Transcript Spreadsheet Modeling & Decision Analysis:

Chapter 3
Modeling and Solving LP
Problems in a Spreadsheet
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-1
Introduction
 Solving
LP problems graphically is only
possible when there are two decision
variables
 Few real-world LP have only two
decision variables
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-2
Spreadsheet Solvers
 The
company that makes the Solver in
Excel, Lotus 1-2-3, and Quattro Pro is
Frontline Systems, Inc.
web site:
http://www.frontsys.com
 Other
packages for solving MP problems:
AMPL
CPLEX
LINDO
MPSX
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-3
The Steps in Implementing an LP
Model in a Spreadsheet
1. Organize the data for the model on the spreadsheet.
2. Reserve separate cells in the spreadsheet to
represent each decision variable in the model.
3. Create a formula in a cell in the spreadsheet that
corresponds to the objective function.
4. For each constraint, create a formula in a separate
cell in the spreadsheet that corresponds to the lefthand side (LHS) of the constraint.
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-4
The Blue Ridge Hot Tubs
Example...
MAX: 350X1 + 300X2
S.T.: 1X1 + 1X2 <= 200
9X1 + 6X2 <= 1566
12X1 + 16X2 <= 2880
X1, X2 >= 0
} profit
} pumps
} labor
} tubing
} nonnegativity
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-5
Implementing the Model
See file Fig3-1.xls
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-6
How Solver Views the Model
 Target
cell - the cell in the spreadsheet
that represents the objective function
 Changing cells - the cells in the
spreadsheet representing the decision
variables
 Constraint cells - the cells in the
spreadsheet representing the LHS
formulas on the constraints
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-7
Goals For Spreadsheet Design
 Communication - A spreadsheet's primary business
purpose is that of communicating information to managers.
 Reliability - The output a spreadsheet generates
should be correct and consistent.
 Auditability - A manager should be able to retrace the
steps followed to generate the different outputs from the
model in order to understand the model and verify results.
 Modifiability - A well-designed spreadsheet should be
easy to change or enhance in order to meet dynamic user
requirements.
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-8
Spreadsheet Design Guidelines








