9. transportation model

2-1

Transportation Model

TRANSPORTATION MODELTRANSPORTATION MODEL

Requirements - List of origins and each one’s capacity. List of destinations and each one’s demand. Unit cost of shipping.

Assumptions - Items to be shipped are homogeneous. Shipping cost per unit is the same. Only one route between origin and destination. Demand and supply is equal.

TRANSPORTATION MODEL SCENARIOTRANSPORTATION MODEL SCENARIO

D(demand)

B(demand)

C(demand)

A(demand)

1(supply)

2(supply)

3(supply)

A TRANSPORTATION TABLEA TRANSPORTATION TABLEWarehouse

4 7 7 1100

12 3 8 8200

8 10 16 5150

450

45080 90 120 160

A B C D

1

2

3

FactoryFactory 1can supply 100units per period

Total supplycapacity perperiod

Total demandper period

Demand

Warehouse B can use 90 units per period

NETWORK PRESENTATIONNETWORK PRESENTATION

Transportation problem can be represented as a network. Circles represent origins and destinations, and the arcs between them represent the decision variables, i.e. the amounts shipped.

Supply Origin

DemandDestination

D1

D2

D3

410

6

816

6

14 1810

100

300

300

200

300

200

S1

S2

S3

LINEAR PROGRAMMING FORMULATIONLINEAR PROGRAMMING FORMULATION

Let xij be unknown number of units shipped from origin i to destination j (i,j = 1,2,3)

x11 + x21 + x31 > 200x12 + x22 + x32 > 300x13 + x23 + x33 > 200

Demand constraintsSupply constraints

x11 + x12 + x13 < 100x21 + x22 + x33 < 300x31 + x32 + x33 < 300

min z = 4x11 + 10x12 + 6x13 + 8x21 + 16x22 + 6x23

+ 14x31 + 18x32 + 10x33

st.

xij > 0

Non-negative constraints

TRANSPORTATION PROBLEMTRANSPORTATION PROBLEM

Des Moines(100 units)capacity

Cleveland(200 units)required

Boston(200 units)required

Evansville(300 units)capacity

Ft. Lauderdale(300 units)capacity

Albuquerque(300 units)required

From(Sources)

To(Destinations)

Albuquerque Boston ClevelandDes Moines

Evansville

Fort Lauderdale

$5

$8

$9

$4

$4

$7

$3

$3

$5

TRANSPORTATION PROBLEMTRANSPORTATION PROBLEM

Des Moines(D)

Evansville(E)

Ft Lauderdale(F)

WarehouseReq.

Albuquerque(A)

Boston(B)

Cleveland(C)

FactoryCapacity

5 4 3

3

57

48

9

UNIT SHIPPING COST: 1 UNIT, FACTORY TO UNIT SHIPPING COST: 1 UNIT, FACTORY TO WAREHOUSEWAREHOUSE

Des Moines(D)

Evansville(E)

Ft Lauderdale(F)

WarehouseReq.

Albuquerque(A)

Boston(B)

Cleveland(C)

FactoryCapacity

300 200 200 700

300

300

100

TOTAL DEMAND & TOTAL SUPPLYTOTAL DEMAND & TOTAL SUPPLY

Des Moines(D)

Evansville(E)

Ft Lauderdale(F)

WarehouseReq.

Albuquerque(A)

Boston(B)

Cleveland(C)

FactoryCapacity

300 200 200 700

300

300

1005 4 3

3

57

48

9

TRANSPORTATION TABLETRANSPORTATION TABLE

• Start in the upper left-hand cell and allocate units to shipping routes as follows:

– Exhaust the supply (factory capacity) of each row before moving down to the next row.

– Exhaust the demand (warehouse) requirements of each column before moving to the next column to the right.

– Check that all supply and demand requirements are met.

NORTH - WEST CORNER METHOD NORTH - WEST CORNER METHOD

Des Moines(D)

Evansville(E)

Ft Lauderdale(F)

WarehouseReq.

Albuquerque(A)

Boston(B)

Cleveland(C)

FactoryCapacity

300 200 200 700

300

300

1005 4 3

3

57

48

9

100

200 100

100 200

NORTH - WEST CORNER METHOD (CONTD.) NORTH - WEST CORNER METHOD (CONTD.)

Total = $4,200

Des Moines(D)

Evansville(E)

Ft Lauderdale(F)

WarehouseReq.

Albuquerque(A)

Boston(B)

Cleveland(C)

FactoryCapacity

2000

700

800

5005 4 3

3

57

48

9

500

600 200

400 300

400 900 700

Total Cost: 10,200

LEAST COST METHOD LEAST COST METHOD

Des Moines(D)

Evansville(E)

Ft Lauderdale(F)

WarehouseReq.

Albuquerque(A)

Boston(B)

Cleveland(C)

FactoryCapacity

2000

700

800

5005 4 3

3

57

48

9

100

800

400 900 700

Total Cost: 9,100

400

700

SAME EXAMPLE WITH NWM SAME EXAMPLE WITH NWM

EXCEL TEMPLATE EXCEL TEMPLATE