Question

A car rental company allows its customers to pick up a rental car at one location and return it to any of its locations. Currently, two locations (1 and 2) have 16 and 18 surplus cars, respectively; and four locations (3, 4, 5, and 6) each need 10 additional cars. Because 34 surplus cars are available at locations 1 and 2 combined, and 40 cars are needed at locations 3, 4, 5, and 6 combined, some locations will not receive as many cars as they need. Management wants to make sure all surplus cars are sent where they are needed, and that each location needing cars receives at least five. Management wants to do this with the lowest total miles added to the cars. The distance (in miles) from locations 1 and 2 to the other locations are summarized in the following table:

Cost	Location 3	Location 4	Location 5	Location 6
Location 1	5.4	1.7	2.3	3.0
Location 2	2.4	1.8	1.9	3.1

    

(1.1) Formulate a linear programming model for this problem

(1.2) Create a spreadsheet model for this problem and solve it using Solver. Submit your Excel file on iLearn.

(1.3) What is the optimal solution?

(1.4) Which location is receiving the fewest cars?

(1.5) Suppose location 6 must have at least eight cars shipped to it. What impact does this have on the optimal objective function value?

Answer 1

Let Xij is the amount of car the location j receives from location i \

j={3,4,5,6}

i= {1,2}

Objective

Minimize Z (CarMiles)= 5.4X13+2.4X23+1.7X14+1.8X24+2.3X15+1.9X25+3X16+3.1X26

Constraints

Number of cars supplied from each location should be less than the number cars available at each location

X13+X14+X15+X16=16

X23+X24+X25+X26=18

Each location combined receive at least 5 cars

X13+X23>=5

X14+X24>=5

X15+X25>=5

X16+X26>=5

Total demand is 40 but as the available is 34 We cab tweak the constraint to less than for demand

X13+X23+X14+X24+X15+X25+X16+X26<=40

Xij>=0 (Non-negativity constraints)

1.2: Spreadsheet Model

1.3: Optimal solution is shown in the spreadsheet

Total car miles 69.6

1.4 Location 3,5 and 6 receives few cars

1.5 When we change the model to at least receive by location 6 is 8

We increase the car miles

Please rate me

Thanks