1)
An electronics company produces two types of television sets, colour and black-and-white. The production of
a colour set requires 10 hours of skilled and 100 hours of unskilled labour. The production of a black-andwhite set requires 5 hours of skilled and 150 hours of unskilled labour. The company has 100 hours of skilled
labour and 1,500 hours of unskilled labour normally available per month to produce television sets. The
maximum number black-and-white and colour sets that can be sold each month are 45 and 70,
respectively. The profit margin from the sale of a colour set is $20, whereas it is $15 from a black-and-white
set. The company has set the following goals:
1. Avoid the over utilization of skilled labour since it is hard to obtain in the labour market.
2. Minimize the under-utilization of unskilled labour.
3. Meet the demand as much as possible.
4. Limit over utilization of unskilled labour to 100 hours.
Formulate the above as a goal programming problem and solve using Excel.
2)
A department store plans to schedule its annual advertising. The total budget is set at $200,000. The store
can purchase local radio spots at $100 per spot, local television spots at $500 per spot and local newspaper
advertising at $200 per ad. The payoff from each advertising medium is a function of its audience size and
audience characteristics. The generally accepted objective criterion for advertising is audience points,
reflected in the following table:
Medium
Points
Radio
30 per spot
Television
150 per spot
Newspaper
150 per ad
The president of the firm has established the following goals for the campaign:
1. The total budget should not exceed $200,000.
2. Meet the contract with the local television station that requires that the firm spend at least $30,000.
3. The corporate advertising policy prohibits annual newspaper ad expenditures in excess of $50,000.
4. Maximize the audience points for the advertising campaign.
The president has established unit weights on the goals of 10, 6, 3 and 1 for the goals 1 through 4,
respectively. Formulate the above as a goal programming problem and solve using Excel.
3)
The Midtown City Council is reviewing housing proposals for a new development area. There is some dispute
among various interest groups as to what goals should be sought. The zoning committee has recommended
three types of housing: one-family houses, deluxe condominiums and apartments. The zoning committee has
compiled the following data for each type of housing:
One-family Condominiums Apartments
Acres per unit
Families housed per unit
Tax base generated per unit
.25
.20
.125
1
4
6
$50,000
$100,000
$150,000
Taxes required for city services
$4,000
$8,000
$10,000
There are 50 acres available for zoning. The League for Better Housing has conducted a campaign to gain
housing for at least 500 families. The Taxpayers’ Uniion has strongly lobbied for an added tax base of
$5,000,000. The Gray Panthers have disrupted the city council meetings to demand that taxes for city services
be no more than $250,000. The city council hired a public opinion survey company to assess the priorities of
the citizens. The poll results are as follows:
Priority Priority Priority
1
2
3
Housing for 500
families
55%
35%
10%
Tax base of $5,000,000
40
30
30
Taxes for services of
$250,000
15
20
65
Based on this survey the city council has established the following goals:
1. Provide housing for at least 500 families.
2. Establish at least $5,000,000 worth of new tax base.
3. Limit taxes for city services to $250,000.
4. Reserve at least 5 acres for a neighborhood park area.
It is assumed that the first goal is met fully before the second, the second met fully before the third and the
third met fully before the fourth. Formulate the above as a goal programming problem and solve using Excel.
4)
Acme Sawmill can produce plywood, chipboard and pulp for sale, realizing profit margins of $10, $9 and $6 per
ton, respectively. The mill can run any number of operations at the same time, but the setup costs for each of
the operations differ. While the pulp production only costs $2,000 to set up the plywood production costs
$50,000 and the chipboard production costs $25,000. Plywood consists of 95% wood and 5% resin
glue. Chipboard consists of 91% wood, 5% resin glue and 4% other additives. Pulp consists of 86% wood and
14% other additives. Demand limits the amount of plywood produced to 10,000 tons, while as much as 5,000
tons each may be produced of chipboard and pulp. For the next month of production there are 15,000 tons of
wood and 500 tons each of resin glue and other additives available. Finally, Acme’s customers always place
orders for whole tons of plywood and chipboard. No partial tons may be accepted.
The operations manager at Acme has set the following goals for the upcoming month:
1. Achieve at least $70,000 profit.
2. Avoid having to special order more glue and additives.
Although it is an inconvenience to have to reorder glue and additives the target profit is a more important
consideration. Accordingly, the operations manager has placed subjective weights of 5 on profit deviations
and 1 on resource deviations. Formulate the above as a goal programming problem and solve using Excel.
Answers:
1) Note that since the demand goal stated "meet the demand as much as possible" production deviation
either above or below demand is considered undesirable for both types of televisions. Also note that all
apparent "hard" constraints are superceded by goal constraints.
