SMQ 3043LINEAR PROGRAMMINGSTEPPING STONE METHOD

NOR IZZATI BT ABDUL HAMID D20081032229

STEPPING STONE METHOD To

FromAlbuquerque Boston Cleveland

Des Moines RM 5 RM 4 RM 3

Evansville RM 8 RM 4 RM 3

Fort Lauderdale RM 9 RM 7 RM 5

The objective of such a transportation problem is to

schedule shipments from sources to destinations so that total

transportation costs are minimized.

xij = the costs of shipping from i sources (factories) to j destinations (warehouses)

Complete ModelMinimize: Z = 5x11 + 4x12 + 3x13 + 8x21 + 4x22 + 3x23 + 9x31 + 7x32 + 5x33

Subject to: x11 + x12 + x13 ≤ 100 x21 + x22 + x23 ≤ 300x31 + x32 + x33 ≤ 300x11 + x21 + x31 ≥ 300x12 + x22 + x32 ≥ 200x13 + x23 + x33 ≥ 200

xij ≥ 0 , i = 1, 2, 3 , j = 1, 2, 3

D

E

F

A

B

C

(Sources)Factories

(Destinations)

WarehousesCapacities

≤ 100 units

≤ 300 units

≤ 300 units

Requirements≥ 300 units

≥ 200 units

≥ 200 units

supply

demand

To

From

A B C Supply

D

E

F

Demand

100 0 0

0100200

0 100 200

100 0

300 100 0

300 200 0

300

200

0

200

100

0

200

0

X12 → X11 → X21 → X22 X31 → X32 → X22 → X214 – 5 + 8 – 4 = + 3 9 – 7 + 4 – 8 = – 2

X23 → X22 → X32 → X33 X13 → X11 → X21 → X22 → X32 → X333 – 4 + 7 – 5 = + 1 3 – 5 + 8 – 4 + 7 – 5 = + 4

TO

FROMA B C

D100

E200 100

F100 200

45 3

8 4 3

9 7 5

–

–

+

+

X12 → X11 → X21 → X22 X32 → X22 → X21 → X314 – 5 + 8 – 4 = + 3 7 – 4 + 8 – 9 = + 2

X23 → X21 → X31 → X33 X13 → X11 → X31 → X333 – 8 + 9 – 5 = – 1 3 – 5 + 9 – 5 = + 2

TO

FROMA B C

D100

E100 200

F100 200

45 3

8 4 3

9 7 5

–

–

+

+

X21 → X23 → X33 → X31 X32 → X33 → X23 → X228 – 3 + 5 – 9 = + 1 7 – 5 + 3 – 4 = +1

X13 → X11 → X31 → X33 X12 → X11 → X31 → X33 → X23 → X223 – 5 + 9 – 5 = + 2 4 – 5 + 9 – 5 + 3 – 4 = + 2

TO

FROMA B C

D100

E200 100

F200 100

45 3

8 4 3

9 7 5

X11 = 100X22 = 200X23 = 100X31 = 200X33 = 100

Z = 5(100) + 4(200) + 9(200) + 3(100) + 5(100)Z = RM3900