TEST OR

  1. A power system has 3 coal-fired generation units that operate together to satisfy the load demand. Build a LP model to describe the generation unit economic dispatch problem. If two units at most are needed to satisfy the demand, revise the LP model to describe the new problem of unit commitment.
  2. In constrained optimization in economics, the shadow price is the change in the objective value of the optimal solution of an optimization problem obtained by relaxing the constraint by one unit. In other words, it is the marginal utility of relaxing the constraint, or, equivalently, the marginal cost of strengthening the constraint.In a business application, a shadow price is the maximum price that management is willing to pay for an extra unit of a given limited resource. Is the description of shadow price right? If not, give your reason.
  3. You have a constraint that limits the amount of labor available to 40 hours per week. If your shadow price is $10/hour for the labor constraint, and the market price for the labor is $11/hour. Should you pay to obtain additional labor?
  4. Give the definitionto the convex set.
  5. Sensitivity analysis (SA) is the study of how the variation in the output of a model can be attributed to different variations in the inputs of the model. Put another way, it is a technique for systematically changing variables in a model to determine the effects of such changes. Is that right?
  6. Consider the following LP model of a production plan of tables and chairs:

Max 3T + 2C(profit)

Subject to the constraints:

2T + C ≤100(carpentry hrs)

T + C ≤80(painting hrs)

T≤40

T, C ≥0(non-negativity)

1) Draw the feasible region.

2) Find the optimal solution.

3) Does the optimal solution change if the profit contribution for tables changed from $3 to $4 per table? (Clues: There is no effect on the feasible region. The slope of the level profit line changes. If the slope changes enough, a different corner point will become optimal.)

4) What if painting hours available changed from 80 to 100? (Clues: The constraint line shifts, which could change the feasible region. Slope of constraint line does not change. Corner point locations can change. The optimal solution can change.)

  1. Find the shortest path from node 1 to node 4 in the following graph using the Dijkstra’s algorithm. Why does this algorithm fail to obtain the correct answer?
  1. Pizza King and Noble Greek are two competingrestaurants. Each must determine simultaneously whether toundertake small, medium, or large advertising campaigns.Pizza King believes that it is equally likely that Noble Greekwill undertake a small, a medium, or a large advertisingcampaign. Given the actions chosen by each restaurant,Pizza King’s profits are asshown in the following table.

For themaximin, maximax, and minimax regret criteria, determinePizza King’s choice of advertising campaign.

  1. Suppose we are offered achoice between the following two lotteries:

L1: With probability 1, we receive $1 million.

L2: With probability .10, we receive $5 million.

With probability .89, we receive $1 million.

With probability .01, we receive $0.

Which lottery do we prefer? Now consider the followingtwo lotteries:

L3: With probability .11, we receive $1 million.

With probability .89, we receive $0.

L4: With probability .10, we receive $5 million.

With probability .90, we receive $0.

Which lottery do we prefer?

Suppose weprefer L1 to L2. Show that L3 must have a larger expectedutility than L4.

  1. You are given a choice between lottery 1 and lottery 2.

You are also given a choice between lottery 3 and lottery 4.

Lottery 1: A sure gain of $240

Lottery 2: 25% chance to gain $1,000 and 75%chance to gain nothing

Lottery 3: A sure loss of $750

Lottery 4: A 75% chance to lose $1,000 and a25% chance of losing nothing

84% of all people prefer lottery 1 over lottery 2, and 87%choose lottery 4 over lottery 3.

1）Explain why the choice of lottery 1 over lottery 2 andlottery 4 over lottery 3 contradicts expected utility maximization.

2) Can you explain this anomalous behavior?