Homework Assignment 9 Solutions
The problems may be solved with the AMPL model and data files listed below:
| Problem | Model File | Data File |
| 1) | dietmodel.txt | diet1.txt |
| 2) | dietmodel.txt | diet2.txt |
| 3) | gtmodel.txt | GT1.txt |
| 4) | gtmodel.txt | GT2.txt |
The following is a transcript of an AMPL session in which we use the model and data files described above to solve the homework problems. Command that you would type are shown in bold.
% ampl ampl: model dietmodel.txt; ampl: data diet1.txt; ampl: expand; minimize Total_Cost: Buy['milk'] + 2.5*Buy['cheese'] + 0.75*Buy['apples']; s.t. Diet['protein']: -40*Buy['milk'] - 20*Buy['cheese'] - 10*Buy['apples'] <= -80; s.t. Diet['vitamin_A']: -5*Buy['milk'] - 40*Buy['cheese'] - 30*Buy['apples'] <= -60; s.t. Diet['vitamin_B']: -20*Buy['milk'] - 30*Buy['cheese'] - 40*Buy['apples'] <= -50; s.t. Diet['vitamin_C']: -30*Buy['milk'] - 50*Buy['cheese'] - 60*Buy['apples'] <= -30; ampl: solve; CPLEX 6.6.0: CPLEX 6.6.0: optimal solution; objective 2.869565217 3 simplex iterations (1 in phase I) ampl: display Buy; Buy [*] := apples 1.73913 cheese 0 milk 1.56522 ;Before reading in the data file for problem (diet2.txt), type reset data;.
ampl: reset data; ampl: data diet2.txt; ampl: display amt; amt := protein apples 10 protein cheese 20 protein milk 40 protein powerbar 10 vitamin_A apples 30 vitamin_A cheese 40 vitamin_A milk 5 vitamin_A powerbar 0 vitamin_B apples 40 vitamin_B cheese 30 vitamin_B milk 20 vitamin_B powerbar 0 vitamin_C apples 60 vitamin_C cheese 50 vitamin_C milk 30 vitamin_C powerbar 30 ; ampl: expand; minimize Total_Cost: Buy['milk'] + 2.5*Buy['cheese'] + 0.75*Buy['apples'] + 1.5*Buy['powerbar']; s.t. Diet['protein']: -40*Buy['milk'] - 20*Buy['cheese'] - 10*Buy['apples'] - 10*Buy['powerbar'] <= -80; s.t. Diet['vitamin_A']: -5*Buy['milk'] - 40*Buy['cheese'] - 30*Buy['apples'] <= -60; s.t. Diet['vitamin_B']: -20*Buy['milk'] - 30*Buy['cheese'] - 40*Buy['apples'] <= -50; s.t. Diet['vitamin_C']: -30*Buy['milk'] - 50*Buy['cheese'] - 60*Buy['apples'] - 30*Buy['powerbar'] <= -30; ampl: solve; CPLEX 6.6.0: CPLEX 6.6.0: optimal solution; objective 2.869565217 3 simplex iterations (1 in phase I) ampl: display Buy; Buy [*] := apples 1.73913 cheese 0 milk 1.56522 powerbar 0 ; We can use the let command in AMPL to change the cost of powerbars: ampl: let cost['powerbar'] := 0.75; ampl: expand Total_Cost; minimize Total_Cost: Buy['milk'] + 2.5*Buy['cheese'] + 0.75*Buy['apples'] + 0.75*Buy['powerbar']; ampl: solve; CPLEX 6.6.0: CPLEX 6.6.0: optimal solution; objective 2.869565217 0 simplex iterations (0 in phase I) ampl: display Buy; Buy [*] := apples 1.73913 cheese 0 milk 1.56522 powerbar 0 ; ampl: reset; ampl: model gtmodel.txt; ampl: data GT1.txt; ampl: expand; minimize cost: 13*x['IE_Junction','A_Station'] + 35*x['IE_Junction','Fine_Place'] + 42*x['IE_Junction','Goodville'] + 9*x['IE_Junction','Somewhere_Street'] + 6*x['Centerville','A_Station'] + 61*x['Centerville','Fine_Place'] + 18*x['Centerville','Goodville'] + 30*x['Centerville','Somewhere_Street'] + 15*x['Wayover_City','A_Station'] + 10*x['Wayover_City','Fine_Place'] + 5*x['Wayover_City','Goodville'] + 9*x['Wayover_City','Somewhere_Street'] + 20*x['Mockingbird','A_Station'] + 15*x['Mockingbird','Fine_Place'] + 12*x['Mockingbird','Goodville'] + 40*x['Mockingbird','Somewhere_Street']; s.t. supply['IE_Junction']: x['IE_Junction','A_Station'] + x['IE_Junction','Fine_Place'] + x['IE_Junction','Goodville'] + x['IE_Junction','Somewhere_Street'] <= 4; s.t. supply['Centerville']: x['Centerville','A_Station'] + x['Centerville','Fine_Place'] + x['Centerville','Goodville'] + x['Centerville','Somewhere_Street'] <= 1; s.t. supply['Wayover_City']: x['Wayover_City','A_Station'] + x['Wayover_City','Fine_Place'] + x['Wayover_City','Goodville'] + x['Wayover_City','Somewhere_Street'] <= 2; s.t. supply['Mockingbird']: x['Mockingbird','A_Station'] + x['Mockingbird','Fine_Place'] + x['Mockingbird','Goodville'] + x['Mockingbird','Somewhere_Street'] <= 