Solution Key Problem Set 4
The following linear programming problem has been solved by The Management Scientist. Use the
output to answer the questions.
LINEAR PROGRAMMING PROBLEM
MAX 25X1+30X2+15X3
S.T.
1) 4X1+5X2+8X3<1200
2) 9X1+15X2+3X3<1500
OPTIMAL SOLUTION
Objective Function Value = 4700.000
Variable
X1
X2
X3
Value
140.000
0.000
80.000
Reduced Cost
0.000
10.000
0.000
Constraint
1
2
Slack/Surplus
0.000
0.000
Dual Price
1.000
2.333
OBJECTIVE COEFFICIENT RANGES
Variable
Lower Limit
Current Value
X1
19.286
25.000
X2
No Lower Limit
30.000
X3
8.333
15.000
Upper Limit
45.000
40.000
50.000
RIGHT HAND SIDE RANGES
Constraint
Lower Limit
1
666.667
2
450.000
Upper Limit
4000.000
2700.000
a.
b.
c.
d.
e.
f.
Current Value
1200.000
1500.000
Give the complete optimal solution.
Which constraints are binding?
What is the dual price for the second constraint? What interpretation does this have?
Over what range can the objective function coefficient of x2 vary before a new solution
point becomes optimal?
By how much can the amount of resource 2 decrease before the dual price will change?
What would happen if the first constraint's right-hand side increased by 700 and the
second's decreased by 350?
ANS:
a.
b.
c.
d.
e.
f.
x1 = 140, x2 = 0, x3 = 80, s1 = 0, s2 = 0, obj. func. = 4700
Constraints 1 and 2 are binding.
Dual price 2 = 2.33. A unit increase in the right-hand side of constraint 2 will increase the
value of the objective function by 2.33.
As long as c2
1050
change.
Question Number 2
The binding constraints for this problem are the first and second.
Min
x1 + 2x2
s.t.
a.
b.
c.
d.
e.
1) x1 + x2
2) 2x1 + x2
3) 2x1 + 5x2
4) x1 , x2
Keeping c2 fixed at 2, over what range can c1 vary before there is a change in the optimal
solution point?
Keeping c1 fixed at 1, over what range can c2 vary before there is a change in the optimal
solution point?
If the objective function becomes Min 1.5x1 + 2x2, what will be the optimal values of x1,
x2, and the objective function?
If the objective function becomes Min 7x1 + 6x2, what constraints will be binding?
Find the dual price for each constraint in the original problem.
ANS:
a.
b.
c.
d.
e.
c1
c2
x1 = 250, x2 = 50, obj. func. = 475
Constraints 1 and 2 will be binding.
Dual prices are .33, 0, .33 (The first and third values are negative.)
Question Number 3
Write the LP formulation for this transportation problem.
ANS:
Min
5x1A + 6x1B + 4x2A + 2x2B + 3x3A + 6x3B + 9x4A + 7x4B
s.t.
x1A + x1B
x2A + x2B
x3A + x3B
x4A + x4B
200
x1A + x2A + x3A + x4A = 250
x1B + x2B + x3B + x4B = 250
all xij
Question Number 4
Draw the network for this transportation problem.
Min
2xAX + 3xAY + 5xAZ+ 9xBX + 12xBY + 10xBZ
s.t.
xAX + xAY + xAZ
xBX + xBY + xBZ
xAX + xBX
= 300
xAY + xBY
= 300
xAZ + xBZ
= 300
xij
ANS:
Question Number 5
Canning Transport is to move goods from three factories to three distribution centers. Information about
the move is given below. Give the network model and the linear programming model for this problem.
Source
A
B
C
Supply
200
100
150
Destination
X
Y
Z
Demand
50
125
125
Shipping costs are:
Source
A
B
C
Destination
X
Y
Z
3
2
5
9
10
-5
6
4
(Source B cannot ship to destination Z)
ANS:
Min
3xAX + 2xAY + 5xAZ + 9xBX + 10xBY + 5xCX + 6xCY + 4xCZ
s.t.
xAX + xAY + xAZ
xBX + xBY
xCX + xCY + xCZ
xDX + xDY + xDZ
xAX + xBX + xCX + xDX = 250
xAY + xBY + xCY + xDY = 125
xAZ + xBZ + xCZ + xDZ = 125
xij
Question Number 6
Write the linear program for this transshipment problem.
ANS:
Min
3x16 + 2x14 + 3x15 + 5x24 + 6x25 + 2x32 + 8x34 + 10x35 + 5x46 + 9x47 + 12x56 + 15x57
s.t.
x16 + x14 + x35
x24 + x25 x23
x32 + x34 + x35
x46 + x47
x14 + x24 + x34) = 0
x56 + x57
x15 + x25 + x35) = 0
x16 + x46 + x56 = 600
x56 + x57 = 600
Question Number 7
RVW (Restored Volkswagens) buys 15 used VW's at each of two car auctions each week held at different
locations. It then transports the cars to repair shops it contracts with. When they are restored to RVW's
specifications, RVW sells 10 each to three different used car lots. There are various costs associated with
the average purchase and transportation prices from each auction to each repair shop. Also there are
transportation costs from the repair shops to the used car lots. RVW is concerned with minimizing its
total cost given the costs in the table below.
a. Given the costs below, draw a network representation for this problem.
Repair Shops
Used Car Lots
S1
S2
L1
L2
L3
Auction 1
550
500
S1
250
300
500
Auction 2
600
450
S2
350
650
450
b.
c
c.
Formulate this problem as a transshipment linear programming model.
Report the optimal solution using excel solver.
ANS:
a.
b.
Denote A1 as node 1, A2 as node 2, S1 as node 3, S2 as node 4,
L1 as node 5, L2 as node 6, and L3 as node 7
Min
50x13 + 500x14 + 600x23 + 450x24 + 250x35 + 300x36
+ 500x37 + 350x45 + 650x46 + 450x47
s.t.
x13 + x14
 15
x23 + x24
 15
x13 + x23  x35  x36  x37 = 0
x14 + x24  x45  x46  x47 = 0
x35 + x45
= 10
x36 + x46
= 10
x37 + x4
= 10
xij  0 for all i,j