51. The law firm Smith and jones 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 casework it will need its new lawyers to handle each month for the following year:

Month Casework (hr.) Month Casework(hr.)

Jan 650 Jul 750

Feb 450 Aug 900

Mar 600 Sept 800

Apr 500 Oct 650

May 700 Nov 700

Jun 650 Dec 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.

1. Formulate a Linear programming model for this problem
2. Solve this model by using the computer

52. In problem 51 the optimal solution results in a fractional (i.e., non-integer) number of lawyers being hired. Explain how you would 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 51.

67. 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 applejuice. 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 warehouse/distribution centers in Virginia, Kentucky, and Lousianna. 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:

…………………… …… .……….. Plant ……………………………………

Farm 4. Indiana 5. Georgia Supply (bushels)

1. Ohio .41 .57 24,000

2.Pennsylvania .37 .48 18,000

3. New York .51 .60 32,000

Plant Capacity 48,000 35,000

………………………………Distribution Centers………………………….

Plant 6. Virginia 7. Kentucky 8. Lousiana

4. Indiana .22 .10 .20

5. Georgia .15 .16 .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.

10. Solve the following mixed integer linear programming model by using the computer:

Maximize Z =

Subject to

5x1+3x2+6x3≤20

X1+3x2≤12

X1, ≥ 0

X2≥0 and integer

14. 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.

24. 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:

……………Distance (miles) ……………………………….

District Central School West School South School Student Population

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 students to bus from each district to each school to minimize the total busing miles traveled.

1. Formulate a linear programming model for this problem
2. Solve the model by using the computer.

Price: $17.0

Solution: The downloadable solution consists of 7 pages, 1000 words and 6 charts.
Deliverable: Word Document