EMGT 501 Mid-Term Exam Due Day: Oct. 18 (Noon)

Note: (a) Do not send me after copying your computer results of QSB. Answer what are your decision variables, formulation and solution, only. See my HW answer on my HP. (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 series of slides. (d) Do not discuss on the exam with other students. (e) Return your answer attached to your e-mail.

1. Joyce and Marvin run a day care for preschoolers 1. Joyce and Marvin run a day care for preschoolers. They are trying to decide what to feed the children for lunches. They would like to keep their costs down, but also need to meet the nutritional requirements of the children. They have already decided to go with peanut butter and jelly sandwiches, and some combination of graham crackers, milk, and orange juice. The nutritional content of each food choice and its cost are given in the table below.

Calories from Fat Total Calories Vitamin C (mg) Protein (g) Cost (¢) Food Item Bread (1 slice) 10 70 3 5 Peanut butter (1tbsp) 75 100 4 4 Strawberry jelly (1 tbsp) 50 3 7 Graham cracker (1 cracker) 20 60 1 8 Milk (1 cup) 70 150 2 8 15 Juice (1 cup) 100 120 1 35

The nutritional requirements are as follows The nutritional requirements are as follows. Each child should receive between 400 and 600 calories. No more than 30 percent of the total calories should come from fat. Each child should consume at least 60 milligrams (mg) of vitamin C and 12 grams (g) of protein. Furthermore, for practical reasons, each child needs exactly 2 slices of bread (to make the sandwich), at least twice as much peanut butter as jelly, and at least 1 cup of liquid (milk and/or juice). Joyce and Marvin would like to select the food choices for each child which minimize cost while meeting the above requirements. (a) Formulate a linear programming model for this problem. (b) Solve the problem.

2. Consider the following problem. 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: Eq. Z (0) 1 2 (1) -1 (2) Basic Variable Right Side Coefficient of: 1 x 2 x 3 x 4 x 5 x 6 x 2 x 4 x (a) Solve the problem. (b) What is B-1 ? How about B-1b and CBB-1b ?

3. Alfred Lowenstein is the president of the research division for Better health, Inc., a major pharmaceutical company. His most important project coming up is the development of a new drug to combat AIDS. He has identified 10 groups in his division which will need to carry out different phases of this research and development project. Referring to the work to be done by the respective groups as activities A, B, …, J, the precedence relationships for when these groups need to do their work are shown in the following project network.

A E I C F START Finish D G J B H Duration Activity Estimated Mean Estimated Variance A 4 months 5 months B 6 months 10 months C 4 months 8 months D 3 months 6 months E 8 months 12 months F 4 months 6 months G 3 months 5 months H 7 months 14 months I 5 months 8 months J 5 months 7 months

Find the mean critical path for this project. Use this mean critical path to find the approximate probability that the project will be completed within 22 months. Now consider the other three paths through this project network. For each of these paths, find the approximate probability that the path will be completed within 22 months. What should Alfred tell his CEO about the likelihood that the drug will be ready within 22 months?

4. Consider the game having the following payoff table. Player 2 Strategy 1 2 3 4 1 5 -3 3 -1 Player 1 2 -2 4 3 2 3 3 2 -4 4 (a) Formulate the problem of finding optimal mixed strategies according to the maxmin criterion as a linear programming problem. (b) Use the simplex method to find these optimal mixed strategies.