Organize the data, then build the model around the data.
Do not embed numeric constants in formulas.
Things which are logically related should be physically
related.
Use formulas that can be copied.
Column/rows totals should be close to the columns/rows
being totaled.
The English-reading eye scans left to right, top to
bottom.
Use color, shading, borders and protection to distinguish
changeable parameters from other model elements.
Use text boxes and cell notes to document various
elements of the model.
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-9
Make vs. Buy Decisions:
The Electro-Poly Corporation
 Electro-Poly
is a leading maker of slip-rings.
 A $750,000 order has just been received.
Model 1
Model 2
Model 3
3,000
2,000
900
Hours of wiring/unit
2
1.5
3
Hours of harnessing/unit
1
2
1
Cost to Make
$50
$83
$130
Cost to Buy
$61
$97
$145
Number ordered
 The
company has 10,000 hours of wiring
capacity and 5,000 hours of harnessing
capacity.
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-10
Defining the Decision Variables
M1 = Number of model 1 slip rings to make in-house
M2 = Number of model 2 slip rings to make in-house
M3 = Number of model 3 slip rings to make in-house
B1 = Number of model 1 slip rings to buy from competitor
B2 = Number of model 2 slip rings to buy from competitor
B3 = Number of model 3 slip rings to buy from competitor
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-11
Defining the Objective Function
Minimize the total cost of filling the order.
MIN: 50M1 + 83M2 + 130M3 + 61B1 + 97B2 + 145B3
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-12
Defining the Constraints
 Demand
Constraints
M1 + B1 = 3,000 } model 1
M2 + B2 = 2,000 } model 2
M3 + B3 =
 Resource
900 } model 3
Constraints
2M1 + 1.5M2 + 3M3 <= 10,000 } wiring
1M1 + 2.0M2 + 1M3 <= 5,000 } harnessing
 Nonnegativity
Conditions
M1, M2, M3, B1, B2, B3 >= 0
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-13
Implementing the Model
See file Fig3-17.xls
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-14
An Investment Problem:
Retirement Planning Services, Inc.
 A client
wishes to invest $750,000 in the
following bonds.
Company
Return
Years to
Maturity
Acme Chemical
8.65%
11
1-Excellent
DynaStar
9.50%
10
3-Good
Eagle Vision
10.00%
6
4-Fair
Micro Modeling
8.75%
10
1-Excellent
OptiPro
9.25%
7
3-Good
Sabre Systems
9.00%
13
2-Very Good
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
Rating
2-15
Investment Restrictions
 No
more than 25% can be invested in
any single company.
 At least 50% should be invested in longterm bonds (maturing in 10+ years).
 No more than 35% can be invested in
DynaStar, Eagle Vision, and OptiPro.
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-16
Defining the Decision Variables
X1 = amount of money to invest in Acme Chemical
X2 = amount of money to invest in DynaStar
X3 = amount of money to invest in Eagle Vision
X4 = amount of money to invest in MicroModeling
X5 = amount of money to invest in OptiPro
X6 = amount of money to invest in Sabre Systems
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-17
Defining the Objective Function
Maximize the total annual investment return.
MAX: .0865X1 + .095X2 + .10X3 + .0875X4 + .0925X5 + .09X6
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-18
Defining the Constraints
 Total
amount is invested
X1 + X2 + X3 + X4 + X5 + X6 = 750,000
 No
more than 25% in any one investment
Xi <= 187,500, for all i
 50%
long term investment restriction.
X1 + X2 + X4 + X6 >= 375,000
 35% Restriction on DynaStar, Eagle Vision, and
OptiPro.
X2 + X3 + X5 <= 262,500
 Nonnegativity
conditions
Xi >= 0 for all i
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-19
Implementing the Model
See file Fig3-20.xls
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-20
A Transportation Problem:
Tropicsun
Supply
Groves
Distances (in miles)
Capacity
21
Mt. Dora
275,000
Processing
Plants
1
Ocala
4
50
200,000
40
35
400,000
30
Eustis
Orlando
2
600,000
5
22
55
20
Clermont
300,000
3
Leesburg
25
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
6
225,000
2-21
Defining the Decision Variables
Xij = # of bushels shipped from node i to node j
Specifically, the nine decision variables are:
X14 = # of bushels shipped from Mt. Dora (node 1) to Ocala (node 4)
X15 = # of bushels shipped from Mt. Dora (node 1) to Orlando (node 5)
X16 = # of bushels shipped from Mt. Dora (node 1) to Leesburg (node 6)
X24 = # of bushels shipped from Eustis (node 2) to Ocala (node 4)
X25 = # of bushels shipped from Eustis (node 2) to Orlando (node 5)
X26 = # of bushels shipped from Eustis (node 2) to Leesburg (node 6)
X34 = # of bushels shipped from Clermont (node 3) to Ocala (node 4)
X35 = # of bushels shipped from Clermont (node 3) to Orlando (node 5)
X36 = # of bushels shipped from Clermont (node 3) to Leesburg (node 6)
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-22
Defining the Objective Function
Minimize the total number of bushel-miles.
MIN: 21X14 + 50X15 + 40X16 +
35X24 + 30X25 + 22X26 +
55X34 + 20X35 + 25X36
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-23
Defining the Constraints
 Capacity
constraints
X14 + X24 + X34 <= 200,000
X15 + X25 + X35 <= 600,000
X16 + X26 + X36 <= 225,000
 Supply
} Ocala
} Orlando
} Leesburg
constraints
X14 + X15 + X16 = 275,000
X24 + X25 + X26 = 400,000
X34 + X35 + X36 = 300,000
} Mt. Dora
} Eustis
} Clermont
 Nonnegativity conditions
Xij >= 0 for all i and j
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-24
Implementing the Model
See file Fig3-24.xls
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-25
A Blending Problem:
The Agri-Pro Company
 Agri-Pro
has received an order for 8,000 pounds of
chicken feed to be mixed from the following feeds.
Percent of Nutrient in
Nutrient
Feed 1
Feed 2
Feed 3
Feed 4
Corn
30%
5%
20%
10%
Grain
10%
3%
15%
10%
Minerals
20%
20%
20%
30%
Cost per pound
$0.25
$0.30
$0.32
$0.15
 The
order must contain at least 20% corn, 15%
grain, and 15% minerals.
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-26
Defining the Decision Variables
X1 = pounds of feed 1 to use in the mix
X2 = pounds of feed 2 to use in the mix
X3 = pounds of feed 3 to use in the mix
X4 = pounds of feed 4 to use in the mix
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-27
Defining the Objective Function
Minimize the total cost of filling the order.
MIN: 0.25X1 + 0.30X2 + 0.32X3 + 0.15X4
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-28
Defining the Constraints
 Produce
8,000 pounds of feed
X1 + X2 + X3 + X4 = 8,000
 Mix
consists of at least 20% corn
(0.3X1 + 0.5X2 + 0.2X3 + 0.1X4)/8000 >= 0.2
 Mix
consists of at least 15% grain
(0.1X1 + 0.3X2 + 0.15X3 + 0.1X4)/8000 >= 0.15
 Mix
consists of at least 15% minerals
(0.2X1 + 0.2X2 + 0.2X3 + 0.3X4)/8000 >= 0.15
 Nonnegativity
conditions
X1, X2, X3, X4 >= 0
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-29
A Comment About Scaling
Notice that the coefficient for X2 in the ‘corn’
constraint is 0.05/8000 = 0.00000625
 As Solver solves our problem, intermediate
calculations must be done that make coefficients
large or smaller.
 Storage problems may force the computer to use
approximations of the actual numbers.
 Such ‘scaling’ problems sometimes prevents
Solver from being able to solve the problem
accurately.
 Most problems can be formulated in a way to
minimize scaling errors...

