Question

3. Oranges are grown, picked and then stored in warehouses in Tampa and Fresno. Theses warehouses supply oranges to markets in New York, Chicago, and Boston. The following table shows the shipping costs per truckload (in $100s), supply, and demand. Because of an agreement between distributors, shipments are prohibited from Tampa to New York: To From New York Chicago Boston Supply Tampa 12 16 14 350 Fresno 16 12 20 500 Demand 250 210 180 (a) Formulate a linear programming model for this problem. (b) Find the optimal objective value and the optimal solution point in (a) by using the computer. (c) What is the new optimal solution if the prohibition of the shipment from Tampa to New York were removed from the original model formulation in (a)? (d) Formulate a linear programming model for the original problem with the amount of orange supply from Fresno as 200. (e) Find the optimal objective value and the optimal solution point in (d) by using the computer

Answer 1

Formulate the linear programming model and consider the following decision variables:

Decision variable:

Objective function is to minimize the total cost associated with shipment:

Min Z = 12XTN +16 XTC + 14XTB + 16XFN + 12XFC + 20XFB

Supply constraints are subjected to:

XTC + XTB <= 350

XFN + XFC + XFB <= 500

Note: XTN is not included in the first supply constraint equation because of the agreement between the distributors.

Demand constraints are subjected to:

XNT + XNF <= 250

XCT + XCF <= 210

XBT + XBF <= 180

Non-negativity constraints are:

Formulate an Excel spreadsheet for warehouses and markets and calculate the total cost of shipment, as shown below:

The following spreadsheet is obtained:

Select “Solver” from the “Data” menu bar, as shown below:

A pop-up window would appear after clicking on “Solver”. Fill in the objective function and constraints as shown below:

Click on “Solve” option.

The following results are obtained:

Hence, the minimum cost required for the shipment of oranges from warehouses to fulfill the demand of all four markets is $9120