Question

^{Des Moines25 Jefferson 30 City Kansas 15 City 10 20 Omaha 5 St. Louis 10 Supplies Demands Develop a linear programming model for this problem; be sure to define the variables in your model. b. a. Solve the linear program to determine the optimal solution.}

Answer 1

The supply available is equal to the demand

Let the number of units shipped from Jefferson City to Des Moines, Kansas City and St. Louis be a, b and c respectively

Let the number of units shipped from Omaha to Des Moines, Kansas City and St. Louis be x, y and z respectively

Objective is to minimize the cost

Hence, objective function is

Min Z = 14a + 9b + 7c + 8x + 10y + 5z

Constraints

a + b + c = 30 .. supply at Jefferson City

x + y + z = 20 ... supply at Omaha

a + x = 25 ... demand at Des Moines

b + y = 15 ... demand at Kansas City

c + z = 10 ... demand at St. Louis

a, b, c, x, y, z > = 0 ... non negativity constraint

Let us solve this problem using excel solver

Configure Solver as below -

Solver Parameters Set Objective $B$18 To: O Max O Min Value Of By Changing Variable Cells Subject to the Constraints $B$11:SB$15$D$11:$D$15 Add 10 Constraint RHS Change 11 ab c 30 12 | x + y + z-20 Delete 13 ax 25 25 14by 15 15 Reset AlI 15 Cz 10 10 16 Load/Save 17 Cost Make Unconstrained Variables Non-Negative Select a Solving Method: Simplex LPOptions 21 Solving Method 23 Select the GRG Nonlinear engine for Solver Problems that are smooth nonlinear. Select the LP Simplex engine for linear Solver Problems, and select the Evolutionary engine for Solver problems that are non- smooth. 24 25 26 27 28 Close Solve 29

The green cells need to be optimized to find the min cost in blue cell

Run the solver to get -

5 15 10 20 0 0 US 30 abcx +xyz by abc B- 0 1 2 3 4 5 6 7 123456789

Hence, 5, 15 and 10 units should be shipped from Jefferson City to Des Moines, Kansas City and St. Louis respectively

20, 0 and 0 units should be shipped from Omaha to Des Moines, Kansas City and St. Louis respectively

Minimum cost = 435