1. In each part below draw the network corresponding to the following
node-arc incidence matrices:
(a)
X12 X14
x23 x24 x52 x53 x54
-1 -1
1
-1 -1 1
1
1
1
1
1
-1 -1 -1
(b)
X12 X13 X14 X23
x24 x25 x32 x35 x43 x45
-1 -1 -1
1
-1
1
1
-1 -1
1
-1 -1
1
1
1
-1
1
-1
1
2. Consider the constraints
X13 + x14 - x41  6
+ x32  5
X41 + X42 + x43 - X14  4
X14 + X34 2
every xij  0
X12 +
X12
Draw a network corresponding to these restrictions. (HINT: convert the
inequalities to equalities by adding slack variables; then add a
redundant restriction corresponding to a fictitious node.)
3. The Widget Company maintains manufacturing and distribution
facilities for its Four Star Widget at three plants in the U.S.: Boston,
Chicago, and Denver. Each plant can make and sell the Widget but because
of capacity imbalances, may have to resort to internal redistribution
(i.e., shipments) in order to satisfy anticipated demand. Specifically,
the following information is available for the coming month's situation:
Marketing Report:
PLANT
B
C
D
EXPECTED DEMAND
(Boston)
(Chicago)
(Denver)
UNIT SELLING PRICE
1000
800
1200
$80
$60
$70
This data is fairly accurate since orders are compiled mostly in advance
and the price has been established by previous experience.
Manufacturing Report:
REGULAR TIME
PLANT
B
C
D
UNIT COST
$50
$40
$45
CAPACITY
900
1000
800
OVERTIME
UNIT COST
$75
$55
$70
CAPACITY
400
500
300
The overtime capacity is in addition to the regular figures.
Shipping Cost Per Unit Report:
TO PLANT
B
C
D
B
C
D
FROM PLANT
0
12
18
12
0
10
18
10
0
Given this data, the following model will find a manufacturing and
shipping schedule that maximizes the monthly profit.
Convert this model to a network model and draw its NETFORM.
In this case, the objective to the system is clearly stated as part of
the problem statement. Interestingly, however, it should be noted that
unlike the preceding examples there is a disproportionality in this
problem structure corresponding to the relevant manufacturing expenses
for different levels of output. Although such a disproportionality may
initially appear to hamper formulating the problem as an LP problem we
shall show how to overcome this obstacle by a judicious choice of
variables.
Let xij be the number of units produced on regular time at plant i and
shipped and sold at plant j. while
xij = number of units produced on overtime at plant i and shipped and
sold at plant j where i and j both range over values B, C, D and
i=j implies production and sales at the same plant.
The linear programming formulation then becomes:
Maximize
30xBB - 2xBC + 2xBD + 20xCC + 28xCB + 20xCD
+ 25xDD + 17xDB + 5xDC + 5xBB – 27xBC - 23xBD
+ 5xCC + 13xCB + 5xCD + 0xDD - 8xDB - 20xDC
subject to:
xBB + xBC + xBD
xBB + xBC + xBD
xCC + xCB + xcd
xCC + xCB + xCD
xDD + xDB + xDC
xDD + xDB + xDC
xBB + xCB
xCC + xBC
xDD + xBD
all xij,
+ xDB
+ xDC
+ xCD
xij 






900
400
1000
500
800
300
+ xBB + xCB + xDB = 1000
+ xCC + xBC + xDC = 800
+ xDD + xBD + xCD = 1200
0.
The coefficients in the objective (profit) function are found by
subtracting the unit shipping costs from the unit price. The first six
constraints represent capacity limitations on production while the last
three represent demand requirements. Incidentally, it should be observed
that more than enough capacity exists throughout the system (i.e., 3900
units) to more than satisfy total demand (i.e., 3000 units) even though
this may not be true for each plant (e.g.), Denver).
3. Suppose a firm having two plants and three demand points is planning
a production schedule for four periods. Assume over periods 1, 2, 3, and
4 that Plant 1 has available supplies of 4, 5, 6, and 7 respectively and
that Plant 2 has available supplies of 8, 9, 10, and 11, respectively.
Assume over periods 1, 2, 3, and 4 that Demand Point 1 has
demands 3, 3, 10, and 10 respectively; similarly, Demand Point 2 has
demands 4, 2, 6, and 6 respectively; and Demand Point 3 has demands 2,
6, 4, and 4 respectively. Let cij be the cost of shipping a unit from
Plant i to Demand Point j in any period.
(a)
Construct a transportation model to represent this problem.
(b)
Is there a feasible solution to this numerical example? Justify
your conclusion. Give a general rule for discerning whether such a
problem has a feasible solution.
(c)
Explain how to modify the formulation when the total supply
exceeds total demand.
(d)
Suppose each unit of inventory held at the end of a period
incurs the cost h. Will adding this cost change the optimal solution in
part (a)? In part (c)?
(e)
Is there an advantage to using a transshipment model rather than
a pure transportation formulation?
4. Suppose an LP formulation has the following coefficient matrix:
1
1
1
0
0
0
0
1
1
1
1
0
0
0
1
1
1
1
0
1
1
0
0
0
0
0
0
0
1
1
0
1
0
0
0
0
0
0
1
1
1
0
R1
R2
R3
R4
R5
R6
Using elementary row operations, attempt to convert this formulation
into a network structure.
5. Colonel Cutlass, having just taken command of the brigade, has
decided to assign people to his staff based on previous experience. His
list of major staff positions is adjutant (personnel officer),
intelligence officer, operations officer, supply officer, and training
officer. He has 5 people he feels could occupy these 5 positions, Below
are their years of experience in several fields:
Muddle
Whiteside
Kid
Klutch
Whiz
Adjutant
Intelligence
Operations
3
2
3
3
0
5
3
0
0
3
6
5
4
3
0
Supply
2
3
2
2
1
Training
2
2
2
2
0
Formulate the problem as an assignment problem to fill all positions and
to maximize the total number of people years experience and draw its
NETFORM.