## Quantitative Analysis for Management Decision Making (MBA 661) Assignment

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1). Dorian Auto manufactures luxury cars and trucks. The company believes that its most likely customers are high-income women and To reach these groups, Dorian auto has embarked on an ambitious TV advertising campaign and has decided to purchase 1-minute commercial spots on two types of programs: comedy show and football games. Each comedy commercial is seen by 7 million high income women and 2 million high-income men.

Each football commercial is seen by 2 million high-income women and 12 million high-income men. A 1-minute comedy advertising costs \$50,000 and a 1-minute football advertising costs \$100,000. Dorian would like the commercials to be seen by at least 28 million high-income women and 24 million high-income men. Use linear programing to determine how Dorian auto can meet its advertising requirements at minimum cost.

A.  Formulate the linear Programing Problem

B.  Solve the problem using graphic method.

C.  Find the dual problem for the primal you formulated in ‘A’ above

2).  Solve the following LPP using Graphic method and formulate the respective dual forms for each.

3).  My diet requires that all the food that I eat come from one of the four basic food groups (chocolate cake, ice cream, soda, and cheesecake). At present, the following four foods are available for consumption: brownies, chocolate ice cream, cola, and pineapple cheese-cake. Each brownie costs \$50, each scoop of chocolate ice cream costs \$20, each bottle of coca cola costs \$30, and each piece of pineapple cheese-cake \$80. Each day, I must in-gest at least 500 calories, 6 oz of chocolate, 10 oz of sugar, and 8 oz of fat. The nutritional content per unit of each food is shown in the following

 Type of food Calories Chocolate (Ounce) Sugar (Ounce) Fat (Ounce) Brownies 400 3 2 2 Chocolate  ice cream (1 scoop) 200 2 2 4 Cola (1 bottle) 150 0 4 1 pineapple cheese-cake (1 piece) 500 0 4 5

A. Formulate the linear Programing Problem.

B. Solve the problem using Simplex method.

C.  Find the dual form of this problem

D. Find the solution of the dual from the optimal tableau of the solution of the primal problem.

E. Find the shadow prices and interpret them.

F.  For which values of the objective function coefficient of the non-basic variables would the optimal solution mix remain unchanged?

G.  For which values of the objective function coefficient of the basic variables would the optimal solution mix remain unchanged?

H.  For which values of the right hand side values would the optimal solution mix remain unchanged?

4).  Beerco manufactures ale and beer from corn, hops, and malt. Currently, 40 lb of corn, 30 lb of hops, and 40 lb of malt are available. A barrel of ale sells for \$40 and requires 1 lb of corn, 1 lb of hops, and 2 lb of malt. A barrel of beer sells for \$ 50 and requires 2 lb of corn, 1 lb of hops, 1 lb of malt. Beerco can sell all ale and beer that is produced. If Beerco wants to maximize its revenue, solve the following questions and give the company advices based on your solutions.

A.  Develop the linear programing problem.

B.  Develop the dual LPP and find the solution for it using simplex method.

C.  Find the range of values of the price of ale for which the current basis remains optimal.

D.  Find the range of values of the price of beer for which the current basis remains optimal.

E.  Find the range of values of the amount of available corn for which the current basis remains optimal.

F.  Find the range of values of the amount of available hops for which the current basis remains optimal.

G.  Find the range of values of the amount of available malt for which the current basis remains optimal.

5). Two reservoirs are available to supply the water needs of three cities. Each reservoir can supply up to 50 million gallons of water per day. Each city would like to receive 40 million gallons per day. For each million gallons per day of unmet demand, there is a penalty. A city 1, the penalty is \$20; at city 2, the penalty is \$22; and a city 3, the penalty is \$23. The cost of transporting 1 million gallons of water from each reservoir to each city is shown in the table below.

 From To City 1 City 2 City 3 Reservoir 1 \$7 \$8 \$10 Reservoir 2 \$9 \$7 \$8

A.  Formulate a balanced transportation problem that can be used to minimize the sum of shortage and transport cost.

B. Find the initial basic feasible solution using Northwest corner rule.

C.  Find an optimal solution using stepping stone method.

6). A bank has two sites at which checks are processed. Site 1 can process 10,000 checks per day and site 2 can process 6,000 checks per day. The bank processes three types of checks: vendor, salary, and personal. The processing cost per check depends on the site (see in table below). Each day, 5,000 checks of each type must be processed.

A.  Formulate a balanced transportation problem that can be used to minimize the sum of shortage and transport cost.

B.  Find the initial basic feasible solution using intuitive method

C.  Find the initial basic feasible solution using Vogel’s Approximation method

D.  Find an optimal solution using MODI method.

7).  Doc Councilman is putting together a relay team for the 400-meter Each swimmer must swim 100 meters of breaststroke, backstroke, butterfly, or freestyle. Doc believes that each swimmer will attain the times given in the table below. To minimize the team’s time for the race, which swimmer should swim which stroke?

 Swimmer Time (Seconds) Free Breast Fly Back Gary Hall 54 54 51 53 Mark Spitz 51 57 52 52 Jim Montgomery 50 53 54 56 Chet Jastremski 56 54 55 53

8).  A company is taking bids on four construction jobs. Three people have placed bids on the Their bids (in thousands of dollars) are given in the table below (* indicates that the person did not bid on the given job). Person 1 can do only one job, but person 2 and 3 can each do as many as two jobs. Determine the minimum cost assignment of persons to jobs.

 From To 1 2 3 4 1 50 46 42 40 2 51 48 44 * 3 * 47 45 45

9). Consider the project below. List of all the activities with their respective optimistic, most likely and pessimistic time estimates are given.

 Activity to tm tp (1, 2) 4 6 8 (1, 3) 2 4 8 (2, 4) 1 3 7 (3, 4) 6 9 12 (3, 5) 5 10 15 (3, 6) 7 12 18 (4, 7) 5 9 12 (5, 7) 1 2 3 (6, 8) 2 3 6 (7, 9) 10 15 20 (8, 9) 6 9 11

A. Construct the network diagram.

B.  Find the critical path

C.  Find the early event time, late event time and total float for each activity.

10).  A project consists of the following set of activities.

 Activity Immediate predecessors Duration (Days) A – 3 B – 3 C – 1 D A, B 3 E A, B 3 F B, C 2 G D, E 4 H E 3

A. Construct the network diagram.

B.  Find the critical path

C.  Find the early event time, late event time and total float for each activity.

11).  Three strategies and three states of nature are given and payoffs represent (i) What is the optimal strategy if we apply the criterion of:

 Strategies States of nature N1 N2 N3 S1 47 49 33 S2 32 25 41 S3 51 30 14

A.  Pessimistic,

B.  Optimistic,

C.  Equal probability,

D.  Regret,

E.  Hurwicz The decision maker’s criterion of realism (α) being 0.7