Chapter 9
If one would take statistics about which
mathematical problem is using most of the
computer time in the world (not including
data base handling problems like sorting
and searching) the answer would probably
be linear programming.—Laslo Lavasz
1
Linear Programming
Applications and
Computer Solutions
Product-Mix Selection
2
 Let XE, XL, XR, XS, and XM denote the number of extra-large, large,
regular, small, and miniature modules to assemble.
Maximize P =
58XE + 43XL + 25XR + 17XS + 28XM
Subject to:
58XE + 43XL + 25XR + 17XS + 28XM < 50,000 (PC bd)
25XE + 15XL + 10XR + 5XS + 1XM < 10,000 (res. A)
52XE + 48XL + 40XR + 60XS + 75XM < 25,000 (res. B)
1.50XE +1.25XL + 1.00XR +.75XS + 1.50XM < 2,000 (assem.)
XR ≥ 200 (reg. qty)
XS ≥ 100 (sm. qty)
2XE < XL
(mix 1)
XM < .50(XE + XL + XR + XS) (mix 2)
where
XE, XL, XR, XS, and XM ≥ 0
 The resource constraints all take the form: amt. used < amt. avail.
 The quantity constraints take form: number made ≥ minimum quantity.
 The mix 1 constraint translates: at least 2 extra-large modules made for
every large one.
 The mix 2 constraint translates: miniatures cannot exceed half the total
of the other sizes combined.
Product-Mix Selection
 The above problem must be solved with the
simplex method. That is nearly always
done with computer assistance.
 The constraint expressions must first be
modified so that all Xs appear on the left:
 2XE – XL < 0 for mix 1
 –.50XE –.50XL –.50XR –.50XS + XM < 0
for mix 2
 The Xs should all align vertically.
3
Entering Data with QuickQuant
 From the QuickQuant menu, select linear
programming.
 That brings to screen the following.
4
Entering Data with QuickQuant
 After supplying basic information, the
variables are named.
5
Entering Data with QuickQuant
 Then the objective coefficients are entered.
That is followed by entering the constraints.
6
Solving the Problem with
QuickQuant
 After entering the data, Run is pulled down
in the menu bar and Quick Solve is selected.
7
Product-Mix Selection Solution
 QuickQuant provides the following
solution.
8
Slack and Surplus Variables
 QuickQuant automatically assigns slack
and surplus variables to the constraints.
S1 is the unused quantity (slack) of
chips A.
S5 is the surplus regular modules
beyond the minimum.
9
Portfolio Selection
10
 A portfolio manager wants to determine how much to invest in
company bonds A, B, C, D, E, or F with respective yields 8.5, 9, 10,
9.5, 8.5 and 9%. Letting Xi = the dollar amount invested in company i
bonds, she wants to maximize interest income. Her objective is to
Maximize P = .085XA + .090XB + .100XC + .095XD + .085XE + .090XF
Total available funds are $100,000, and that constraint is:
XA + XB + XC + XD + XE + XF = 100,000 (funds)
No bond investment can exceed $25,000. For bond A that constraint is:
XA < 25,000
(limitation A)
Similar but separate constraints apply to the other five bonds. At least
half the funds must be placed in longer maturities (B, E, and F):
XE + XL + XR ≥ 50,000
(long maturity)
No more than 30% of all funds can be place in junk (C and D):
XC + XD < 30,000 (junk)
Non-negativity conditions apply.
The optimal solution is:
XA = 20,000
XB = 25,000
XC = 25,000
XD = 5,000
XE = 0
XA = 25,000
P = 9,175
Transportation Problem:
Shipment Scheduling
 The following capacity, demand, and unit costs
apply for plants and warehouses.
To Warehouse
From Plant Frankfurt
11
New York
Phoenix
Yokohama
Capacity
Juarez
$19
$7
$3
$21
100
Seoul
15
21
18
6
300
Tel Aviv
11
14
15
22
200
Demand
150
100
200
150
600
 The linear program involves one variable for each
cell in the above:
Xij = quantity shipped from plant i to warehouse j
i = J, S, T and j = F, N, P, Y
Transportation Problem:
Shipment Scheduling
 The following objective applies.
