Introduction to Operations Research

ANSWERS

Transportation, Assignment, and Transshipment Problems

Prof. dr. E-H. Aghezzafir. Rodrigo Rezende Amaral

Exercise 1

Optimize using the stepping stone algorithm.

(MODI – Modified Distribution Method)

Exercise 1

xij: units produced in month i to meet the demand of month j

Balanced Transportation Problem:

J F M Dummy Supply Row Penalty

J400 420 440 0

030 5

F425 420 440 0

010 20

M420 415 410 0

015 20

Demand 0 0 0 0Column Penalty x 5 x x

Exercise 1

Vogel’s method:

Exercise 1

J F M Dummy Supply Row Penalty

J400 420 25 440 0 0

030 5

F25 425 420 25 440 0

010 20

M25 420 415 410 5 0

015 20

Demand 0 0 0 0Column Penalty x 5 x x

J → M: +440 - 410 + 415 - 420 = 25J → D: +0 - 0 + 420 - 420 = 0F → J: +425 - 400 + 420 - 420 = 25F → M: +440 - 410 + 415 - 420 = 25M → J: +420 - 415 + 420 - 400 = 25M → D: +0 - 0 + 420 - 415 = 5

Stepping stone:

All indices ≥ 0 → Solution is optimal

Exercise 1

J F M Dummy Supply Row Penalty

J400 420 440 0

030 5

F425 420 440 0

010 20

M420 415 410 0

015 20

Demand 0 0 0 0Column Penalty x 5 x x

Transportation Simplex (MODI):

( Iij ) = cij = cij – ui – vj

Exercise 2

Answer:

- 200 word processing files should be stored on the hard disk

- 100 word processing files in the computer memory

- 100 packaged programs on tape

- 100 data files on tape.

Exercise 3

Exercise 3

Hungarian Method:

Exercise 3

Hungarian Method:

-1

+1

Exercise 3

Hungarian Method:

Exercise 3

Hungarian Method:

Exercise 3

Hungarian Method - Two solutions

Exercise 3

Hungarian Method - Two solutions

Exercises 5 + 6

Exercises 5 + 6

5

5

7

3

3

7

Total cost: 98

Exercise 7

Exercise 7

Exercise 7

Exercise 7

Exercise 7

Exercise 7

Exercise 7

Exercise 7

Exercise 7

Exercise 7

3-1#; 4-2#; 5-3#; 1-4*; 6-5#; 7-6#; 2-7*

# starts in NY; * starts in Chicago

Exercise 7

+2+5

+3

+5+5+3

+2

3-1#; 4-2#; 5-3#; 1-4*; 6-5#; 7-6#; 2-7*

Exercise 8

Exercise 8

1P 1S 2P 2S 3P 3S Supply

12 2 0 0 1 1

35

2-∞ -∞ 2 2 3 3

35

3-∞ -∞ -∞ -∞ 3 3

35

DUMMY-∞ 0 -∞ 0 -∞ 0

5

Demand 20 15 15 20 25 15

xi-jk: units produced in month ito meet the demand of month jfrom customer type k

i ϵ {1,2,3,D}j ϵ {1,2,3}k ϵ {P,S}

Exercise 10

Exercise 10

Exercise 10

Extra Exercise