Solution Summary

 

The optimal solution to extended problem #3:

 

School

Teacher

1

2

3

4

5

6

7

8

9

10

1

 

 

 

A

 

 

 

 

P

 

2

P

 

 

 

 

 

 

 

A

 

3

 

 

 

 

P

 

 

A

 

 

4

 

 

 

 

 

P

A

 

 

 

5

 

 

 

 

 

A

P

 

 

 

6

 

 

A

P

 

 

 

 

 

 

7

A

 

 

 

 

 

 

 

 

P

8

 

 

 

 

A

 

 

P

 

 

9

 

A

P

 

 

 

 

 

 

 

10

 

P

 

 

 

 

 

 

 

A

 

With the optimal objective value of 986.

 

It is not logical to compare the optimal value found in this problem to any other problem, since the objective function has incorporated the reduction of utility by the reimbursement of travelling cost to teachers, which is not accounted for in any other problems.

 

Mathematical Model

 

Assumptions

In additions to the assumptions of the core problem, we also assume that a fixed conversion factor of 3 utility per dollars paid is applied by all schools.

 

Index Sets and Variables

The index sets and variables are the same as defined for the core problem.

 

Data

The conversion factor is defined as a scalar, such that any unanticipated changes can be easily handled.

 

In addition, the data regarding the travelling cost must be supplied in order to determine the best assignment of the teachers to the school

Travelling cost to be reimbursed to each teacher:

Morning session:

School

1

2

3

4

5

6

7

8

9

10

1

6

5

3

2

1

6

3

2

1

4

2

9

8

2

5

3

4

2

4

1

4

3

11

9

9

8

7

6

5

4

4

9

4

5

1

1

3

3

4

5

4

2

5

Teacher

5

7

6

5

2

5

4

3

5

8

7

6

5

4

10

2

4

7

8

7

6

5

7

5

4

2

5

5

4

6

8

1

11

8

9

8

7

6

5

4

3

2

1

10

9

5

4

3

8

7

1

9

9

8

4

10

9

6

8

5

7

4

5

6

3

2

 

 

Travelling cost to be reimbursed to each teacher:

Afternoon session:

School

1

2

3

4

5

6

7

8

9

10

1

8

5

6

7

9

8

7

5

5

6

2

3

4

7

8

6

2

2

4

8

3

3

14

12

25

3

5

9

3

5

7

8

4

5

1

1

3

3

4

5

4

2

5

Teacher

5

7

6

5

2

5

4

3

5

8

7

6

5

4

10

2

4

7

8

7

6

5

7

5

4

2

5

5

4

6

8

1

11

8

9

8

7

6

5

4

3

2

1

10

9

5

4

3

8

7

1

9

9

8

4

10

9

6

8

5

7

4

5

6

3

2

 

 

Program

The objective function is modified as follows:

z = S(ikj)(xikj*uikj - cost*xikj*tikj), where cost is the conversion factor.

 

The second part of the objective function as defined above represents the reduction in utility by the reimbursement of cost.