Question

1. Rockbottom Sand and Gravel Company has contracts to deliver the following amounts of broken rock to six customers:

Customer                                 A          B          C          D          E          F

Contracted amount (tons)         1          2          6          2          3          4

Rockbottom has five trucks that may be used to make these deliveries. With the use of dividers, a truck can deliver to multiple customers, as long as it does not exceed the truck’s capacity. However, a customer’s order cannot be broken up and delivered by multiple trucks. The available trucks and their capacities are:

Truck #                         1          2          3          4          5

Capacity (tons)              4          8          4          8          6

The costs, in $000s, for each truck to deliver the required load to each customer are:

     Customer

            Truck                A          B          C          D          E          F          

                 1                  17        19        21        20        20        21

                 2                  15        18        20        18        19        23

                 3                  18        19        22        22        21        22

                 4                  15        16        19        18        18        20

                 5                  16        15        20        22        19        20

Develop a linear program to determine the delivery plan that will allow Rockbottom to minimize the cost of fulfilling their contracts.

Answer 1

Based on the given data, we tabulate the data as shown below:

Let the No. of units transported by Truck 1 to Customer A be X1a, to Cstomer B be X1b and so on. Hence, we get the decision variables as X1a, X1b, X1c, X1d, X1d, X1e, X1f, X2a, X2b, X2c, X2d, X2e, X2f, X3a, X3b, X3c, X3d, X3e, X3f, X4a, X4b, X4c, X4d, X4e, X4f, X5a, X5b, X5d, X5d, X5e, X5f

Total Supply = Capacity = 4 + 8 + 4 + 8 + 6 = 30

Total Demand = 1 + 2 + 6 + 2 + 3 + 4 = 18

Hence, Total Capacity > Total Demand. There will be "<=" sign in Capacity Constraints

Total Cost = 17*X1a + 19*X1b + 21*X1c + 20*X1d + 20*X1e + 21*X1f + 15*X2a + 18*X2b + 20*X2c + 18*X2d + 19*X2e + 23*X2f + 18*X3a + 19*X3b + 22*X3c + 22*X3d + 21*X3e + 22*X3f + 15*X4a + 16*X4b + 19*X4c + 18*X4d + 18*X4e + 20*X4f + 16*X5a + 15*X5b + 20*X5c + 22*X5d + 19*X5e + 20X5f

Subject to Constraints:

Capacity Constraints:

X1a + X1b + X1c + X1d + X1e + X1f <= 4...........Constraint for Capacity of Truck 1

X2a + X2b + X2c + X2d + X2e + X2f <= 8...........Constraint for Capacity of Truck 2

X3a + X3b + X3c + X3d + X3e + X3f <= 4...........Constraint for Capacity of Truck 3

X4a + X4b + X4c + X4d + X4e + X4f <= 8...........Constraint for Capacity of Truck 4

X5a + X5b + X5c + X5d + X5e + X5f <= 6...........Constraint for Capacity of Truck 5

Demand Constraints:

X1a + X2a + X3a + X4a + X5a = 1...........Constraint for Customer A demand

X1b + X2b + X3b + X4b + X5b = 2...........Constraint for Customer B demand

X1c + X2c +X3c + X4c + X5c = 6.............Constraint for Customer C demand

X1d + X2d + X3d + X4d + X5d = 2...........Constraint for Customer D demand

X1e + X2e + X3e + X4e + X5e = 3...........Constraint for Customer E demand

X1f + X2f + X3f + X4f + X5f = 4................Constraint for Customer F demand

X1a, X1b, X1c, X1d, X1d, X1e, X1f, X2a, X2b, X2c, X2d, X2e, X2f, X3a, X3b, X3c, X3d, X3e, X3f, X4a, X4b, X4c, X4d, X4e, X4f, X5a, X5b, X5d, X5d, X5e, X5f >= 0............Non-negative constraint

-----------------------------------------------------------------------------------------------------------------------

In case of any doubt, please ask through the comment section before Upvote/downvote.

If you liked the answer, please give an upvote. This will be quite encouraging for me, thank-you!!!