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Homework 05 Quantitative Business Analysis Spring 2020 Due: March 16, 3:00 pm

Problem 1. Michael would like to drink 3 pints of home-brewed craft beer today and an additional 4 pints tomorrow. There are two neighbors on his block who brew beer. Louise is able to sell up to 5 pints of her beer. She’s charging $3.00 a pint today and will charge $2.70 a pint tomorrow. Serge is able to sell up to 4 pints of his beer, at $2.90 a pint today and at $2.80 a pint tomorrow. Transportation models are quite flexible. The “origins” and “destinations” don’t necessarily need to be physical plants or warehouses. In this problem, there is a time component that can be modeled as different destinations – consider the beer that Michael receives today as one destination and the beer he receives tomorrow as the second destination. a)

Draw a network diagram representing the problem.

b) Formulate a linear optimization model that can be used to help Michael decide how much beer to purchase from each neighbor and when. (You do not need to set your model up in Excel and you do not need to solve it.)

Today

Tomorrow

Louise

$3.00

$2.70

Serge

$2.90

$2.80

Min(Louise,serge) =$2.90

Min(Louise,serge) =2.70

It would be optimal for Michael to purchase 3 pints of beer from Serge today & should purchase 4 pints from Louise tomorrow. Today’s value is $2.90 per pint while tomorrows value is $2.70 per pint in comparison to a higher price by the competitor. Problem 2. Tri-County Utilities, Inc., supplies natural gas to customers in a three-county area. The company purchases natural gas from two companies: Southern Gas and Northwest Gas. Demand forecasts for the coming winter season are as follows: Hamilton County, 400 units; Butler County, 200 units; and Clermont County, 300 units. Contracts to provide the following quantities have been written: Southern Gas, 500 units; and Northwest Gas, 400 units. Distribution costs for the counties vary, depending upon the location of the suppliers. The distribution costs per unit (in thousands of dollars) are as follows: From

To Hamilton

Butler

Clermont

Southern Gas

10

20

15

Northwest Gas

12

15

18

a)

Develop a network representation of this problem.

b) Develop a linear programming model that can be used to determine the plan that will minimize total distribution costs. The variables X, Y, Z represents the amount of X units shipped from supply node Y to demand node Z. X11 represents that the units shipped from supply node “southern” to demand node “Hamilton”. X12 represents that the units shipped from supply node “southern” to demand node “Butler”. X13 represents that the units shipped from supply node “southern” to demand node “Clermont”. X21 represents that the units shipped from supply node “Northwest” to demand node “Hamilton”. X22represents that the units shipped from supply node “Northwest” to demand node “Butler”. X23 represents that the units shipped from supply node “Northwest” to demand node “Clermont”.

Objective function = minimize total distribution costs Min = 10*x11 + 20x*12 + 15*x13 + 12*x21 + 15*x22 + 18*x23 Constraints: X11 + x12 + x13 = 0; X13 c) Enter your model in Excel and use Solver to find the optimal solution. Describe the distribution plan and show the total distribution cost. d) Recent residential and industrial growth in Butler County has the potential for increasing demand by as much as 100 units. Which supplier should Tri-County contract with to supply the additional capacity? x11 + x12 + x13 < 600 x21 + x22 + x23 < 500 x11 + x21 = 400 x12 + x22 = 300 x13 + x23 = 300 x11 = 300, x12 = 0, x13 = 300, x21 = 100, x22=300, x23 = 0 c = 10x11 + 20x12 + 15x13 + 12x21 + 15x22 + 18x23 c = 10(300) + 20(0) + 15(300) + 12(100) + 15(300) + 18(0) = $13,200 e) Assume that Southern Gas cannot provide any gas to Hamilton County. Revise your LP in part b and resolve the problem.

Problem 3. Forbelt Corporation has a one-year contract to supply motors for all refrigerators produced by the Ice Age Corporation. Ice Age manufactures the refrigerators at four locations around the country: Boston, Dallas, Los Angeles, and St. Paul. Plans call for the following number (in thousands) of refrigerators to be produced at each location:

Forbelt’s three plants are capable of producing the motors. The plants and production capacities (in thousands) are as follows:

Because of varying production and transportation costs, the profit that Forbelt earns on each lot of 1000 units depends on which plant produced the lot and which destination it was shipped to. The following table gives the accounting department estimates of the profit per unit (shipments will be made in lots of 1000 units): Produced At

Shipped to Boston

Dallas

Los Angeles

St. Paul

Denver

7

11

8

13

Atlanta

20

17

12

10

Chicago

8

18

13

16

With profit maximization as a criterion, Forbelt’s management wants to determine how many motors should be produced at each plant and how many motors should be shipped from each plant to each destination. a) Develop a network representation of this problem.

b) Enter your model in Excel and use Solver to find the optimal solution Problem 4. Scott and Associates, Inc., is an accounting firm that has three new clients. Project leaders will be assigned to the three clients. Based on the different backgrounds and experiences of the leaders, the various leader–client assignments differ in terms of projected completion times. The possible assignments and the estimated completion times in days are as follows: Project leader

Client 1

2

3

Jackson

10

16

32

Ellis

14

22

40

Smith

22

24

34

a)

Develop a network representation of this problem.

b) Formulate the problem as a linear program, and solve. What is the total time required? Min. 10*J1 + 16*J2 + 32*J3 + 14*E1 + 22*E2 + 40*E3 + 22*S1 + 24*S2 + 34*S3 Constraints; J1 + J2 + J3 ≤ 1 E1 + E2 + E3 ≤ 1 S1 + S2 + S3 ≤ 1 J1 + E1 + S1 = 1 J2 + E2 + S2 = 1 J3 + E3 + S3 = 1 J1 ≥ 0; J2 ≥ 0; J3 ≥ 0 E1 ≥ 0; E2 ≥ 0; E3 ≥ 0 S1 ≥ 0; S2 ≥ 0; S3 ≥ 0 c = 10*J1 + 16*J2 + 32*J3 + 14*E1 + 22*E2 + 40*E3 + 22*S1 + 24*S2 + 34*S3; J1 + J2 + J3 = 0; E1 >= 0; E2 >= 0; E3 >= 0; S1 >= 0; S2 >= 0; S3 >= 0;

Jackson should work for Client 2, Ellis would work for Client 1 and Smith should work for Client 3. The total time required is 64 days....