Let: C = # of color televisions produced
B = # of black and white televisions produced
ds+ = deviation above skilled labor utilization target
ds- = deviation below skilled labor utilization target
du+ = deviation above unskilled labor utilization target
du- = deviation below unskilled labor utilization target
dc+ = deviation above color television demand target
dc- = deviation below color television demand target
db+ = deviation above black and white television demand
target
db- = deviation below black and white television demand
target
do+ = deviation above unskilled labor overutilization target
do- = deviation below unskilled labor overutilization target
Minimize Z = ds+ + du- + dc+ + dc- + db+ + db- + do+
s.t.
10 C + 5 B - ds+ + ds- =
100
100 C + 150 B - du+ + du- = 1,500
(Skilled use goal)
(Unskilled use goal)
C - dc+ + dc- =
70
(Color demand goal)
B - db+ + db- =
45
(B&W demand goal)
du+ - do+ + do- =
100 (Unskilled overuse goal)
C, B, ds+, ds-, du+, du-, dc+, dc-, db+, db-, do+, do- > 0
where: and
C, B are integer
2) Note that an arbitrarily high number has been chosen for the RHS of the final goal constraint to "maximize
the audience points." Also note that all apparent "hard" constraints are superceded by goal constraints.
Let: R = # of local radio spots purchased
T = # of local television spots purchased
N = # of newspaper ads purchased
db+ = deviation above budget target
db- = deviation below budget target
dc+ = deviation above television contract target
dc- = deviation below television contract target
dp+ = deviation above newspaper policy target
dp- = deviation below newspaper policy target
da+ = deviation above audience points target
da- = deviation below audience points target
Minimize Z = 10 db+ + 6 dc- + 3 dp+ + da-
s.t.
100 R + 500 T + 200 N - db+ +
=
db-
200,000
(Budget goal)
500 T - dc+ + dc- =
30,000
(TV contract
goal)
200 N - dp+ + dp- =
50,000
(Newspaper
goal)
30 R + 150 T + 150 N - da+ +
= 1,000,000 (Audience goal)
dawhere:
R, T, N, db+, db-, dc+, dc-, dp+, dp-, da+, da- > 0 and
R, T, N are integer
3) Note that a heuristic is employed in assigning objective coefficients in order to prioritize goal compliance.
The variable representing the deviation with the lowest priority is assigned a weight of "1" and the weights are
increased by a factor of 10 for each successively more important goal. The resulting solution, however, does
not allow compliance with goal 3 before that for goal 4. There is no feasible solution that allows compliance
with goal 3 if goals 1 and 2 are both satisfied. Also note that only the total acres available for building remains
as a "hard" constraint.
Let: S = # of single-familiy homes built
C = # of deluxe condominiums built
A = # of apartments built
D = # of acres of land not used for building
df+ = deviation above families housed target
df- = deviation below families housed target
dt+ = deviation above tax base target
dt- = deviation below tax base target
ds+ = deviation above city services tax target
ds- = deviation below city services tax target
dp+ = deviation above park set-aside target
dp- = deviation below park set-aside target
Minimize Z = 1,000 df- + 100 dt- + 10 ds+ + dps.t.
.25 S + .2 C + .125 A + D = 50
(Acres available)
S + 4 C + 6 A - df+ + df- = 500
(Families goal)
5 S + 10 C + 15 A - dt+ + dt- = 500
(Tax base goal)
4 S + 8 C + 10 A - ds+ + ds- = 250
(City service tax goal)
D - dp+ + dp- =
where:
5
S, C, A, D, df+, df-, dt+, dt-, ds+, ds-, dp+, dp- > 0 and
S, C, A are integer
4)
Let:
(Park goal)
X1 = # of tons of plywood produced
X2 = # of tons of chipboard produced
X3 = # of tons of pulp produced
X4 = {
1 if plywood is produced
0 if otherwise
X5 = {
1 if chipboard is produced
0 if otherwise
X6 = {
1 if pulp is produced
0 if otherwise
dp+ = deviation above profit target
dp- = deviation below profit target
dg+ = deviation above glue requirement target
dg- = deviation below glue requirement target
da+ = deviation above additive requirement target
da- = deviation below additive requirement target
Minimize Z = 5 dp- + dg+ + da+
s.t.
(Wood
available)
.95 X1 + .91 X2 + .86 X3 < 15,000
X1 - 10,000 X4 <
0
(Ply demand)
X2 - 5,000 X5 <
0 (Chip demand)
X3 - 5,000 X6 <
0 (Pulp demand)
10 X1 + 9 X2 + 6 X3 - 50,000 X4 - 25,000 X5 - 2,000
= 70,000
X6 - dp+ + dp-
(Profit goal)
.05 X1 + .05 X2 - dg+ + dg- =
500
(Glue goal)
.04 X2 + .14 X3 - da+ + da- =
500
(Additives
goal)
X1, X2, X3, X4, X5, X6, dp+, dp-, dg+, dg-, da+, da- > 0,
where: X1, X2 are integer, and
X4, X5, X6 are binary