Rent”R cars is a multisite car rental company. It is trying out a new “return th
ID: 464994 • Letter: R
Question
Rent”R cars is a multisite car rental company. It is trying out a new “return the car to the location most convenient for you” policy to improve customer service. But this means that the company has to constantly move cars around the city to maintain required levels of vehicle availability. The supply can demand for economy cars, and the cost of moving these vehicles between sites are shown below: move to from D E F G Supply A $9 $8 $8 $13 50 B $20 $17 $8 $2 40 C $8 $18 $3 $10 75 Demand 50 60 25 30 Formulate the problem use the transportation model: Denote the shipment amount From A to D, E, F, G as X11, X12, X13, X14 From B to D, E, F, G as X21, X22, X23, X24 From C to D, E, F, G as X31, X32, X33, X34 a) What is the objective function? b) Provide ONE supply constraint. c) Provide ONE demand constraint. d) Provide the nonnegtivity constraint. Answer the questions in sequence
Explanation / Answer
Destination Sites
Source Sites
D
E
F
G
Supply of cars
A
$9
$8
$8
$13
50
B
$20
$17
$8
$2
40
C
$8
$18
$3
$10
75
Demand of Cars
50
60
25
30
Decision Variables:
Let Xij = Cars moving from ith source to jth destination,
Where, i = 1,2,3 and j = 1,2,3,4
X11, X12, X13, X14 represents shipment amount of cars from source site A to destination site D, E, F, G
From B to D, E, F, G as
X21, X22, X23, X24 represents shipment amount of cars from source site B to destination site D, E, F, G
X31, X32, X33, X34 represents shipment amount of cars from source site C to destination site D, E, F, G
a)
Objective Function:
Rent”R cars objective is to move optimal amount of cars from sources to destination, while minimizing the total transportation cost. The objective function is formulated as:
Minimize Z = $9X11 + $8X12 + $8X13 + $13X14 + $20X21 + $17X22 + $8X23 + $2X24 + $8X31 + $18X32 + $3X33 + $10X34
Subject to Constraints:
b)
Supply constraint of Sources:
Maximum supply of cars from source A is 50 cars, thus the supply constraint is:
Total number of cars moving from source A <= 50 cars,
X11 + X12 + X13 + X14 <= 50
Similarly other supply constraints are:
X21 + X22 + X23 + X24 <= 40
X31 + X32 + X33 + X34 <= 75
c)
Demand Constraint of Destination:
The destination D requires exact 50 numbers of cars to be moved to D.
Total number of cars moved to destination D = 50 cars
X11 + X21 + X31 = 50
Similarly other supply constraints are:
X12 + X22 + X31 = 60
X13 + X23 + X33 = 25
X14 + X24 + X34 = 30
d)
Non-negativity constraint:
As the amount of cars moving from source I to destination j might be zero or more than zero, but never negative, the constraint is:
Xij >=0, for all i and j
Destination Sites
Source Sites
D
E
F
G
Supply of cars
A
$9
$8
$8
$13
50
B
$20
$17
$8
$2
40
C
$8
$18
$3
$10
75
Demand of Cars
50
60
25
30