Download BA 275, Fall 1998 Quantitative Business Methods

BA 555 Practical Business Analysis
Agenda
 Linear Programming (LP)

Sensitivity Analysis
 Simulation Using @Risk
1
Sensitivity Analysis (p.70)
 How will a change in a coefficient of the
objective function affect the optimal
solutions?
 How will a change in the right-hand-side
value for a constraint affect the optimal
solution?
MAX
3 A + 4 B
SUBJECT TO
2)
2 A + 2 B <=
3)
2 A + 4 B <=
END
80
120
2
Range of Optimality (p.70)
 The range of values over which an objective
function coefficient may vary without causing
any change in the values of the decision
variables in the optimal solution.
MAX
3 A + 4 B
SUBJECT TO
2)
2 A + 2 B <=
3)
2 A + 4 B <=
END
VARIABLE
A
B
CURRENT
COEF
3.000000
4.000000
80
120
OBJ COEFFICIENT RANGES
ALLOWABLE
ALLOWABLE
INCREASE
DECREASE
1.000000
1.000000
2.000000
1.000000
3
Range of Feasibility (p.70)
 The range of values over which a right-hand side
may vary without changing the value and
interpretation of the dual price (shadow price).
MAX
3 A + 4 B
SUBJECT TO
2)
2 A + 2 B <=
3)
2 A + 4 B <=
END
OBJECTIVE FUNCTION VALUE
80 1)
120
VARIABLE
A
B
140.0000
VALUE
REDUCED COST
OBJECTIVE
FUNCTION VALUE
20.000000
.000000
20.000000
.000000
1)
140.0000
VARIABLE
VALUE
REDUCED
ROW
SLACK OR SURPLUS
DUAL PRICES
A
20.000000
2)
.000000
1.000000.0
B
20.000000
3)
.000000
.500000.0
ROW
2
3
CURRENT
RHS
80.000000
120.000000
RIGHTHAND SIDE RANGES
ALLOWABLE
INCREASE
40.000000
40.000000
ROW
SLACK OR SURPLUS
2)
.000000
ALLOWABLE
3)
.000000
DECREASE
20.000000
40.000000
DUAL P
1.0
.5
4
Reduced Cost (p.70)
 The amount by which an objective function coefficient
would have to improve (increase for a maximization
problem, decrease for a minimization problem),
before it would be possible for the corresponding
variable to assume a positive value in the optimal
solution.
OBJECTIVE FUNCTION VALUE
1)
VARIABLE
A
B
ROW
2)
3)
140.0000
VALUE
20.000000
20.000000
SLACK OR SURPLUS
.000000
.000000
REDUCED COST
.000000
.000000
DUAL PRICES
1.000000
.500000
5
LINDO: The Model and Report
LP OPTIMUM FOUND AT STEP
MAX 30 C + 40 D
Objective:
(carpentry)
(varnishing)
(demand for desks)
(non-negativity)
OBJECTIVE FUNCTION VALUE
1)
240
6 C + 4 D <= 36 VARIABLE
C
4 C + 8 D <= 40
D
D <= 8
C >= 0
D >= 0 ROW
VALUE
4.000000
3
s.t.
2)
3)
4)
LP OPTIMUM FOUND AT STEP
240
VARIABLE
C
D
VALUE
4.000000
3
ROW
2)
3)
4)
VARIABLE
REDUCED COST
.000000
.000000
SLACK OR SURPLUS
.000000
.000000
5.000000
NO. ITERATIONS=
DUAL PRICES
2.500000
3.750000
.000000
2
RANGES IN WHICH THE BASIS IS UNCHANGED:
OBJECTIVE FUNCTION VALUE
1)
REDUCED COST
.000000
.000000
SLACK OR SURPLUS
.000000
.000000
5.000000
NO. ITERATIONS=
2
2
DUAL PRICES
2.500000
3.750000
.000000
C
D
ROW
2
3
4
CURRENT
COEF
30.000000
40.000000
OBJ COEFFICIENT RANGES
ALLOWABLE
ALLOWABLE
INCREASE
DECREASE
30.000000
10.000000
20.000000
20.000000
CURRENT
RHS
36.000000
40.000000
8.000000
RIGHTHAND SIDE RANGES
ALLOWABLE
INCREASE
24.000000
26.666670
INFINITY
ALLOWABLE
DECREASE
16.000000
16.000000
5.000000
2
6
RANGES IN WHICH THE BASIS IS UNCHANGED:
OBJ COEFFICIENT RANGES
EXCEL: The Model
7
EXCEL: The Answer Report
Microsoft Excel 11.0 Answer Report
Worksheet: [LP Example 10.xls]Example10
Report Created: 11/7/2006 11:03:04 AM
Target Cell (Max)
Cell
Name
$D$11 Maximize profit
Original Value
240
Final Value
240
Adjustable Cells
Cell
Name
$B$6 Production C
$C$6 Production D
Original Value
4
3
Final Value
Constraints
Cell
Name
$D$19 Carpentry Total hours
$D$20 Varnishing Total hours
$C$6 Production D
Cell Value
4
3
Formula
Status
Slack
36 $D$19<=$F$19 Binding
0
40 $D$20<=$F$20 Binding
0
3 $C$6<=$C$8
Not Binding
5
8
EXCEL: The Sensitivity Report
Microsoft Excel 11.0 Sensitivity Report
Worksheet: [LP Example 10.xls]Example10
Report Created: 11/7/2006 11:03:04 AM
Adjustable Cells
Cell
Name
$B$6 Production C
$C$6 Production D
Final Reduced Objective
Value
Cost
Coefficient
4
0
30
3
0
40
Allowable Allowable
Increase
Decrease
30
10
20
20
Constraints
Cell
Name
$D$19 Carpentry Total hours
$D$20 Varnishing Total hours
Final Shadow Constraint Allowable Allowable
Value
Price
R.H. Side
Increase
Decrease
36
2.5
36
24
16
40
3.75
40 26.66666667
16
Dual Prices in LINDO
9
EXCEL: The Limit Report
Microsoft Excel 11.0 Limits Report
Worksheet: [LP Example 10.xls]Limits Report 1
Report Created: 11/7/2006 10:26:28 AM
Target
Cell
Name
$D$11 Maximize profit
Adjustable
Cell
Name
$B$6 Production C
$C$6 Production D
The values in the Lower Limit column indicate the
smallest value each decision variable can assume
while the values of all other decision variables remain
Constant and all the constraints are satisfied.
Value
240
Value
4
3
Lower Target
Limit Result
0
120
0
120
Upper Target
Limit Result
4
240
3
240
The values in the Upper Limit column indicate the
largest value each decision variable can assume
while the values of all other decision variables remain
constant and all the constraints are satisfied.
10
Simulation (pp. 81 – 104)
Uncertainty
11
Simulation: Preparation (p.81)
 An experiment is the process by which an observation (or
measurement) is obtained.

