Consider the following network representation of a transportation problem.

55

30

25

20

40

15

9

8

8

11

24

A network diagram between five locations is shown. The left-hand side of the graph is labeled "Supplies" and lists Jefferson City and Omaha. The right-hand side of the graph is labeled "Demands" and lists Des Moines, Kansas City, and St. Louis. Lines are shown between various locations. The following list contains the numbers placed on the graph.

• Jefferson City: 55
• Omaha: 30
• Des Moines: 25
• Kansas City: 20
• St. Louis: 40

• Jefferson CityDes Moines: 15
• Jefferson CityKansas City: 9
• Jefferson CitySt. Louis: 8

• OmahaDes Moines: 8
• OmahaKansas City: 11
• OmahaSt. Louis: 24

The supplies, demands, and transportation costs per unit are shown on the network.

(a)

Develop a linear programming model for this problem; be sure to define the variables in your model.

Let

• x₁₁ = amount shipped from Jefferson City to Des Moines
• x₁₂ = amount shipped from Jefferson City to Kansas City
• x₁₃ = amount shipped from Jefferson City to St. Louis
• x₂₁ = amount shipped from Omaha to Des Moines
• x₂₂ = amount shipped from Omaha to Kansas City
• x₂₃ = amount shipped from Omaha to St. Louis

Min

x11

+

x12

+

x13

+

x21

+

x22

+

x23

s.t.

x11

+

x12

+

x13

x21

+

x22

+

x23

x11

+

x21

=

x12

+

x22

=

x13

+

x23

=

x₁₁, x₁₂, x₁₃, x₂₁, x₂₂, x₂₃ 0

(b)

Solve the linear program to determine the optimal solution.

	Amount	Cost
Jefferson CityDes Moines
Jefferson CityKansas City
Jefferson CitySt. Louis
OmahaDes Moines
OmahaKansas City
OmahaSt. Louis
Total