Page 302 -

282 CHAPTER 7 TRANSPORTATION, ASSIGNMENT AND TRANSSHIPMENT PROBLEMS

Table 7.1 Transportation Cost Per Unit for the Foster Electronics Transportation
Problem (Euros)
Destination

Origin Boston Dubai Singapore London
Czech Republic 3 2 7 6
Brazil 7 5 2 3
China 2 5 4 5

Because the objective of the transportation problem is to minimize the total trans-
portation cost, we can use the cost data in Table 7.1 or on the arcs in Figure 7.1 to
develop the following cost expressions:

Transportations costs for units shipped from Czech Republic ¼ 3x 11 þ 2x 12 þ 7x 13 þ 6x 14
Transportation costs for units shipped from Brazil ¼ 7x 21 þ 5x 22 þ 2x 23 þ 3x 24
Transportation costs for units shipped from China ¼ 2x 31 þ 5x 32 þ 4x 33 þ 5x 34
The sum of these expressions provides the objective function showing the total
transportation cost for Foster Electronics.
Transportation problems need constraints because each origin has a limited
supply and each destination has a specific demand. We will consider the supply
constraints first. The capacity at the Czech Republic plant is 5000 units. With the
total number of units shipped from the Czech Republic plant expressed as x 11 +
x 12 + x 13 + x 14 , the supply constraint for the Czech Republic plant is:
x 11 þ x 12 þ x 13 þ x 14 5000 Czech Republic supply
With three origins (plants), the Foster transportation problem has three supply
constraints. Given the capacity of 6000 units at the Brazil plant and 2500 units at
the China plant, the two additional supply constraints are:

x 21 þ x 22 þ x 23 þ x 24 6000 Brazil supply
x 31 þ x 32 þ x 33 þ x 34 2500 China supply

To obtain a feasible With the four distribution centres as the destinations, four demand constraints
solution, the total supply are needed to ensure that destination demands will be satisfied:
must be greater than or
equal to the total x 11 þ x 21 þ x 31 ¼ 6000 Boston demand
demand.
x 12 þ x 22 þ x 32 ¼ 4000 Dubai demand
x 13 þ x 23 þ x 33 ¼ 2000 Singapore demand
x 14 þ x 24 þ x 34 ¼ 1500 London demand
Combining the objective function and constraints into one model provides a 12-
variable, seven-constraint linear programming formulation of the Foster Electronics
transportation problem:

Min 3x 11 þ 2x 12 þ 7x 13 þ 6x 14 þ 7x 21 þ 5x 22 þ 2x 23 þ 3x 24 þ 2x 31 þ 5x 32 þ 4x 33 þ 5x 34
s:t:
x 11 þ x 12 þ x 13 þ x 14 5000
x 21 þ x 22 þ x 23 þ x 24 6000
x 31 þ x 32 þ x 33 þ x 34 2500
x 11 þ x 21 þ x 31 ¼ 6000
x 12 þ x 22 þ x 32 ¼ 4000
x 13 þ x 23 þ x 33 ¼ 2000
x 14 þx 24 þ x 34 ¼ 1500
x ij 0 for i ¼ 1; 2; 3; j ¼ 1; 2; 3; 4

Copyright 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has
deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.