EMGT 501

Mid-Term Exam

Due Day: Oct. 20 (Noon)

Note:

(a) Do not send me after copying your computer results. Answer formulations and these solutions.

(b) Put your mailing address so that I will be able to return your exam result via US postal service.

(c) Answer on a PPS that has many slides.

1. Slim-Down Manufacturing makes a line of nutritionally

complete, weight-reduction beverages. One of their products

is a strawberry shake which is designed to be a complete meal.

The strawberry shake consists of several ingredients. Some

information about each of these ingredients is given below.

Ingredient

Calories from Fat(per tbsp)

Total Calories(per tbsp)

Vitamin Content(mg/tbsp)

Thickeners(mg/tbsp)

Cost(¢/tbsp)

Strawberry flavoring 1 50 20 3 10Cream 75 100 0 8 8Vitamin supplement 0 0 50 1 25Artificial sweetener 0 120 0 2 15Thickening agent 30 80 2 25 6

The nutritional requirements are as follows. The beverage must total between 380 and 420 calories (inclusive). No more than 20 percent of the total calories should come from fat. There must be at least 50 milligrams (mg) of vitamin content. For taste reasons, there must be at least 2 tablespoons (tbsp) of strawberry flavoring for each tablespoon of artificial sweetener. Finally, to maintain proper thickness, there must be exactly 15 mg of thickeners in the beverage.

Management would like to select the quantity of each ingredient for the beverage which would minimize cost while meeting the above requirements.

(a) Formulate a linear programming model for this problem.(b) Solve this model by your computer.

2. Consider the following problem.Maximizesubject to

Let and denote the slack variables for the respective constraints. After you apply the simplex method, a portion of the final simplex tableau is as follows:

,234 4321 xxxxZ

423

524

4321

4321

xxxx

xxxx

.0,0,0,0 4321 xxxx

5x 6x

(a) Solve the problem

(b) What is B-1 ? How about B-1b and CBB-1b ?

Eq. ZZ (0) 1 1 1

(1) 0 1 -1(2) 0 -1 2

BasicVariable

RightSide

Coefficient of:

1x 2x 3x 4x 5x 6x

2x4x

3. The Lockhead Aircraft Co. is ready to begin a project to develop a new fighter airplane for the U.S. Air Force. The company’s contract with the Department of Defense calls for project completion within 100 weeks, with penalties imposed for late delivery.

The project involves 10 activities (labeled A, B, …, J), where their precedence relationships are shown in the following project network.

START FINISH

B

A

D

F

E

C

G

H

I

J

Using the PERT three-estimate approach, the usual three estimates of the duration of each activity have been obtained as given below.

ActivityA 28 weeks 32 weeks 36 weeksB 22 weeks 28 weeks 32 weeksC 26 weeks 36 weeks 46 weeksD 14 weeks 16 weeks 18 weeksE 32 weeks 32 weeks 32 weeksF 40 weeks 52 weeks 74 weeksG 12 weeks 16 weeks 24 weeksH 16 weeks 20 weeks 26 weeksI 26 weeks 34 weeks 42 weeksJ 12 weeks 16 weeks 30 weeks

OptimisticEstimate

Most LikelyEstimate

PessimisticEstimate

(a) Find the estimate of the mean and variance of the duration of each activity.

(b) Find the mean critical path.

(c) Find the approximate probability that the project will finish within 100 weeks.

(d) Is the approximate probability obtained in part (c) likely to be higher or lower than the true value?

4. Consider the game having the following payoff table.

(a) Formulate the problem to find an optimal mixed strategy according to the minimax criterion as a linear programming problem.

(b) Show its dual formulation.

(c) Use the simplex method to find these optimal mixed strategies of the primal and dual models.

1 2 3 4 51 1 -3 2 -2 12 2 3 0 3 -23 0 4 -1 -3 24 -4 0 -2 2 1

Player 1

Player 2Strategy

5. The Hit-and-Miss Manufacturing Company produces items that have a probability p of being defective. These items are produced in lots of 150. Past experience indicates that p for an entire lot is either 0.05 or 0.25. Furthermore, in 80 percent of the lots produced, p equals 0.05 (so p equals 0.25 in 20 percent of the lots). These items are then used in an assembly, and ultimately their quality is determined before the final assembly leaves the plant. Initially the company can either screen each item in a lot at a cost of $10 per item and replace defective items or use the items directly without screening. If the latter action is chosen, the cost of network is ultimately $100 per defective item. Because screening requires scheduling of inspectors and equipment, the decision to screen or not screen must be made 2 days before the screening is to take place. However, one item can be taken from the lot and sent to a laboratory for inspection, and its quality (defective or nondefective) can be reported before the screen/no screen decision must be made. The cost of this initial inspection is $125.

(a) Develop a decision analysis formulation of this problem by identifying the alternative actions, the states of nature, and the payoff table if the single item is not inspected in advance.

(b) Assuming the single item is not inspected in advance, use Bayes’ decision rule to determine which decision alternative should be chosen.

(c) Find EVPI. Does this answer indicate that consideration should be given to inspecting the single item in advance?

(d) Assume now that the single item is inspected in advance. Find the posterior probabilities of the respective states of nature for each of the two possible outcomes of this inspection.

(e) Find EVE. Is inspecting the single item worthwhile?

(f) Determine the optimal policy.

EMGT 501

HW #5

Answer

15.3-11

(a)

(b)

Alternative Success UnsuccessDepelop new product 1,500,000 -1,800,000Don't develop new product 0 0Prior probabilities 0.667 0.333

State of Nature

Choose to develop new product (expected payoff is $400,000).

Alternative Success UnsuccessDepelop new product 1,500,000 -1,800,000 400,000 MaximumDon't develop new product 0 0 0Prior probabilities 0.667 0.333

State of Nature ExpectedPayoff

(c)

Alternative Success UnsuccessDepelop new product 1,500,000 -1,800,000Don't develop new product 0 0Prior probabilities 0.667 0.333Maximum Payoff 1,500,000 0

State of Nature

Expected Payoff with Perfect Information = 1,000,000

This indicates that consideration should be given to conducting the market survey.

EVPI = EP (with perfect info) - EP (without more info) = 1,000,000 - 400,000 = $600,000

(d)

Data:

Success UnsuccessSuccess 0.6666667 0.8 0.2Unsuccess 0.3333333 0.3 0.7

P(Finding | State)FindingState of

NaturePrior

Probability

PosteriorProbabilities

Finding P(Finding) Success UnsuccessSuccess 0.6666667 0.842105 0.157894737Unsuccess 0.3333333 0.363636 0.636363636

P(State | Finding)State of Nature

(e)

not. if success, pred. if Optimal

0)]succ. no pred.|,([

000,600$

)000,800,1(6364.0)000,500,1(3636.0

)]succ. no pred.|,([

0)]succ. pred.|,([

000,979$

)000,800,1(1579,0)000,500,1(8421.0

)]succ. pred.|,([

21

2

1

2

1

aa

apE

apE

apE

apE

product.market survey,conduct not Do

: isstrategy optimal So

000,214$000,300$survey ofCost

000,220$000,400000,620EVE So

000,620$)0(367.0)000,979(633.0

infogiven payoff Expected

633.0313.03

28.0

)(p)|success pred.(p)(p)|success pred.(p

success) pred.(p

2211