Minimize C =19XJF + 7XJN + 3XJP +21XJY
+15XSF +21XSN +18XSP + 6XSY
+11XTF +14XTN +15XTP +22XTY
Subject to:
XJF + XJN + XJP + XJY = 100 (Juarez Capacity)
XSF +XSN + XSP + XSY = 300 (Seoul Capacity)
XTF +XTN + XTP + XTY = 200 (Tel Aviv Capacity)
XJF + XSF + XTF
= 150 (Frankfurt Demand)
XJN + XSN + XTN
= 100 (New York Demand)
XJP + XSP + XTP
= 200 (Phoenix Demand)
XJY + XSY + XTY
= 150 (Yokohama Demand)
where
all Xij’s > 0
12
Solution to
Transportation Problem
 The linear program was solved on the
computer. The following shipment
quantities apply.
C = 6,250
To Warehouse
From Plant Frankfurt
13
New York
Phoenix
Yokohama
Juarez
0
0
100
0
Seoul
50
0
100
150
Tel Aviv
100
100
0
0
Budgeting Advertising
Expenditures
 Real Reels is deciding how many ads to
place in Playboy (P), True (T), and Esquire
(E). Respective costs are $10,000, $5,000,
and $6,000.
 The respective variables are XP, XT, and XE.
 The objective is to
Maximize P = 1XP + .9XT + .28XE
with the coefficients are the number of
exposures (millions of gear users) per ad.
14
Budgeting Advertising
Expenditures
 There is a budget maximum of $100,000. A
maximum of 5 ads may be placed in True
and minimum of 2 for each other magazine.
 The following constraints apply:
10XP + 5XT + 6XE < 100 (budget)
1XT
< 5 (True max.)
1XP
> 2 (Playboy min.)
1XE > 2 (Esquire min.)
all Xs > 0
15  Solution: X = 6.3, X = 5, X = 2, P = 11.36
P
T
E
Assignment Problem
 The following data apply for persons and jobs.
Time to Complete One Job
Individual
Drilling
Grinding
Lathe
Ann
5 min.
10 min.
10 min.
Bud
10
5
15
Chuck
15
15
10
 The linear program involves one variable for each
cell in the above:
Xij = Fraction of time person i is assigned to job j
i = A, B, C and j = D, G, L
16
Assignment Problem
17
 The following objective applies.
Minimize C = 5XAD + 10XAG + 10XAL
+10XBD + 5XBG + 15XBL
+
11XCD + 14XCG + 15XCL
Subject to:
XAD + XAG + XAL = 1 (Ann’s Availability)
XBD + XBG + XBL = 1 (Bud’s Availability)
XCD + XCG + XCL = 1 (Chuck’s Availability)
XAD + XBD + XCD = 1 (drill-press requirement)
XAG + XBG + XCG = 1 (grinder requirement)
XAL + XBL + XCL = 1 (lathe requirement)
where
all Xij’s > 0
 Solution:
XAD = 1 (Ann to Drilling) XBG = 1 (Bud to Grinding)
XCL = 1 (Chuck to Lathe) C = 20
Liquid Blending
 Chanel 2000 makes aftershave and cologne. The following data apply.
Agents (%)
Selling Order
Product
Emulsion Evaporatives Price Quantity
Aftershave
--
20
$10
1,500
Cologne
30
--
20
500
 The following data apply to the raw materials.
Agents (%)
Ingredient
Oil
Rinse
Emulsion
50
Evaporatives
0
Cost
$ 2
100
25
30
Available
2,000
500
Stabilizer
10
50
4
1,000
 Let Xij = Volume (liters) of ingredient i used in blending product j
with i = O, R, S and j = A, C
18
Liquid Blending





19
Revenue = 10(XOA + XRA + XSA) + 20(XOC + XRC + XSC)
Cost = 2(XOA + XOC) + 30(XRA + XRC) + 4(XSA + XSC)
Using Profit = Revenue – Cost, collecting terms, the objective is to
Maximize P = 8XOA - 20XRA + 6XSA + 18XOC - 10XRC + 16XSC
There are three constraints for resource availabilities:
XOA + XOC < 2,000
(available oil)
XRA + XRC <
500
(available rinse)
XSA + XSC < 1,000
(available stabilizer)
There are two constraints for product quantity requirements:
XOA + XRA + XSA > 1,500
(aftershave volume)
XOC + XRC + XSC > 500
(cologne volume)
There are two proportional ingredient requirements:
.50XOC + 1.00XRC + .10XSC > .30(XOC + XRC + XSC)
(emulsions in cologne)
.25XRA + .50XSA > .20(XOA + XRA + XSA)
(evaps. in aftershave)
These simplify to:
.20XOC + .70XRC - .20XSC > 0
(emulsions in cologne)
-.20XOA + .05XRA + .30XSA > 0
(evaporatives in aftershave)
Solution:
XOA = 500 XRA = 0 XSA = 1,000
XOC = 1,500 XRC = 0 XSC =
0 P = 37,000
Solving Linear Programs
with a Spreadsheet
Step 1: Write out the formulation table.
