What is the maximum profit? $ Steinwelt Piano manufactures uprights and consoles
ID: 3122075 • Letter: W
Question
What is the maximum profit? $ Steinwelt Piano manufactures uprights and consoles in two plants, Plant I and Plant II. The output of Plant I is at most 300/month, whereas the output of Plant II is at most 250/month. These pianos are shipped to three warehouses that serve as distribution centers for the company. To fill current and projected orders, Warehouse A requires a minimum of 200 pianos/month, Warehouse B requires at least 150 pianos/month, and Warehouse C requires at least 200 pianos/month. The shipping cost of each piano from Plant I to Warehouse A, Warehouse B, and Warehouse C is $45, $45, and $65, respectively, and the shipping cost of each piano from Plant II to Warehouse A, Warehouse B, and Warehouse C is $65, $55, and $35, respectively. Use the method of this section to determine the shipping schedule that nutrition will enable Steinwelt to meet the warehouses' requirements while keeping the shipping costs to a minimum. Plant I to Warehouse A pianos. Plant I to Warehouse B pianos. Plant I to Warehouse C pianos. Plant II to Warehouse A pianos. Plant II to Warehouse B pianos. Plant II to Warehouse C pianos. What is the minimum cost? $Explanation / Answer
WAREHOUSES
A
B
C
SUPPLY
PLANT I
45
45
65
300
PLANTII
65
55
35
250
DEMAND
200
150
200
Initial solution using Vogel’s method is shown below
WAREHOUSES
Row differences
A
B
C
SUPPLY
PLANT I
45 [200]
45 [100]
65
300
20
0
PLANTII
65
55 [50]
35 [200]
250
10
10
DEMAND
200
150
200
Column differences
20
10
30
20
10
-----
Checking the optimality of solution using Modified Distribution method
Step 1: Make equations of the form Ui + Vj = Cij for allocated cells. Where Ui is the row number, Vj – column number, Cij – cost of (I,j)th cell.
From the above table, equations are
U1 +V1 = 45
U1+V2 = 45
U2 +V2 = 55
U2 +V3 = 35
Solving the equations by putting U1 =0, we have V1 = 45, V2 =45, U2 =10, V3 = 25
Step 2: construct Ui+Vj table for unoccupied cells
WAREHOUSES
V1= 45
V2 = 45
V3 = 25
U1= 0
X
X
25
U2=10
55
X
X
Step 3: construct Cij table for unoccupied cells
WAREHOUSES
A
B
C
PLANT I
X
X
65
PLANT II
65
X
X
Step 4 : construct D ij = Cij- (Ui+Vj )table for unoccupied cells
WAREHOUSES
A
B
C
PLANT I
X
X
40
PLANT II
10
X
X
Since all values in Dij 0, the solution obtained in the initial stage is optimum. The solution is
WAREHOUSES
A
B
C
SUPPLY
PLANT I
45 [200]
45 [100]
65
300
PLANTII
65
55 [50]
35 [200]
250
DEMAND
200
150
200
20
10
-----
Therefore,
Plant I to Warehouse A = 200 pianos
Plant I to Warehouse B = 100 pianos
Plant I to Warehouse C = 0 pianos
Plant II to Warehouse A = 0 pianos
Plant II to Warehouse B = 50 pianos
Plant II to Warehouse C = 200 pianos
Minimum cost is = 45*200 + 45 * 100 + 55* 50 + 35 * 200 = 23250 $
WAREHOUSES
A
B
C
SUPPLY
PLANT I
45
45
65
300
PLANTII
65
55
35
250
DEMAND
200
150
200
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