solve using VAM method, this is a linear programming problem. MG Auto, of Exampl
ID: 365333 • Letter: S
Question
solve using VAM method, this is a linear programming problem.
MG Auto, of Example 5.1-1, produces four car models: M1, M2, M3, and M4. The Detroit plant produces models M1, M2, and M4. Models M1 and M2 are also produced in New Orleans. The Los Angeles plant manufactures models M3 and M4. The capacities of the various plants and the demands at the distribution centers are given in Table 5.29 5-13. The mileage chart is the same as given in Example 5.1-1, and the transportation rate remains at 8 cents per car mile for all models. Additionally, it is possible to satisfy a percentage of the demand for some models from the supply of others according to the specifications in Table 5.30 (a) Formulate the corresponding transportation model (b) Determine the optimum shipping schedule. (Hint: Add four new destinations corre- demands at the new destinations are determined from the given percentages.) TABLE 5.27 Transportation Cost/Crate for Problem 5-11 sponding to the new combinations [M1, M2], [M3, M4], [M1, M3], and [M2, M4].The Retailer 4 Orchard 1 $1 Orchard 2 $2 Orchard 3 $1 S2 $4 $3 $1 $5 $2 $2 S3 TABLE 5.28 Mileage Chart and Supply and Demand for Problem 5-12 Dealer 2 5 Supply 200 100 150 140 150 160 35 400 80 200 130 150 140 150 70 Center 1 100 Center 2 Center 3 50 40 Demand 100 200Explanation / Answer
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Vogel’s Approximation method:
Steps:
100
150
200
140
35
50
70
60
65
80
40
90
100
150
130
Row penalty
100
150
200
140
35
100-35=65
50
70
60
65
80
10
40
90
100
150
130
50
Column Penalty
50-40=10
20
40
75
45
Step 2:
Supply
Row penalty
100
150
200
140
35
400
100-35=65
50
70
60
65 (160)
80
200-160=40
10
40
90
100
150
130
150
50
Demand
100
200
150
160-160=0
140
Column Penalty
50-40=10
20
40
75
45
Remove column 4:
Supply
Row penalty
100
150
200
35
400
65
50
70
60
80
40
10
40
90
100
130
150
50
Demand
100
200
150
140
Column Penalty
50-40=10
20
40
45
The maximum penalty is 65
The row has 35 as the minimum value with the demand 140 < supply 400
so allocate 45 and subtract it from both the demand and supply and update the table as follows:
Supply
Row penalty
100
150
200
35 (140)
400-140=260
65
50
70
60
80
40
10
40
90
100
130
150
50
Demand
100
200
150
140-140=0
Column Penalty
50-40=10
20
40
45
Remove column 4
Supply
Row penalty
100 (100)
150
200
260-100=160
50 (Max)
50
70
60
40
10
40
90
100
150
50
Demand
100-100=0
200
150
Column Penalty
10
20
40
Remove column 1:
Supply
Row penalty
150 (160)
200
160-160=0
50 (Max)
70
60
40
10
90
100
150
10
Demand
200-160=40
150
Column Penalty
20
40
Remove row 1:
Supply
Row penalty
70
60(40)
40 -40 =0
10
90
100
150
10
Demand
40
150-40=110
Column Penalty
20
40(Max)
Remove row 1:
Supply
90
100
150
Demand
40
110
As there is only one row left, we just allocate the supply and demand by matching
Demands of 40+110 = supply of 150
Supply
90 (40)
100 (110)
150-40-110=0
Demand
40-40=0
110-110=0
Now multiply each allocation to get the total cost:
Total cost = 100*110 + 90*40 + 60*40 + 150*160 + 100*100 + 35*140 + 65*160
Total Cost = $66,300
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100
150
200
140
35
50
70
60
65
80
40
90
100
150
130
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