Problems:
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If using QM for Windows to solve the problem, please copy and paste the appropriate solution windows, as well as including the model formulation, in a Word document. You can do a screen capture of the appropriate QM windows and paste those into a Word document. Please follow appropriate "copy and paste" procedures to avoid creating a very large file. Data from QM for Windows can also be saved as an HTML file. You can only save data one window at a time. Highlight the QM for Windows output window you want to save and select "Save as HTML." You can then edit the HTML file in Word, or you can copy it directly to Word. Please show all work, where appropriate.
1. The law firm of Smith and Smith is recruiting at law schools for new lawyers for the coming year. The firm has developed the following estimate of the number of hours of case work it will need its new lawyers to handle each month for the following year:
|
Month
|
Casework (hr.)
|
Month
|
Casework (hr.)
|
|
January
|
650
|
July
|
750
|
|
February
|
450
|
August
|
900
|
|
March
|
600
|
September
|
800
|
|
April
|
500
|
October
|
650
|
|
May
|
700
|
November
|
700
|
|
June
|
650
|
December
|
500
|
Each new lawyer the firm hires is expected to handle 150 hours per month of casework and to work all year. All casework must be completed by the end of the year. The firm wants to know how many new lawyers it should hire for the year.
a) formulate a linear programming model for this problem
b) Solve this model by using the computer
2. In problem 1, the optimal solution results in a fractional (i.e., non-integer) number of lawyers being hired. Explain how would you go about logically determining a new solution with a whole (integer) number of lawyers being hired and discuss the difference in results between this new solution and the optimal non-integer solution obtained in Problem 1.
3. Valley Fruit Products Company has contracted with apple growers in Ohio, Pennsylvania and New York to purchase apples that the company then ships to its plants in Indiana and Georgia, where they are processed into apple juice. Each bushel of apples produces 2 gallons of apple juice. The juice is canned and bottled at the plants and shipped by rail and truck to warehouses/distribution centers in Virginia, Kentucky and Louisiana. The shipping costs per bushel from the farms to the plants and the shipping costs per gallon from the plants to the distribution centers are summarized in the following tables:
|
|
Tables
|
|
Farm
|
4. Indiana
|
5. Georgia
|
Supply (bushels)
|
|
1. Ohio
|
0.41
|
0.57
|
24,000
|
|
2. Pennsylvania
|
0.37
|
0.48
|
18,000
|
|
3. New York
|
0.51
|
0.60
|
32,000
|
|
Plant Capacity
|
48,000
|
35,000
|
|
|
|
Distribution Centers
|
|
Farm
|
6.. Virginia
|
7. Kentucky
|
8. Louisiana
|
|
1. Indiana
|
0.22
|
0.10
|
0.20
|
|
2. Georgia
|
0.15
|
0.16
|
0.18
|
|
Demand (gal.)
|
9,000
|
12,000
|
15,000
|
Formulate and solve a linear programming model to determine the optimal shipments from the farms to the plants and from the plants to the distribution centers in order to minimize total shipping costs.
4. Solve the following mixed integer linear programming model by using the computer
Maximize Z = 5X1 +6X2 + 4X3
Subject to
5X1 +3X2 + 6X3 ≤ 20
X1 + 3X2 + ≤12
X1, X3 ≥0
X2 ≥0 and integer
5.The Texas Consolidated Electronics Company is contemplating a research and development program encompassing eight research projects. The company is constrained from embarking on all projects by the number of available management scientists (40) and the budget available for R&D projects ($300,000). Further, if project 2 is selected, project 5 must also be selected (but not vice versa). Following are the resource requirements and the estimated profit for each project.
|
Project
|
Expense
($1,000s)
|
Management
Scientists Required
|
Estimated Profit
($1,000,000s)
|
|
1
|
$60
|
7
|
0.36
|
|
2
|
110
|
9
|
0.82
|
|
3
|
53
|
8
|
0.29
|
|
4
|
47
|
4
|
0.16
|
|
5
|
92
|
7
|
0.56
|
|
6
|
85
|
6
|
0.61
|
|
7
|
73
|
8
|
0.48
|
|
8
|
65
|
5
|
0.41
|
Formulate the integer programming model for this problem, and solve it by using the computer.
6. Brooks City has three consolidated high schools, each with a capacity of 1,200 students. The school board has partitioned the city into five busing districts - north, south, east, west, and central - each with different high school student populations. The three schools are located in the central, west and south districts. Some students must be bused outside their districts, and the school board wants to minimize the total bus distance traveled by these students. The average distances from each district to the three schools and the total student population in each district are as follows:
|
District
|
Distance (miles)
|
Student
Population
|
|
Central
School
|
West
School
|
South
School
|
|
North
|
8
|
11
|
14
|
700
|
|
South
|
12
|
9
|
-
|
300
|
|
East
|
9
|
16
|
10
|
900
|
|
West
|
8
|
-
|
9
|
600
|
|
Central
|
-
|
8
|
12
|
500
|
The school board wants to determine the number of student to bus from each district to each school to minimize the total busing miles traveled.
a) formulate a linear programming model for this problem
b) solve the model by using the computer