Question

The new manager of Hervis Rent-a-Car (Hervis operates in 100 different cities, with a total of 1.000 cars) has to decide every week how to relocate its car fleet between the different cities where there is an office of the firm. At the end of the week, some stores have a shortage of cars, while in others there is a surplus. Weekly demand does not change, but the number of cars at the end of the week changes, due to the random behavior of the customers that go from one city to another. The last manager used the following method for re-assigning cars: Start from the west coast, and keep covering shortage with the surplus of the closest cities. But the new manager thinks that a model can be build to tackle the problem more optimally. At the end of each week the shortage or surplus in each city is known, together with the number of cars need at the beginning of each week. The cost of relocating a car from one city to another is also known. Formulate the problem as a linear program, defining the indexes, parameters and variables.

Answer 1

Let the demand indexes be i where i = {1,2,….99,100}. These are the shortage numbers and the cars will be needed at these locations. They can be represented on the horizontal axis.

Let the surplus indexes be j where j = {1,2,…99,100}. These are the surplus numbers and cars from these locations will be sent to the demand locations. They can be represented on the vertical axis.

A model can be visually represented as follows

From/To	1	2	3	..	..	99	100	Surplus
1	C11	C12	C13	C1..	C1..	C199	C1100	S1
2	C21	C22	C23	C2..	C2..	C299	C2100	S2
3	C31	C33	C33	C3..	C3..	C399	C3100	S3
..	C..1	C..2	C..3	C....	C....	C..99	C..100	S..
..	C..1	C..2	C..3	C....	C....	C..99	C..100	S..
99	C991	C992	C993	C99..	C99..	C9999	C99100	S99
100	C1001	C1002	C1003	C100..	C100..	C10099	C100100	S100
Demand	D1	D2	D3	D..	D..	D99	D100

There the Cij represents the cost to ship a car from i to j. The ".." values represent continuation.

Then let the number of cars shipped between i and j be denoted by X and we can define our decision variable as Xij = number of cars sent from i to j.

The formulation can be generally written as

Minimize Z where

Z = Σ(Xij*Cij)

Subject to

ΣXij = Di for all j

ΣXij = Sj for all i

Xij >= 0

This is the LP formulation to solve the problem at Hervis Rent-A-Car optimally.