Consider a very small truckload carrier with 4 trucks, currently at locations A,
ID: 375369 • Letter: C
Question
Consider a very small truckload carrier with 4 trucks, currently at locations A, B, C, and D respectively. There are 8 truckloads available to be transported with origin-destination pairs as follows:
The travel costs between the locations are as follows:
1. Suppose that each truck must be assigned to exactly one load (and thus some loads will not be handled). Write down an integer linear programming formulation to minimize the total distance traveled. Your formulation may ignore what happens to the trucks after their deliveries. You do not have to write the objective and all the constraints explicitly, given that you define the sets, parameters, and decision variables you are using very clearly. Find an optimal truck assignment using a solver of your choice. State which solver you used. Clearly present your solution and optimal objective value.
2. Consider the same data as given above. Now suppose that each load must be handled. (Thus a truck may be assigned to more than one load; however, in that case you must specify in what sequence each truck must handle its assigned loads. Also, a truck does not have to be used.) Write an integer linear programming formulation to minimize the total distance traveled. Your formulation may ignore what happens to the trucks after their last deliveries. You do not have to write the objective and all the constraints explicitly, given that you define the sets, parameters, and decision variables you are using very clearly. Find an optimal truck schedule, using a solver of your choice. State which solver you used. Clearly present your solution and optimal objective value.
Load Number Origin Destination 1 E A 2 F B 3 G C 4 J D 5 I A 6 H B 7 E C 8 I DExplanation / Answer
In part A the question says each load may not be handled and since one truck will handle exactly one load we can have only 4 deliveries of load. Now this we are going to decide on the basis of lowest cost of transportation. Here if we summarize the data we have then we will find that total number of origins are 6 out of which 2 origins (E & I) have 2 loads and 4 origins (F,G,H &J) have 1 load each. Now since 4 trucks are located at A , B , C & D let us find out for each truck what would be the most cost effective deliveries. E F G H I J Cost of returning to origin A 12 25 31 14 26 32 Cost of going from origin to destination A 12 25 31 14 26 32 B 20 19 32 24 13 37 C 31 10 28 33 41 22 D 53 18 46 27 39 14 A to origin to A 24 50 62 28 52 64 A to origin to B 32 44 63 38 39 69 A to origin to C 43 35 59 47 67 54 A to origin to D 65 43 77 41 65 46 Here as we can see the most cost effective will be A to E to A. E F G H I J Cost of returning to origin B 20 19 32 24 13 37 Cost of going from origin to destination A 12 25 31 14 26 32 B 20 19 32 24 13 37 C 31 10 28 33 41 22 D 53 18 46 27 39 14 B to origin to A 32 44 63 38 39 69 B to origin to B 40 38 64 48 26 74 B to origin to C 51 29 60 57 54 59 B to origin to D 73 37 78 51 52 51 Here as we can see the most cost effective route will be B to F to B E F G H I J Cost of returning to origin C 31 10 28 33 41 22 Cost of going from origin to destination A 12 25 31 14 26 32 B 20 19 32 24 13 37 C 31 10 28 33 41 22 D 53 18 46 27 39 14 C to origin to A 43 35 59 47 67 54 C to origin to B 51 29 60 57 54 59 C to origin to C 62 20 56 66 82 44 C to origin to D 84 28 74 60 80 36 Here as we can see the most cost effective route will be C to F to C E F G H I J Cost of returning to origin D 53 18 46 27 39 14 Cost of going from origin to destination A 12 25 31 14 26 32 B 20 19 32 24 13 37 C 31 10 28 33 41 22 D 53 18 46 27 39 14 D to origin to A 65 43 77 41 65 46 D to origin to B 73 37 78 51 52 51 D to origin to C 84 28 74 60 80 36 D to origin to D 106 36 92 54 78 28 Here as we can see the most cost effective route will be D to J to D We can see there is a tie as B to F to C , C to F to C and D to F to C are all most cost effective . So let us take all possibilities together. E F G H I J A to origin to A 24 50 62 28 52 64 A to origin to B 32 44 63 38 39 69 A to origin to C 43 35 59 47 67 54 A to origin to D 65 43 77 41 65 46 B to origin to A 32 44 63 38 39 69 B to origin to B 40 38 64 48 26 74 B to origin to C 51 29 60 57 54 59 B to origin to D 73 37 78 51 52 51 C to origin to A 43 35 59 47 67 54 C to origin to B 51 29 60 57 54 59 C to origin to C 62 20 56 66 82 44 C to origin to D 84 28 74 60 80 36 D to origin to A 65 43 77 41 65 46 D to origin to B 73 37 78 51 52 51 D to origin to C 84 28 74 60 80 36 D to origin to D 106 36 92 54 78 28 So we have all four loads and routes defined as A to E to A , C to F to C , B to I to B and D to J to D and optimum objective value = 24+20+26+28=98. Here I have excel as a solver to solve the question. I am finishing here as time is a constraint . I shall try the 2nd part if the question comes back to me.