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-30
Re-Defining the Decision
Variables
X1 = thousands of pounds of feed 1 to use in the mix
X2 = thousands of pounds of feed 2 to use in the mix
X3 = thousands of pounds of feed 3 to use in the mix
X4 = thousands of pounds of feed 4 to use in the mix
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-31
Re-Defining the Objective
Function
Minimize the total cost of filling the order.
MIN: 250X1 + 300X2 + 320X3 + 150X4
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-32
Re-Defining the Constraints
 Produce
8,000 pounds of feed
X1 + X2 + X3 + X4 = 8
 Mix
consists of at least 20% corn
(0.3X1 + 0.5X2 + 0.2X3 + 0.1X4)/8 >= 0.2
 Mix
consists of at least 15% grain
(0.1X1 + 0.3X2 + 0.15X3 + 0.1X4)/8 >= 0.15
 Mix
consists of at least 15% minerals
(0.2X1 + 0.2X2 + 0.2X3 + 0.3X4)/8 >= 0.15
 Nonnegativity
conditions
X1, X2, X3, X4 >= 0
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-33
A Comment About Scaling
 Earlier
the largest coefficient in the
constraints was 8,000 and the smallest
is 0.05/8 = 0.00000625.
 Now the largest coefficient in the
constraints is 8 and the smallest is
0.05/8 = 0.00625.
 The problem is now more evenly scaled.
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-34
The Assume Linear Model Option
The
Solver Options dialog box has an option
labeled “Assume Linear Model”.
When you select this option Solver performs some
tests to verify that your model is in fact linear.
These test are not 100% accurate & often fail as a
result of a poorly scaled model.
If Solver tells you a model isn’t linear when you
know it is, try solving it again. If that doesn’t work,
try re-scaling your model.
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-35
Implementing the Model
See file Fig3-33.xls
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-36
A Production Planning Problem:
The Upton Corporation

Upton is planning the production of their heavy-duty air
compressors for the next 6 months.
Month
1
2
3
4
5
6
Unit Production Cost
$240
$250
$265
$285
$280
$260
Units Demanded
1,000
4,500
6,000
5,500 3,500
4,000
Maximum Production 4,000
3,500
4,000
4,500 4,000
3,500
Minimum Production
1,750
2,000
2,250 2,000
1,750




