Download BA 275, Fall 1998 Quantitative Business Methods

BA 555 Practical Business Analysis
Agenda
 Linear Programming (LP)



Introduction
Examples
LINDO and Excel-Solver
1
Decision-making under Uncertainty
 Decision-making under uncertainty entails
the selection of a course of action when we
do not know with certainty the results that
each alternative action will yield.
 This type of decision problems can be solved
by statistical techniques along with good
judgment and experience.
 Example: buying stocks/mutual funds.
2
Decision-making under Certainty
 Decision-making under certainty entails the
selection of a course of action when we know the
results that each alternative action will yield.
 This type of decision problems can be solved by
linear/integer programming technique.
 Example: A company produces two different auto
parts A and B. Part A (B) requires 2 (2) hours of
grinding and 2 (4) hours of finishing. The company
has two grinders and three finishers, each of which
works 40 hours per week. Each Part A (B) brings a
profit of $3 ($4). How many items of each part
should be manufactured per week?
3
Steps in Quantifying and Solving a
Decision Problem Under Certainty
 Formulate a mathematical model:
 Define decision variables,
 State an objective,
 State the constraints.
 Input the model to a LP/ILP solver, e.g., LINDO or
EXCEL Solver.
 Obtain computer printouts and perform sensitivity
analysis.
 Report optimal strategy.
4
Example 1 (p. 61)
 A company produces two different auto parts
A and B. Part A (B) requires 2 (2) hours of
grinding and 2 (4) hours of finishing. The
company has two grinders and three
finishers, each of which works 40 hours per
week. Each Part A (B) brings a profit of $3
($4). How many items of each part should be
manufactured per week?
5
Solving a LP problem:
LINDO or EXCEL Solver
 Install LINDO or EXCEL Solver (do at least one.)
 LINDO: http://www.lindo.com/. Go to DOWNLOAD
HOMEPAGE. On the left-hand-side, chose LINDO
FOR WINDOWS (not LINDO API, not LINGO.)
 Its syntax is given on pp. 78 – 80 of the class packet.

EXCEL Solver: Under Tools / Add-Ins. Check the
SOLVER ADD-INS box. Click OK.
 It is supported by the textbook (Chapter 4, pp. 209 –
281)
6
Example 2 Logistics (p.62)
Warehouse
W1
W2
Requirement
Per-Unit Shipping cost
Store A Store B Store C
$9
$8
$6
$7
$4
$3
140
50
110
Capacity
100
200
300
7
Example 8 Purchasing (p.68)
Component
1
2
Capacity
1
$12
$10
600
Supplier
2
$13
$11
1000
3
$14
$10
800
8
Example 3 Media Selection (p.63)
The Westchester Chamber of Commerce periodically sponsors public service seminars and
programs. Currently, promotional plans are under way for this year’s program. Advertising
alternatives include television, radio, and newspaper. Audience estimates, costs and maximum
media usage limitations are as shown.
Constraint
Audience per advertisement
Cost per advertisement
Maximum media usage
Television
100,000
$2,000
10
Radio
18,000
$300
20
Newspaper
40,000
$600
10
To ensure a balanced use of advertising media, radio advertisements must not exceed 50% of the
total number of advertisements authorized. In addition, television should account for at least
10% of the total number of advertisements authorized. If the promotional budget is limited to
$18,200, how many commercial messages should be run on each medium to maximize total
audience contact? Formulate a linear programming model to answer the above question.
9
Example 4 Portfolio Selection (p.64)
National Insurance Associates carries an investment portfolio of a variety of stocks, bonds, and
other investment alternatives. Currently $200,000 of funds are available and must be considered
for new investment opportunities. The four stock options National is considering and the
relevant financial data are as follows:
Stock
Financial Data
Price per share
Annual rate of return
Risk measure per dollar invested
A
$100
0.12
0.10
B
$ 50
0.08
0.07
C
$ 80
0.06
0.05
D
$ 40
0.10
0.08
Top management has stipulated the following investment guidelines:
1. The annual rate of return for the portfolio must be at least 9%.
2. No one stock can account for more than 50% of the total dollar investment.
Formulate a linear programming model that minimizes risk.
10
Example 6 Blending (p.66)
 Ajax Fuels, Inc., is developing a new additive for airplane fuels. The
additive is a mixture of three ingredients: A, B, and C. For proper
performance, the total amount of additive (amount of A + amount of B +
amount of C) must be at least 10 ounces per gallon of fuel. However,
because of safety reasons, the amount of additive must not exceed 15
ounces per gallon of fuel. The mix or blend of the three ingredients is
critical. At least 1 ounce of ingredient A must be used for every ounce
of ingredient B. The amount of ingredient C must be greater than onehalf the amount of ingredient A. If the costs per ounce for ingredients
A, B, and C are $0.10, $0.03, and $0.09, respectively, find the
minimum-cost mixture of A, B, and C for each gallon of airplane fuel.
11
Example 5 Production Scheduling
(p.65)
Q1
Q2
Q3
Q4
Demand
2000
4000
3000
1500
Capacity
4000
3000
2000
4000
Inv. Cost
Per unit
$250
$250
$300
$300
Prod. Cost
Per unit
$10,000
$11,000
$12,100
$13,310
Initial inventory = 100
Ending inventory ≥ 500
12
Example 7 Staff
Scheduling
(p.67)
Example 7: Staff Scheduling
D. V.
Xi = # of officers served in shift i
Shift 8 A.M.
1 X1
2
3
4
5
6 X6
Noon
X1
X2
Time of Day 8 A.M.
Req'd Officers 5
Noon
6
4 P.M.
X2
X3
4 P.M.
10
8 P.M. Midnight 4 A.M.
X3
X4
X4
X5
X5
X6
8 P.M. Midnight 4 A.M.
7
4
6
13
Variation: Staff Scheduling
SHIFT
M
T
W
R
F
1
X1
X1
X1
X1
X1
X2
X2
X2
X2
X2
X3
X3
X3
X3
X3
X4
X4
X4
X4
X5
X5
X5
A
B
16
11
2
3
4
X4
5
X5
X5
6
Req’
17
13
Different schedules
Different benefits
Full-time vs. part-time
Etc.
15
19
14
SA
SU
14
Example 9 Multi-period Financial
Planning (p.69)
Cash Flow at Time Period #
Investment
0
1
2
3
A
B
C
–1
0
–1
0.5
–1
1.2
1
0.5
0
0
1
0
D
E
–1
0
0
0
0
–1
1.9
1.5
Constraints:
Balance cash inflow and cash outflow at all time periods.
15