# Question

Consider the following problem.

Minimize Z = 5x1 + c2x2,

Subject to

and

x1 ≥ 0, x2 ≥ 0,

where x1 represents the level of activity 1 and x2 represents the level of activity 2. The values of c2, a12, and a22 have not been determined yet. Only activity 1 needs to be undertaken soon whereas activity 2 will be initiated somewhat later. There are different scenarios that could unfold between now and the time activity 2 is undertaken that would lead to different values for c2, a12, and a22. Therefore, the goal is to use all of this information to choose a value for x1 now and to simultaneously determine a plan for choosing a value of x2 later after seeing which scenario has occurred.

Three scenarios are considered plausible possibilities. They are listed below, along with the values of c2, a12, and a22 that would result from each one:

Scenario 1: c2 = 4, a12 = 2, and a22 = 3

Scenario 2: c2 = 6, a12 = 3, and a22 = 4

Scenario 3: c2 = 3, a12 = 2, and a22 = 1

These three scenarios are considered equally likely.

Use stochastic programming with recourse to formulate the appropriate model for this problem and then to solve for the optimal plan.

Minimize Z = 5x1 + c2x2,

Subject to

and

x1 ≥ 0, x2 ≥ 0,

where x1 represents the level of activity 1 and x2 represents the level of activity 2. The values of c2, a12, and a22 have not been determined yet. Only activity 1 needs to be undertaken soon whereas activity 2 will be initiated somewhat later. There are different scenarios that could unfold between now and the time activity 2 is undertaken that would lead to different values for c2, a12, and a22. Therefore, the goal is to use all of this information to choose a value for x1 now and to simultaneously determine a plan for choosing a value of x2 later after seeing which scenario has occurred.

Three scenarios are considered plausible possibilities. They are listed below, along with the values of c2, a12, and a22 that would result from each one:

Scenario 1: c2 = 4, a12 = 2, and a22 = 3

Scenario 2: c2 = 6, a12 = 3, and a22 = 4

Scenario 3: c2 = 3, a12 = 2, and a22 = 1

These three scenarios are considered equally likely.

Use stochastic programming with recourse to formulate the appropriate model for this problem and then to solve for the optimal plan.

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