Step 2: Put the formulation table into a
spreadsheet.
Step 3: Use Excel’s Solver to obtain a solution.
20
Step 1: The Formulation Table
(Figure 9-1)
The formulation table arranges the problem in a
tabular format, as shown below for the Microcircuit
Production Plan.
Variables
XE
XL
XR
XS
XM Sign RHS
Objective
58
43
25
17
28
=
P(max)
PC Board
25
15
10
5
1
<
50,000
A Availability
28
24
18
12
5
<
10,000
B Availability
52
48
40
60
75
<
25,000
Assembly Time
1.50 1.25 1.00 0.75 1.50 <
2,000
Regular Quantity
1
>
200
Small Quantity
1
>
100
Oversized Mixture 2
-1
<
0
Miniature Mixture -0.50 -0.50 -0.50 -0.50 1.00 <
0
21
Step 2: The Excel Spreadsheet
(Figure 9-2)
The numbers in the Excel spreadsheet come from
the formulation table.
A
1
2
3
4
5
6
7
8
9
10
11
12
22
B
C
D
E
F
G
H
Sign
=
<
<
<
<
>
>
<
<
RHS
P(max)
50000
10000
25000
2000
200
100
0
0
Microcircuit Production Plan
Variables
Objective
PC Board
A Availability
B Availability
Assembly Time
Regular Quantity
Small Quantity
Oversized Mixture
Miniature Mixture
XE
58
25
28
52
1.50
XL
43
15
24
48
1.25
XR
25
10
18
40
1.00
1
XS
17
5
12
60
0.75
XM
28
1
5
75
1.50
1
2
-0.50
-1
-0.50
-0.50
-0.50
1.00
Step 3: Expanded Spreadsheet
(Figure 9-3)
The expanded spreadsheet contains the formulas necessary
to use Solver. Put =SUMPRODUCT(B4:F4,$B$15:$F$15) in
cell J4 and copy it down to cell J12. Cell J4 gives the value
of the objective function.
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
B
C
D
E
F
G
H
I
J
Microcircuit Production Plan
Variables
Objective
PC Board
A Availability
B Availability
Assembly Time
Regular Quantity
Small Quantity
Oversized Mixture
Miniature Mixture
23
XE
58
25
28
52
1.50
XL
43
15
24
48
1.25
2
-0.50
-1
-0.50
XE
10.00
XL
20.00
XR
25
10
18
40
1.00
1
XS
17
5
12
60
0.75
XM
28
1
5
75
1.50
1
-0.50
Solution
XR
200.00
-0.50
1.00
XS
100.00
XM
10.00
The solution is found
here (the values of the
decision variables).
Sign
=
<
<
<
<
>
>
<
<
4
5
6
7
8
9
10
11
12
RHS
P(max)
50000
10000
25000
2000
200
100
0
0
Profit
PC Board
A Availability
B Availability
Assembly Time
Regular Quantity
Small Quantity
Oversized Mixture
Miniature Mixture
J
=SUMPRODUCT(B4:F4,$B$15:$F$15)
=SUMPRODUCT(B5:F5,$B$15:$F$15)
=SUMPRODUCT(B6:F6,$B$15:$F$15)
=SUMPRODUCT(B7:F7,$B$15:$F$15)
=SUMPRODUCT(B8:F8,$B$15:$F$15)
=SUMPRODUCT(B9:F9,$B$15:$F$15)
=SUMPRODUCT(B10:F10,$B$15:$F$15)
=SUMPRODUCT(B11:F11,$B$15:$F$15)
=SUMPRODUCT(B12:F12,$B$15:$F$15)
8420
3060
5610
16230
330
200
100
0
-155
Using Excel’s Solver to
Solve Linear Programs
Click on Tools on the menu
bar, select the Solver option, and the
Solver Parameters dialog box shown
next appears.
24
1. Enter the
value of the
objective
function, J4, in
the Target Cell
line, either with
or without the $
sign.
Solver Parameters Dialog Box
(Figure 9-5)
NOTE: Normally all these entries appear in the Solver Parameter dialog
box so you only need to click on the Solve button. However, you
should always check to make sure the entries are correct for the
problem you are solving.