Flipping a fair coin 5 times to observe the total number of Heads (H) or
Tails (T).
 An event is the outcome of an experiment.

3 H’s and 2 T’s in 5 trials.
 A variable X is a random variable if the value it assumes,
corresponding to the outcome of an experiment, is a chance or random
event. It may be defined as a specification or description of a
numerical result from a random experiment.

X = total number of T in 5 trials.
 Probability shows you the likelihood or chances for each of the various
potential future events, based on a set of assumptions about how the
world works. Probability tells you what the data will be like when you
know how the world is. (Cf. Statistics helps you figure out what the
world is like after you have seen some data that it generated.)

Pr( X = 5 ) = 0.03125.
12
Probability Distributions (p.81)
 The pattern of probabilities for a random variable is called its
probability distribution. It can be represented by a formula,
table, or graph.
 A Probability
Distribution
Plot
Probability Distribution
of X
0.4
Probability
 A Probability Table
Variable X
Probability
0
0.03125
1
0.15625
2
0.31250
3
0.31250
4
0.15625
5
0.03125
0.3
0.2
0.1
0
0
1
2
3
4
5
X = Total Number of T in 5 trials
 A Probability Density Function
 5  1   1 
P( X  k )     1  
 k  2   2 
k
5 k
In short, a probability distribution tells us (1) what possible
outcomes of a random experiment are, and (2) how likely
each outcome occurs.
13
Game 1 Expected Payoff (p.82)
Game 1. Flip a fair coin once. You get $1 if T occurs.
Expected Payoff
= $1 P(T) + $0 P(H)
= $1 (0.5) + $0 (0.5)
= $0.50
Game 1 Probability Distribution
Probability
0.6
0.4
0.2
0
H
T
14
Game 2 Expected Payoff (p.82)
Game 2. Flip a fair coin 5 times. You get $1 for every T.
Expected Payoff
5
=
 x  P( x  k )
Game 2 Probability Distribution
k 0
5
 (0.5) k (1  0.5) 5 k
x


k 0
k 
5
5!
= x
(0.5) k (1  0.5) 5 k
k!(5  k )!
k 0
=…
=…
= $2.5
5
0.4
Probability
=
0.3
0.2
0.1
0
0
1
2
3
4
Number of Tails
5
15
Simulation
 Simulation is a method for learning about a
real system by experimenting with a model
that represents the system. In other words, a
simulation model is a model that imitates a
real-life situation.
 How does a computer “flip coins?”
16
Excel Function: =Rand()
 Returns an evenly distributed random number greater than or
equal to 0 and less than 1. A new random number is returned
every time the worksheet is calculated.
 To generate a random real number between a and b, use:
RAND()*(b - a) + a
Uniform Distribution (0.0, 1.0)
0
1
17
Simulation Using Excel Functions
(p.82)
Formula in cells B2, B6:B10:
=if(rand() < 0.5, “H”, “T”)
Formula in cell B11:
=countif(B6:B10, “T”)
Formula in Cell C2:
=1 * B2
Formula in cell C6:
=1 * B11
Problem ? hard to keep track of results.
18
A Simulation Model (p.77)
 A simulation model contains the mathematical
expressions and logical relationships that describe
how to compute the value of the output given the
values of the inputs (both controllable and
probabilistic inputs).
Controllable Inputs
(values are selected by decision makers)
Model
(mathematical expressions and
logical relationships)
Output
Probabilistic Inputs
(values are randomly generated)
19
Game 1 Simulation Using @Risk
(p.83)
20
Game 2 Simulation Using @Risk
(p.84)
21
Key Idea: Use Probability Distributions to Describe
Uncertainty/Summarize Experience
Probability Distributions
22
Estimated Unit Sales
Summarize your experience/knowledge on unit
sales using:
 =RiskUniform(0.08, 0.12)
 =RiskNormal(0.10, 0.02)
 =RiskNormal(0.10, 0.001)
 =RiskPert(0.08, 0.10, 0.12)
=RiskTriang(0.08, 0.10, 0.12)
 =RiskDiscrete({0.08,0.10, 0.12},{0.1, 0.7,
0.2})
23
NPV: Simulation Results
24
Other @Risk Functions
25