2,000
Beginning inventory = 2,750 units
Safety stock = 1,500 units
Unit carrying cost = 1.5% of unit production cost
Maximum warehouse capacity = 6,000 units
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-37
Defining the Decision Variables
Pi = number of units to produce in month i, i=1 to 6
Bi = beginning inventory month i, i=1 to 6
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-38
Defining the Objective Function
Minimize the total cost production & inventory costs.
MIN: 240P1+ 250P2 + 265P3 + 285P4 + 280P5 + 260P6 +
3.6(B1+B2)/2 + 3.75(B2+B3)/2 + 3.98(B3+B4)/2 +
4.28(B4+B5)/2 + 4.20(B5+ B6)/2 + 3.9(B6+B7)/2
Note: The beginning inventory in any month is the same as the
ending inventory in the previous month.
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-39
Defining the Constraints
 Production
levels
2,000 <= P1 <= 4,000 } month 1
1,750 <= P2 <= 3,500 } month 2
2,000 <= P3 <= 4,000 } month 3
2,250 <= P4 <= 4,500 } month 4
2,000 <= P5 <= 4,000 } month 5
1,750 <= P6 <= 3,500 } month 6
 Ending
Inventory (EI = BI + P - D)
1,500 <=
1,500 <=
1,500 <=
1,500 <=
1,500 <=
1,500 <=
B1 + P1 - 1,000 <= 6,000
B2 + P2 - 4,500 <= 6,000
B3 + P3 - 6,000 <= 6,000
B4 + P4 - 5,500 <= 6,000
B5 + P5 - 3,500 <= 6,000
B6 + P6 - 4,000 <= 6,000
} month 1
} month 2
} month 3
} month 4
} month 5
} month 6
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-40
Defining the Constraints
 Beginning
(cont’d)
Balances
B1 = 2750
B2 = B1 + P1 - 1,000
B3 = B2 + P2 - 4,500
B4 = B3 + P3 - 6,000
B5 = B4 + P4 - 5,500
B6 = B5 + P5 - 3,500
B7 = B6 + P6 - 4,000
Notice that the Bi can be computed directly from the Pi.
Therefore, only the Pi need to be identified as changing cells.
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-41
Implementing the Model
See file Fig3-31.xls
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-42
A Multi-Period Cash Flow Problem:
The Taco-Viva Sinking Fund - I



Taco-Viva needs to establish a sinking fund to pay $800,000
in building costs for a new restaurant in the next 6 months.
Payments of $250,000 are due at the end of months 2 and
4, and a final payment of $300,000 is due at the end of
month 6.
The following investments may be used.
Investment
A
B
C
D
Available in Month Months to Maturity Yield at Maturity
1, 2, 3, 4, 5, 6
1
1.8%
1, 3, 5
2
3.5%
1, 4
3
5.8%
1
6
11.0%
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-43
Summary of Possible Cash Flows
Cash Inflow/Outflow at the Beginning of Month
Investment
1
2
3
4
5
6
7
A1
-1
1.018
B1
-1 <_____> 1.035
C1
-1 <_____> <_____> 1.058
D1
-1 <_____> <_____> <_____> <_____> <_____> 1.11
A2
-1
1.018
A3
-1
1.018
B3
-1 <_____> 1.035
A4
-1
1.018
C4
-1 <_____> <_____> 1.058
A5
-1
1.018
B5
-1 <_____> 1.035
A6
-1
1.018
Req’d Payments $0
$0
$250
$0
$250
$0
$300
(in $1,000s)
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-44
Defining the Decision Variables
Ai = amount (in $1,000s) placed in investment A at the
beginning of month i=1, 2, 3, 4, 5, 6
Bi = amount (in $1,000s) placed in investment B at the
beginning of month i=1, 3, 5
Ci = amount (in $1,000s) placed in investment C at the
beginning of month i=1, 4
Di = amount (in $1,000s) placed in investment D at the
beginning of month i=1
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-45
Defining the Objective Function
Minimize the total cash invested in month 1.
MIN: A1
+ B1 + C1 + D1
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-46
Defining the Constraints
 Cash
Flow Constraints
1.018A1 – 1A2 = 0
1.035B1 + 1.018A2 – 1A3 – 1B3 = 250
1.058C1 + 1.018A3 – 1A4 – 1C4 = 0
1.035B3 + 1.018A4 – 1A5 – 1B5 = 250
1.018A5 –1A6 = 0
1.11D1 + 1.058C4 + 1.035B5 + 1.018A6 = 300
 Nonnegativity
} month 2
} month 3
} month 4
} month 5
} month 6
} month 7
Conditions
Ai, Bi, Ci, Di >= 0, for all i
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-47
Implementing the Model
See file Fig3-35.xls
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-48
Risk Management:
The Taco-Viva Sinking Fund - II