2. The Target
Cell is to be
maximized so
click on Max in
the Equal To
line.
3. Enter the
decision
variables in the
By Changing
Cells line,
B15:F15.
4. The constraints are entered in the Subject to Constraints box by using the
Add Constraints dialog box shown next (obtained by clicking on the Add
button). If a constraint needs to be changed, click on the Change button. The
25
Change and Add Constraint dialog box function in the same manner.
The Add Constraint Dialog Box
(Figure 9-6)
To represent the constraints in
rows 5 - 8:
1. Enter J5:J8 (or $J$5:$J$8) in
the Cell Reference line. This is
the total amount of these
resources used.
Normally, all
these entries
already appear.
You will need to
use this dialog
box only if you
need to add a
constraint.
3. Enter the
amounts of the
resources available
H5:H8 in the
Constraint line (or
=$H$5:$H$8).
4. Click Add
and repeat
Steps 1 - 3 if
another
constraint is to
be added. If
this is the last
constraint, click
OK.
2. Enter <= as the sign because the resources used must be equal to or less
than the amounts available, given next in Step 3. If another sign is needed, see
the next slide.
26
If you need to change a constraint, the Change Constraint dialog
box functions just like this one.
Dialog Box for Constraint Signs
(Figure 9-7)
To enter different
signs, click on the
down arrow and
three possibilities
are displayed: <=,
=, >=,
27
The Solver Options Dialog Box
(Figure 9-8)
Click on the
Options button
in the Solver
Parameters
dialog box to
check the Solver
Options dialog
box to ensure
that the Assume
Linear Model
and Assume
Non-Negative
boxes are
checked.
28
Solver Results Dialog Box
(Figure 9-9)
Be sure to check
the message in
the Solver
Results dialog
box. In this case
it indicates that
a solution has
been found.
What happens
when Solver
does not find a
solution will be
discussed latter.
Click OK and the
spreadsheet
with the
solution, shown
next, is
obtained.
29
1. To solve
other problems:
Spreadsheet with
Optimal Solution (Figure 9-10)
2. Enter the data: the coefficients of the objective function in cells B4:F4,
the right-hand sides in cells H5:H12, and the exchange coefficients in
cells B5:F12.
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
B
C
D
E
F
G
H
I
J
Microcircuit Production Plan
Variables
Objective
PC Board
A Availability
B Availability
Assembly Time
Regular Quantity
Small Quantity
Oversized Mixture
Miniature Mixture
30
XE
58
25
28
52
1.50
XL
43
15
24
48
1.25
2
-0.50
-1
-0.50
XE
67.54
XL
135.08
XR
25
10
18
40
1.00
1
XS
17
5
12
60
0.75
XM
28
1
5
75
1.50
1
3. To find the
solution, click on
Tools and Solver to
obtain the Solver
Parameters dialog
box and then click
the Solve button.
-0.50
Solution
XR
200.00
-0.50
1.00
XS
100.00
XM
13.39
Sign
=
<
<
<
<
>
>
<
<
RHS
P(max)
50000
10000
25000
2000
200
100
0
0
Profit 16800.65
PC Board 6228.10
A Availability 10000.00
B Availability 25000.00
Assembly Time
565.24
Regular Quantity
200.00
Small Quantity
100.00
Oversized Mixture
0.00
Miniature Mixture
-237.92
4. For bigger problems insert additional rows or
columns. Insert them in the middle of the table and
not at the beginning or the end. Copy the formulas
in column J to any new cells created by inserting
rows. Check to make sure the ranges of the
formulas and signs in the Solver Parameters dialog
box are correct.
Solver’s Answer Report
Solver’s Answer Report gives the values of
the:
• objective function
• decision variables
• slack variables
31
Solver’s Answer Report
To get Solver’s Answer Report, highlight Answer
Report in the Report box of the Solver Results
dialog box before clicking the OK button.
32
Answer Report for Microcircuit
Production Plan (Figure 9-11)
Target Cell (Max)
Cell
Name
$J$4 Profit
Adjustable Cells
Cell
Name
$B$15 XE
$C$15 XL
$D$15 XR
$E$15 XS
$F$15 XM
Original Value
0.00
Original Value
0.00
0.00
0.00
0.00
0.00
Final Value
16800.65
Final Value
67.54
135.08
200.00
100.00
13.39
Objective
function
Decision
variables
Slack
variables
Note: Not binding means the slack variable is positive, binding means it is zero.
