Solutions to Assignment 7
7-11.
MIN 20 X11 + 19 X12 + 22 X13 + 24 X14 + 26 X21 + 24 X22 + 28 X23
+ 23 X24 + 33 X31 + 25 X32 + 29 X33 + 28 X34
SUBJECT TO
2) X11 + X12 + X13 + X14 <= 800
3) X21 + X22 + X23 + X24 <= 600
4) X31 + X32 + X33 + X34 <= 700
5) X11 + X21 + X31 = 300
6) X12 + X22 + X32 >= 500
7) X13 + X23 + X33 >= 400
8) X14 + X24 + X34 >= 600
LP OPTIMUM FOUND AT STEP 6
OBJECTIVE FUNCTION VALUE
1) 40500.00
VARIABLE VALUE REDUCED COST
X11 300.000000 0.000000
X12 100.000000 0.000000
X13 400.000000 0.000000
X14 0.000000 6.000000
X21 0.000000 1.000000
X22 0.000000 0.000000
X23 0.000000 1.000000
X24 600.000000 0.000000
X31 0.000000 7.000000
X32 400.000000 0.000000
X33 0.000000 1.000000
X34 0.000000 4.000000
ROW SLACK OR SURPLUS DUAL PRICES
2) 0.000000 6.000000
3) 0.000000 1.000000
4) 300.000000 0.000000
5) 0.000000 -26.000000
6) 0.000000 -25.000000
7) 0.000000 -28.000000
8) 0.000000 -24.000000
NO. ITERATIONS= 6
RANGES IN WHICH THE BASIS IS UNCHANGED:
OBJ COEFFICIENT RANGES
VARIABLE CURRENT ALLOWABLE ALLOWABLE
COEF INCREASE DECREASE
X11 20.000000 1.000000 INFINITY
X12 19.000000 6.000000 1.000000
X13 22.000000 1.000000 28.000000
X14 24.000000 INFINITY 6.000000
X21 26.000000 INFINITY 1.000000
X22 24.000000 1.000000 4.000000
X23 28.000000 INFINITY 1.000000
X24 23.000000 4.000000 24.000000
X31 33.000000 INFINITY 7.000000
X32 25.000000 1.000000 1.000000
X33 29.000000 INFINITY 1.000000
X34 28.000000 INFINITY 4.000000
RIGHTHAND SIDE RANGES
ROW CURRENT ALLOWABLE ALLOWABLE
RHS INCREASE DECREASE
2 800.000000 400.000000 100.000000
3 600.000000 400.000000 0.000000
4 700.000000 INFINITY 300.000000
5 300.000000 100.000000 300.000000
6 500.000000 300.000000 400.000000
7 400.000000 100.000000 400.000000
8 600.000000 0.000000 400.000000
The relevant cost that must be minimized is the transportation
and
production costs. Thus we enter the sum of these costs as the
cell costs. Thus the cell costs are the cost of producing one
unit at a plant and shipping it to a demand point. Optimal
solution is given in red. Minimum cost is $40,500.
Since we have more capacity than demand, minimizing just the
transportation cost, may force the use of costlier capacity more
fully and leave some inexpensive capacity unused leading to
sub-optimization. If we had just enough capacity to meet all
demand, simply minimizing the transportation costs would be
appropriate since we need all the capacity--
expensive and cheap.
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7-16.
Since there are more developers than plots one of the developers will not get an assignment.
We can handle the situation in two ways: 1) Create a dummy plot (whoever gets the dummy plot
does not get a real plot) or 2) Enter the constraints for each developer as "<=" to allow
not assigning a plot to one of the developers.
MAX 19 A1 + 19 A2 + 29 A3 + 23 A4 + 24 A5 + 23 B1 + 21 B2 + 27 B3
+ 19 B4 + 25 B5 + 19 C1 + 19 C2 + 22 C3 + 20 C5 + 23 D1 + 19 D3 + 21 D4
+ 18 D5
SUBJECT TO
2) A1 + A2 + A3 + A4 + A5 = 1
3) B1 + B2 + B3 + B4 + B5 = 1
4) C1 + C2 + C3 + C4 + C5 = 1
5) D1 + D2 + D3 + D4 + D5 = 1
6) A1 + B1 + C1 + D1 <= 1
7) A2 + B2 + C2 + D2 <= 1
8) A3 + B3 + C3 + D3 <= 1
9) A4 + B4 + C4 + D4 <= 1
10) A5 + B5 + C5 + D5 <= 1
END
LP OPTIMUM FOUND AT STEP 13
OBJECTIVE FUNCTION VALUE
1) 96.00000
VARIABLE VALUE REDUCED COST
A1 0.000000 4.000000
A2 0.000000 4.000000
A3 1.000000 0.000000
A4 0.000000 0.000000
A5 0.000000 1.000000
B1 0.000000 0.000000
B2 0.000000 2.000000
B3 0.000000 2.000000
B4 0.000000 4.000000
B5 1.000000 0.000000
C1 0.000000 0.000000
C2 1.000000 0.000000
C3 0.000000 3.000000
C4 0.000000 19.000000
C5 0.000000 1.000000
D1 1.000000 0.000000
D2 0.000000 23.000000
D3 0.000000 10.000000
D4 0.000000 2.000000
D5 0.000000 7.000000
ROW SLACK OR SURPLUS DUAL PRICES
2) 0.000000 23.000000
3) 0.000000 23.000000
4) 0.000000 19.000000
5) 0.000000 23.000000
6) 0.000000 0.000000
7) 0.000000 0.000000
8) 0.000000 6.000000
9) 1.000000 0.000000
10) 0.000000 2.000000
NO. ITERATIONS= 13
RANGES IN WHICH THE BASIS IS UNCHANGED:
OBJ COEFFICIENT RANGES
VARIABLE CURRENT ALLOWABLE ALLOWABLE
COEF INCREASE DECREASE
A1 19.000000 4.000000 INFINITY
A2 19.000000 4.000000 INFINITY
A3 29.000000 INFINITY 2.000000
A4 23.000000 2.000000 1.000000
A5 24.000000 1.000000 INFINITY
B1 23.000000 1.000000 2.000000
B2 21.000000 2.000000 INFINITY
B3 27.000000 2.000000 INFINITY
B4 19.000000 4.000000 INFINITY
B5 25.000000 INFINITY 1.000000
C1 19.000000 2.000000 0.000000
C2 19.000000 0.000000 2.000000
C3 22.000000 3.000000 INFINITY
C4 0.000000 19.000000 INFINITY
C5 20.000000 1.000000 INFINITY
D1 23.000000 INFINITY 2.000000
D2 0.000000 23.000000 INFINITY
D3 19.000000 10.000000 INFINITY
D4 21.000000 2.000000 INFINITY
D5 18.000000 7.000000 INFINITY
RIGHTHAND SIDE RANGES
ROW CURRENT ALLOWABLE ALLOWABLE
RHS INCREASE DECREASE
2 1.000000 1.000000 0.000000
3 1.000000 0.000000 0.000000
4 1.000000 0.000000 1.000000
5 1.000000 0.000000 1.000000
6 1.000000 1.000000 0.000000
7 1.000000 INFINITY 0.000000
8 1.000000 0.000000 1.000000
9 1.000000 INFINITY 1.000000
10 1.000000 0.000000 0.000000
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