2; s.t. demand['A_Station']: x['IE_Junction','A_Station'] + x['Centerville','A_Station'] + x['Wayover_City','A_Station'] + x['Mockingbird','A_Station'] >= 1; s.t. demand['Fine_Place']: x['IE_Junction','Fine_Place'] + x['Centerville','Fine_Place'] + x['Wayover_City','Fine_Place'] + x['Mockingbird','Fine_Place'] >= 1; s.t. demand['Goodville']: x['IE_Junction','Goodville'] + x['Centerville','Goodville'] + x['Wayover_City','Goodville'] + x['Mockingbird','Goodville'] >= 1; s.t. demand['Somewhere_Street']: x['IE_Junction','Somewhere_Street'] + x['Centerville','Somewhere_Street'] + x['Wayover_City','Somewhere_Street'] + x['Mockingbird','Somewhere_Street'] >= 1; ampl: solve; CPLEX 6.6.0: CPLEX 6.6.0: optimal solution; objective 30 6 simplex iterations (4 in phase I) ampl: display x; x := Centerville A_Station 1 Centerville Fine_Place 0 Centerville Goodville 0 Centerville Somewhere_Street 0 IE_Junction A_Station 0 IE_Junction Fine_Place 0 IE_Junction Goodville 0 IE_Junction Somewhere_Street 1 Mockingbird A_Station 0 Mockingbird Fine_Place 0 Mockingbird Goodville 0 Mockingbird Somewhere_Street 0 Wayover_City A_Station 0 Wayover_City Fine_Place 1 Wayover_City Goodville 1 Wayover_City Somewhere_Street 0 ;The optimal solution sends one locomotive from Centerville to A Station, one from IE Junction to Somewhere Street. It also sends two locomotives from Wayover City: one to Fine Place and one to Goodville. We can display the results in a more compact way by just displaying the non-zero x's:
ampl: display {i in O, j in D: x[i,j] > 0} x[i,j];
x[i,j] :=
Centerville A_Station 1
IE_Junction Somewhere_Street 1
Wayover_City Fine_Place 1
Wayover_City Goodville 1
;
ampl: reset data;
ampl: data GT2.txt;
ampl: expand;
minimize cost:
13*x['IE_Junction','A_Station'] + 35*x['IE_Junction','Fine_Place'] +
42*x['IE_Junction','Goodville'] + 9*x['IE_Junction','Somewhere_Street']
+ 6*x['Centerville','A_Station'] + 61*x['Centerville','Fine_Place'] +
18*x['Centerville','Goodville'] +
30*x['Centerville','Somewhere_Street'] +
15*x['Wayover_City','A_Station'] + 10*x['Wayover_City','Fine_Place'] +
5*x['Wayover_City','Goodville'] +
9*x['Wayover_City','Somewhere_Street'] +
20*x['Mockingbird','A_Station'] + 15*x['Mockingbird','Fine_Place'] +
12*x['Mockingbird','Goodville'] +
40*x['Mockingbird','Somewhere_Street'];
s.t. supply['IE_Junction']:
x['IE_Junction','A_Station'] + x['IE_Junction','Fine_Place'] +
x['IE_Junction','Goodville'] + x['IE_Junction','Somewhere_Street'] <= 4;
s.t. supply['Centerville']:
x['Centerville','A_Station'] + x['Centerville','Fine_Place'] +
x['Centerville','Goodville'] + x['Centerville','Somewhere_Street'] <= 1;
s.t. supply['Wayover_City']:
x['Wayover_City','A_Station'] + x['Wayover_City','Fine_Place'] +
x['Wayover_City','Goodville'] + x['Wayover_City','Somewhere_Street']
<= 2;
s.t. supply['Mockingbird']:
x['Mockingbird','A_Station'] + x['Mockingbird','Fine_Place'] +
x['Mockingbird','Goodville'] + x['Mockingbird','Somewhere_Street'] <= 2;
s.t. demand['A_Station']:
x['IE_Junction','A_Station'] + x['Centerville','A_Station'] +
x['Wayover_City','A_Station'] + x['Mockingbird','A_Station'] >= 3;
s.t. demand['Fine_Place']:
x['IE_Junction','Fine_Place'] + x['Centerville','Fine_Place'] +
x['Wayover_City','Fine_Place'] + x['Mockingbird','Fine_Place'] >= 2;
s.t. demand['Goodville']:
x['IE_Junction','Goodville'] + x['Centerville','Goodville'] +
x['Wayover_City','Goodville'] + x['Mockingbird','Goodville'] >= 0;
s.t. demand['Somewhere_Street']:
x['IE_Junction','Somewhere_Street'] +
x['Centerville','Somewhere_Street'] +
x['Wayover_City','Somewhere_Street'] +
x['Mockingbird','Somewhere_Street'] >= 3;
ampl: solve;
CPLEX 6.6.0:
CPLEX 6.6.0: optimal solution; objective 84
8 simplex iterations (5 in phase I)
ampl: display {i in O, j in D: x[i,j] > 0} x[i,j];
x[i,j] :=
Centerville A_Station 1
IE_Junction A_Station 2
IE_Junction Somewhere_Street 2
Mockingbird Fine_Place 1
Wayover_City Fine_Place 1
Wayover_City Somewhere_Street 1
;