Assume the CFO has assigned the following risk ratings to
each investment on a scale from 1 to 10 (10 = max risk)
Investment
A
B
C
D

Risk Rating
1
3
8
6
The CFO wants the weighted average risk to not exceed 5.
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-49
Defining the Constraints
 Risk
Constraints
1A1 + 3B1 + 8C1 + 6D1
A1 + B1 + C1 + D1
1A2 + 3B1 + 8C1 + 6D1
A2 + B1 + C1 + D1
1A3 + 3B3 + 8C1 + 6D1
A3 + B3 + C1 + D1
1A4 + 3B3 + 8C4 + 6D1
A4 + B3 + C4 + D1
1A5 + 3B5 + 8C4 + 6D1
A5 + B5 + C4 + D1
1A6 + 3B5 + 8C4 + 6D1
A6 + B5 + C4 + D1
<= 5
} month 1
<= 5
} month 2
<= 5
} month 3
<= 5
} month 4
<= 5
} month 5
<= 5
} month 6
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-50
An Alternate Version of the Risk Constraints

Equivalent Risk Constraints
-4A1 - 2B1 + 3C1 + 1D1 <= 0
} month 1
– 2B1 + 3C1 + 1D1 – 4A2 <= 0
} month 2
3C1 + 1D1 – 4A3 – 2B3 <= 0
} month 3
1D1 – 2B3 – 4A4 + 3C4 <= 0
} month 4
1D1 + 3C4 – 4A5 – 2B5 <= 0
} month 5
1D1 + 3C4 – 2B5 – 4A6 <= 0
} month 6
Note that each coefficient is equal to the risk factor for the investment minus 5
(the maximum allowable weighted average risk).
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-51
Implementing the Model
See file Fig3-38.xls
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-52
Data Envelopment Analysis (DEA):
Steak & Burger




Steak & Burger needs to evaluate the performance
(efficiency) of 12 units.
Outputs for each unit (Oij) include measures of: Profit,
Customer Satisfaction, and Cleanliness
Inputs for each unit (Iij) include: Labor Hours, and Operating
Costs
The “Efficiency” of unit i is defined as follows:
nO
Weighted sum of unit i’s outputs
Weighted sum of unit i’s inputs
nO
 Oij w j
j 1
nI
I v
=
j 1
ij j
 Oij w j
j 1
nI
 I ij v j
j 1
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
2-53
Defining the Decision Variables
wj = weight assigned to output j
vj = weight assigned to input j
A separate LP is solved for each unit, allowing each
unit to select the best possible weights for itself.
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
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Defining the Objective Function
Maximize the weighted output for unit i :
nO
MAX:  Oij w j
nO
 Oij w j
j 1
j 1
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
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Defining the Constraints
 Efficiency
cannot exceed 100% for any unit
nO
nI
j 1
j 1
 Okj w j   I kj v j , k  1 to the number of units
 Sum
of weighted inputs for unit i must equal 1
nO n I
  I ij v j  1
nI
j 1 j 1
 I ij v j  1
j 1
 Nonnegativity
Conditions
wj, vj >= 0, for all j
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
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Important Point
When using DEA, output variables should be
expressed on a scale where “more is better”
and input variables should be expressed on a
scale where “less is better”.
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
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Implementing the Model
See file Fig3-41.xls
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
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End of Chapter 3
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning.
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