Answer to Question #226214 in Operations Research for mafi

Question

1.     A transportation problem involves the following costs, supply, and demand.

 

 

To

 

 

From

1

2

3

4

Supply

1

$500

750

300

450

12

2

650

800

400

600

17

3

400

700

500

550

11

Demand

10

10

10

10

 

Required:

             i.           Find the initial solution using the northwest corner method, the minimum cell cost method, and Vogel's Approximation Method. Compute total cost for each.

           ii.           Using the VAM initial solution, find the optimal solution using the modified distribution method (MODI).

Answer 1

i. Find the initial solution using the northwest corner method, the minimum cell cost method, and Vogel's Approximation Method. Compute total cost for each.

Northwest Corner Method

To

From 1 2 3 4 Supply

1 $500 750 300 450 12

10 2

2 650 800 400 600 17

8 9

3 400 700 500 550 11

1 10

Demand 10 10 10 10 40

Z = 500X11 + 750X12 + 800X22 + 400X23 + 500X33 + 550X34

Z = 500 (10) + 750 (2) + 800 (8) + 400 (9) + 500 (1) + 550 (10)

Z = 5,000 + 1,500 + 6,400 + 3,600 + 500 + 5,500

Answer: Total Cost: Z = $ 22,500

Minimum Cell Cost Method

To

From 1 2 3 4 Supply

1 $500 750 300 450 12

10 2

2 650 800 400 600 17

10 7

3 400 700 500 550 11

10 1

Demand 10 10 10 10 40

Z = 300X13 + 450X14 + 800X22 + 600X24 + 400X31 + 550X34

Z = 300 (10) + 450 (2) + 800 (10) + 600 (7) + 400 (10) + 550 (1)

Z = 3,000 + 900 + 8,000 + 4,200 + 4,000 + 550

Answer: Total Cost: Z = $ 20,650

Vogel’s Approximation Method

To

From 1 2 3 4 Supply

1 $500 750 300 450 12

2 10 300

2 650 800 400 600 17

7 10 200

3 400 700 500 550 11

10 1 150

Demand 10 10 10 10 40

Z = 750X12 + 450X14 + 800X22 + 400X23 + 400X31 + 700X32

Z = 750 (2) + 450 (10) + 800 (7) + 400 (10) + 400 (10) + 700 (1)

Z = 1,500 + 4,500 + 5,600 + 4,000 + 4,000 + 700

Answer: Total Cost: Z = $ 20,300

ii. Using the VAM initial solution, find the optimal solution using the modified distribution method (MODI).

SOLUTION:

X12 : u1 + v2 = 750 Let u1 = 0

X14 : u1 + v4 = 450 Let v2 = 750

X22 : u2 + v2 = 800 Let v4 = 450

X23 : u2 + v3 = 400 Let u2 = 50

X31 : u3 + v1 = 400 Let v3 = 350

X32 : u3 + v2 = 700 Let u3 = –50

v1 = 450

vi v1 = 450 v2 = 750 v3 = 350 v4 = 450

To

ui From 1 2 3 4 Supply

u1 = 0 1 $500 750 300 450 12

2 10

u2 = 50 2 650 800 400 600 17

7 10

u3 = –50 3 400 700 500 550 11

10 1

Demand 10 10 10 10 40

X11 : k11 = 500 – 0 – 450 = 50

X13 : k13 = 300 – 0 – 350 = –50

X21 : k21 = 650 – 50 – 450 = 150

X24 : k24 = 600 – 50 – 450 = 50

X33 : k33 = 500 – (–50) – 350 = 200

X34 : k34 = 550 – (–50) – 450 = 150

To

From 1 2 3 4 Supply

1 $500 - 750 + 300 450 12

2 10

2 650 + 800 - 400 600 17

7 10

3 400 700 500 550 11

10 1

Demand 10 10 10 10 40

To

From 1 2 3 4 Supply

1 $500 750 300 450 12

2 10

2 650 800 400 600 17

7 10

3 400 700 500 550 11

10 1

Demand 10 10 10 10 40

 

X13 : u1 + v3 = 300 Let u1 = 0

X14 : u1 + v4 = 450 Let v3 = 300

X22 : u2 + v2 = 800 Let v4 = 450

X23 : u2 + v3 = 400 Let u2 = 100

X31 : u3 + v1 = 400 Let v2 = 700

X32 : u3 + v2 = 700 Let u3 = 0

v1 = 400

vi v1 = 450 v2 = 750 v3 = 350 v4 = 450

To

ui From 1 2 3 4 Supply

u1 = 0 1 $500 750 300 450 12

2 10

u2 = 100 2 650 800 400 600 17

9 8

u3 = 0 3 400 700 500 550 11

10 1

Demand 10 10 10 10 40

X11 : k11 = 500 – 0 – 400 = 100

X12 : k12 = 750 – 0 – 700 = 50

X21 : k21 = 650 – 100 – 400 = 150

X24 : k24 = 600 – 100 – 450 = 100

X33 : k33 = 500 – 0 – 300 = 200

X34 : k34 = 500 – 0 – 450 = 50

Z = 300X13 + 450X14 + 800X22 + 400X23 + 400X31 + 700X32 

Z = 300 (2) + 450 (10) + 800 (9) + 400 (8) + 400 (10) + 700 (1)

Z = 600 + 4,500 + 7,200 + 3,200 + 4,000 + 700

Answer: Z = $ 20,200