of the add-in provides useful information. The cases that involve recourse are the subject of much active research. This subject is beyond this simple introduction, but we consider an example on the next page.

Wait and See

To illustrate stochastic programming, consider the linear programming model below. The solution algorithm is the Jensen LP/IP add-in. The problem was generated randomly using the Math Programming add-in. The solution shown is the optimum solution when all parameters of the model are deterministic.
For this situation we assume that the RHS vector (F15:F19) is uncertain. In particular, we assume that the RHS values have the mean values as given, but the values have Normal distributions with given standard deviations. To create a stochastic model, we choose Function from the menu and fill in the dialog as below. We have specified LP/IP as the algorithm. This means that the LP will be solved for each point of the sample space that is generated by the enumeration or simulation procedure. For the example we use simulation.

The form is placed below the LP model as shown below. The figure shows some of the modifications required for the analysis. The RHS entries shown in column F have been replaced by the equations in column G (the equations are placed in column F, but illustrated in G). Each RHS is the sum of the original value stored in column C and a Normally distributed random variable defined on the simulation form. The random variables all have a mean of 0 and a standard deviation of 10. The function value in G32 is an equation that links to the LP objective in F4. The feasibility value in G33 is a logical statement that points to cell O4 (shown with the LP model above). That cell, filled by the LP/IP add-in, holds the word "Optimal" only when the model has a feasible solution. The particular solution shown has all random variables set to 0, their expected values.

This form illustrates a case where important features of the stochastic model lie in the portion of the worksheet outside the form. The random variables are linked by equations to the RHS cells, and the function value and feasibility state are linked by equations to cells computing the objective function value and optimality state.

To simulate the model choose Moments from the menu and set the number of simulation observations to 100. The results start in row 34 of the figure below. The numbers shown in rows 29 through 33 are for the final simulated observation of the random variables.

Even though the solution is optimum for each simulated RHS, the mean objective function (estimated at 122.7 in cell G36) is lower than the value when the RHS values were set to their expected values (125.6). This is a consequence of "Jensen's Inequality" (named after a different person that the author). The expected value solution will always have a higher objective function (when maximizing) than a solution that explicitly represents uncertainty.

All the observations in the sample of 100 were feasible for the example above. It is interesting to increase the standard deviation of the random variables to 30, thus increasing the variability of the RHS values. The simulated results are below. For this model, the solution is infeasible only if one of the RHS values is negative. With the data provided, the proportion of feasible solutions is 0.73. The mean objective value is decreased with significantly greater variance over the case when the standard deviation values were 10. There is a price to pay for variability, even with the wait and see policy.

No Recourse

With the no recourse situation, we must fix the decision variables before the random variables are realized. We might ask how well the expected value solution serves as the solution to this problem. We create a different model below. The LP model is the same as before, but now we allow the RHS values to vary randomly but do not solve the LP for each sample. For each sample, the solution is fixed at the deterministic solution obtained with the expected RHS. The objective value does not change because the decision vector is fixed. The only question is what is the probability that this solution will be feasible?

We have placed logical expressions in column G to indicate when the constraints are feasible. The logical expression for feasibility in cell G33 is TRUE only when all constraints are all satisfied. The LP solution that is partially shown is optimal when the RHS takes the expected value, so this solution is feasible.

When we simulate the model for this case we do not solve the model for every sample. The word "None" in cell G23 assures this. The results are shown below. The objective value shows no variability since the decision variables remain the same for all samples. The proportion of feasible solutions is, however, only 6%. The probability that this fixed solution is feasible is very small.

These results should not be surprising. With the expected RHS values, four of the five constraints are tight in the deterministic optimal solution. Since the RHS values are normally distributed, there is a 50% that a tight constraint will be violated when the random RHS is realized. Even if the single loose constraint is never violated, the probability that all the tight constraints are satisfied is 0.5 raised to the fourth power or 0.0625. The simulated result is quite close to this value.

One might ask, how should this problem be solved? The answer to this question is part of the subject of stochastic programming.

Chance Constrained Programming

One way to find a solution that has a greater feasibility probability is to make the RHS vector smaller and solve the deterministic problem. A smaller RHS provides a greater probability that a constraint will be satisfied for a sample RHS. Say we set the RHS values so that there is only a 10% probability of violating each constraint individually. This is easily accomplished by setting the values in row 29 to 0.1. The values of the random variables in row 30 are values that have a 90% chance of being exceeded. The solution shown is optimal for the RHS values shown. The objective value is 94.6.

The results shown starting at row 34 are for a simulation of 100 sample points. Even though each constraint has a 90% probability of violation, the probability that all constraints are feasible (the solution is feasible) is approximately 54%.

Combining Wait and See Solutions

One suggestion often made for this kind of problem is to solve the problem as if we could wait and see the random realizations and then combine the wait and see decisions in some way. We do this for the example by creating a new math programming model identical to the wait and see model considered earlier, but create a simulation form that records the values of the decision variables as well as the objective. The form is below. We now have 11 functions defined. The first is the objective function and the remaining ten are the values of the decision variables. All eleven functions have the same feasibility equation.

We simulate the random variables, solve the LP for each sample point and record the observed function values. The results from 100 observations is below. Row 36 shows that six of the decision variables have non-zero values in some of the observations.

To continue we impose the solution shown in row 36 on the LP model.

Using the model previously used to evaluate the mean value solution we simulate this model with the solution fixed as above. Of course, since the solution is fixed there is no variability in the objective function. The results indicate that the solution is feasible for 11% of the simulated RHS vectors.

Simple Recourse

None of the solutions to the No Recourse problem that we have investigated are very satisfactory. When uncertainty affects the RHS values of the constraint coefficients there is always a chance that some constraints will be violated by a fixed solution. Choosing a solution involves a tradeoff between the risk of infeasibility and the expected value of the objective function. The table shows the three solutions considered above in order of decreasing objective value. Other solutions can be generated by choosing different satisfaction probabilities for the chance constrained model.

 Solution Objective Value Feasibility Probability Expected Value RHS 125.6 6% Combined Wait and See 120.1 11% Chance Constrained (90%) 94.5 54%

The Simple Recourse model explicitly penalizes infeasibility and maximizes the sum of the linear objective value and the negative of the expected penalized infeasibility.

Although this model is more satisfactory because it explicitly penalizes infeasibility, it is much more difficult to solve than the other models on this page. Obtaining a solution is probably beyond the algorithms available for Excel. The model is a special case of the single-stage recourse problem which is the described on the next page.

Operations Research Models and Methods
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by Paul A. Jensen
Copyright 2004 - All rights reserved