The Salem City Council is making plans to allocate funds ($900,000) remaining in

Question

The Salem City Council is making plans to allocate funds ($900,000) remaining in this year’s budget. There are nine project considered. The estimated cost of each project, the estimated number of new jobs each would create, and the desirability of each project (the larger the better) are summarized below: Page: 1 of 3 Remark 1: The homework is a groupwork. (You may check for your group information on webonline.) Remark 2: Submit your written homework with the signed honor code to the Teaching Assistant. Academic integrity is expected of all students of C ankaya University at all times, whether in the presence or absence of members of the faculty. Understanding this, please read the following Honor Code carefully. Put the honor code and sign it at the beginning of your written submission. Submissions without the signed honor code will receive zero from the assignment. Remark 3: For each question make sure that you have defined and written: * Decision variables * Objective and objective function * Constraints I(we) hereby declare that I(we) have neither given nor received any aid in this assignment. Signature(s) 1. [13 points] Project 1. Hire new police officers 2. Modernize police headquarters 3. Buy two new police cars 4. Give bonuses to foot patrol police officers 5. Buy new fire truck/support equipment 6. Hire assistant fire chief 7. Restore cuts to sports programs at schools 8. Restore cuts to school music 9. Buy new computers for high school Cost ($1,000) 400 350 50 100 500 90 220 150 140 New jobs 7 41 0 17 1 25 0 19 2 36 1 9 8 28 3 16 2 30 Desirability The council is going to select the project(s) based on their desirability and cannot partially invest in a project. In addition, the council wants to create at least 10 jobs, and wants to invest at least in two projects. (a) [5 points] Formulate an integer programming model for the above problem. Write necessary constraint(s) for the following parts. (Notes: Each part is independent.) (b) [1 point] Although police-related projects (1, 2, 3, 4) have high desirability, the council feels that it must also be fair to other sectors of public service (fire (6) and education (7, 8, 9)). Accordingly, it wishes to fund at most three of the police-related projects. (c) [2 points] Either two police cars should be purchased or the fire truck should be purchased but not both. (d) [1 point] The council believes that if it decides to restore funds cut from the sports programs at the schools (7), or restore funds cut from their music programs (8), then the other should be funded. (e) [2 points] Both sports and music funds must be restored before new computer equipment can be purchased. (f) [1 point] Project 1 must not be selected if both project 2 and project 3 are selected. (g) [2 points] Project 1 must not be selected if either project 2 or project 3 is selected. (h) [1 point] Number of police-related projects should not exceed the number of education-related projects.

Explanation / Answer

(a) Integer programming models are given below:

Decision variables: Let Xi be a binary variable such that Xi =1, if project "i" is selected, otherwise Xi = 0 , where i =1 to 9

(b) Objective: Maximize 41X1 + 17X2 + 25X3 + 19X4 + 36X5 + 9X6 + 28X7 + 16X8 + 30X9

s.t.

400X1 + 350X2 + 50X3 + 100X4 + 500X5 + 90X6 + 220X7 + 150X8 + 140X9 <= 900

7X1 + 1X3 + 2X5 + 1X6 + 8X7 + 3X8 + 2X9 >= 10

X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 >= 2

X1 + X2 + X3 + X4 <= 3

X1, X2, X3, X4, X5, X6, X7, X8, X9 = 0,1

_________________

(c) Objective: Maximize 41X1 + 17X2 + 25X3 + 19X4 + 36X5 + 9X6 + 28X7 + 16X8 + 30X9

s.t.

400X1 + 350X2 + 50X3 + 100X4 + 500X5 + 90X6 + 220X7 + 150X8 + 140X9 <= 900

7X1 + 1X3 + 2X5 + 1X6 + 8X7 + 3X8 + 2X9 >= 10

X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 >= 2

X3 + X5 = 1

X1, X2, X3, X4, X5, X6, X7, X8, X9 = 0,1

_________________

(d) Objective: Maximize 41X1 + 17X2 + 25X3 + 19X4 + 36X5 + 9X6 + 28X7 + 16X8 + 30X9

s.t.

400X1 + 350X2 + 50X3 + 100X4 + 500X5 + 90X6 + 220X7 + 150X8 + 140X9 <= 900

7X1 + 1X3 + 2X5 + 1X6 + 8X7 + 3X8 + 2X9 >= 10

X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 >= 2

X7 - X8 = 0

X1, X2, X3, X4, X5, X6, X7, X8, X9 = 0,1

_________________

(e) Objective: Maximize 41X1 + 17X2 + 25X3 + 19X4 + 36X5 + 9X6 + 28X7 + 16X8 + 30X9

s.t.

400X1 + 350X2 + 50X3 + 100X4 + 500X5 + 90X6 + 220X7 + 150X8 + 140X9 <= 900

7X1 + 1X3 + 2X5 + 1X6 + 8X7 + 3X8 + 2X9 >= 10

X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 >= 2

X7 + X8 - 2X9 >= 0

X1, X2, X3, X4, X5, X6, X7, X8, X9 = 0,1

_________________

(f) Objective: Maximize 41X1 + 17X2 + 25X3 + 19X4 + 36X5 + 9X6 + 28X7 + 16X8 + 30X9

s.t.

400X1 + 350X2 + 50X3 + 100X4 + 500X5 + 90X6 + 220X7 + 150X8 + 140X9 <= 900

7X1 + 1X3 + 2X5 + 1X6 + 8X7 + 3X8 + 2X9 >= 10

X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 >= 2

X2 + X3 + X1 <= 2

X1, X2, X3, X4, X5, X6, X7, X8, X9 = 0,1

_________________

(g) Objective: Maximize 41X1 + 17X2 + 25X3 + 19X4 + 36X5 + 9X6 + 28X7 + 16X8 + 30X9

s.t.

400X1 + 350X2 + 50X3 + 100X4 + 500X5 + 90X6 + 220X7 + 150X8 + 140X9 <= 900

7X1 + 1X3 + 2X5 + 1X6 + 8X7 + 3X8 + 2X9 >= 10

X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 >= 2

X2 + X3 + 2X1 <= 2

X1, X2, X3, X4, X5, X6, X7, X8, X9 = 0,1

_________________

(h) Objective: Maximize 41X1 + 17X2 + 25X3 + 19X4 + 36X5 + 9X6 + 28X7 + 16X8 + 30X9

s.t.

400X1 + 350X2 + 50X3 + 100X4 + 500X5 + 90X6 + 220X7 + 150X8 + 140X9 <= 900

7X1 + 1X3 + 2X5 + 1X6 + 8X7 + 3X8 + 2X9 >= 10

X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 >= 2

X1 + X2 + X3 + X4 - X7 - X8 - X9 <= 0

X1, X2, X3, X4, X5, X6, X7, X8, X9 = 0,1

Project Cost ($1,000) New jobs Desirability Hire new police officers 400 7 41 Modernize police headquarters 350 0 17 Buy two new police cars 50 1 25 Give bonuses to foot patrol police officers 100 0 19 Buy new fire truck/support equipment 500 2 36 Hire assistant fire chief 90 1 9 Restore cuts to sports programs at schools 220 8 28 Restore cuts to school music 150 3 16 Buy new computers for high school 140 2 30

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