33
Constraints
Cell
Name
$J$5 PC Board
$J$6 A Availability
$J$7 B Availability
$J$8 Assembly Time
$J$9 Regular Quantity
$J$10 Small Quantity
$J$11 Oversized Mixture
$J$12 Miniature Mixture
Cell Value
6228.10
10000.00
25000.00
565.24
200.00
100.00
0.00
-237.92
Formula
$J$5<=$H$5
$J$6<=$H$6
$J$7<=$H$7
$J$8<=$H$8
$J$9>=$H$9
$J$10>=$H$10
$J$11<=$H$11
$J$12<=$H$12
Status
Not Binding
Binding
Binding
Not Binding
Binding
Binding
Binding
Not Binding
Slack
43771.90
0.00
0.00
1434.76
0.00
0.00
0.00
237.92
Bond Portfolio Selection
(page 319)
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
34
B
C
D
E
F
G
H
I
J
K
RHS
P(max)
100,000
25,000
25,000
25,000
25,000
25,000
25,000
50,000
30,000
Profit
Funds
Diversification - A
B
C
D
E
F
Maturity
Quality
9,175
100,000
20,000
25,000
25,000
5,000
0
25,000
70,000
30,000
Bond Portfolio Selection
Variables
Objective
Funds
Diversification - A
B
C
D
E
F
Maturity
Quality
XA
0.085
1
1
XB
0.09
1
XC
0.1
1
XD
XE
0.095 0.085
1
1
XF
0.09
1
1
1
1
1
1
XA
20,000
1
XB
25,000
1
1
Solution
XC
XD
25,000 5,000
1
1
1
XE
0
XF
25,000
Sign
=
=
<
<
<
<
<
<
>
<
Real Reels
(page 323)
A
1
2
3
4
5
6
7
8
9
10
11
35
B
C
D
E
F
G
H
RHS
P(max)
100,000
5
2
2
Profit
Budget
Maximum True ads
Minimum Playboy ads
Minimum Esquire ads
11.36
100000.00
5.00
6.30
2.00
Real Reels
Variables
Objective
Budget
Maximum True ads
Minimum Playboy ads
Minimum Esquire ads
XP
1
10000
XT
0.9
5000
1
XE
0.28
6000
1
1
Solution
XA
XB
6.30
5.00
XC
2.00
Sign
=
<
<
>
>
Scent Mixing
(page 328)
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
36
B
C
D
E
F
G
H
I
J
Sign
=
<
<
<
>
>
>
>
RHS
P(max)
2000
500
1000
1500
500
0
0
Profit
Available oil
Available rinse
Available stabilizer
Aftershave volume
Cologne volume
Emulsions in cologne
Evaporatives in aftershave
K
Scent Mixing
Variables
Objective
Available oil
Available rinse
Available stabilizer
Aftershave volume
Cologne volume
Emulsions in cologne
Evaporatives in aftershave
XOA
8
1
XRA
-20
XSA
6
XOC
18
1
1
1
1
0.05
XOA
500
XRA
0
XSC
16
1
1
1
1
1
0.2
-0.2
XRC
-10
1
0.7
1
-0.2
0.3
Solution
XSA XOC XRC
1000 1500 0
XSC
0
37000
2000
0
1000
1500
1500
300
200
Yosemite Ann
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
37
B
C
D
E
F
G
H
I
J
Yosemite Ann
Variables
Objective
Total weight
Calories
Protein
Iron
Vitamin A
Thiamin
Riboflavin
Niacin
Calcium
Ascorbic acid
XC
1.50
1.00
540
20
10
230
0.5
0.5
3
600
2000
XP
2.50
1.00
5720
270
30
0
2.5
2.6
170
720
10
XW
XK
XM
Sign RHS
4.00
1.00
1.50
=
C(min) Cost
1.00
1.00
1.00
=
1 Total weight
6540 5530 4990
>
3000 Calories
150
290
260
>
56 Protein
20
110
10
>
10 Iron
300
700
9200
>
1000 Vitamin A
4.8
2.4
2.9
>
1.4 Thiamin
1.3
1.9
12.1
>
1.6 Riboflavin
12
24
7
>
18 Niacin
830
510
9120
>
800 Calcium
30
0
90
>
60 Ascorbic acid
Solution
XC
XP
XW
XK
XM
0.0283 0.0000 0.0000 0.9348 0.0369
1.0326
1
5369
281
103
1000
2
2
23
830
60