Mathematical Modeling
Tran, Van Hoai
Faculty of Computer Science & Engineering
HCMC University of Technology
2012-2013
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What is it ?
MAXIMIZE 50D + 30C + 6M
SUBJECT TO 7D + 3C + 1.5M ≤ 2000
D
≥ 100
C
≤ 500
D, C,
M
≥0
D, C
integers
(Total profit)
(Raw steel)
(Contract)
(Cushions)
(Nonnegativity)
(Discrete)
Mathematical Modeling = process to translate
observed or desired phenomena into mathematical
expressions
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Modeling profit
• NetOffice: a company to produce
– Desk (D = number of desks)
– Chair (C = number of chairs)
– Molded steel (M = pounds of molded steel)
• Profit (net)
– $50/a desk
– $30/a chair
– $6/a pound molded steel
50D + 30C + 6M
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Modeling functional constraints
• Raw steel
– 7 pounds for a desk
– 3 pounds for a chair
– 1.5 pounds for a pound of molded steel
7D + 3C + 1.5M
– Functional constraint
7D + 3C + 1.5M ≤ 2000
NewOffice only has 2000 pounds of raw steel
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Modeling variable constraints
• Limited number of cushions (lót nệm)
C ≤ 500
• Contract commitments
C ≥ 100
• Trivial constraints
D, C, M ≥ 0
D, C integers
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Solving the model is quite simple
MAXIMIZE 50D + 30C + 6M
SUBJECT TO 7D + 3C + 1.5M ≤ 2000
D
≥ 100
C
≤ 500
D,
C,
M
≥0
D,
C
integers
Spreadsheet, WinQSB, Gurubi, COIN, ILOG,…
D = 100 (desks)
C = 433 (chairs)
M = 2/3 (pound)
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Mathematical models
• Optimization model is to maximize/minimize
a quantity that maybe restricted by a set of
constraints
• Prediction model is to describe/predict events
given a certain conditions
• Deterministic model is in which profit,
cost,…assumed to be known with certainty
• Stochastic model is in which (at least) one
values of parameters determined by
probability distributions
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MS process – step 1:
Defining the problem
• General situation to apply MS/OR
– Designing/implementing new operations
– Evaluating ongoing set of operations
– Determining/recommending corrective action for
operations which producing unsatisfactory results
Good principle
wrong answer to right question is not fatal
Right question to wrong answer is disastrous (thảm khốc)
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Factors to be faced
•
•
•
•
•
•
•
“Fuzzy” (incomplete, conflicting)
“Soft” constraints (goals or restrictions)
Different opinions (worker/manager/owner)
Limited budget for analyses
Limited time for analyses/recommendations
Political “turf wars”
No idea on what is wanted (ask consultant to
tell)
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Suggested approach
1. Observe operations
– Understanding at least as well as those
directly involved
2. Ease into complexity
3. Recognize political realities
4. Decide what is really wanted
Relate
closely
to
models
– Making company be sure of its objective
5. Identify constraints
6. Seek continuous feedback
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Delta Hardware Store
Problem statement
• 3 warehouses
• 1 production
plant
– Do not expand
To find least cost distribution scheme
production
(from its plant, shipments from subcontractor)
capacity
– Subcontract
To meet demands its warehouses
other
manufacturer
(label product
s by Delta)
Google.com
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MS process – step 2:
Building mathematical model
• “Put scattered thoughts, ideas, conflicting
objectives/constraints into logical coherent
decision framework”
• “Mathematical modeling is an art”
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Suggested approach
1.
2.
3.
4.
Identify decision variables
Quantify the objectives/constraints
Construct a model shell
Gather data – Consider time/cost issues
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Decision variables & decision makers
Inputs
Manager
PRODUCTION PROCESS
$
Owner
• “Controllable” or “uncontrollable” depend on
who has control
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Quick guide
• Controllable input = decision variable
• Uncontrollable input = parameter
Hardest part to build mathematical model
1. Ask “Does the decision maker have the
authority to decide the numerical value of
the item?”
– If answer = “yes”, it is decision variable
2. Be very precise in the units (& time frame) of
each decision variable
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Delta Hardware Store
Variable definition
Decision maker has no control over demand,
production capacities, unit costs
X1
X2
X3
X4
X5
X6
Amount of paint shipped from Phoenix to San Jose
Amount of paint shipped from Phoenix to Fresno
Amount of paint shipped from Phoenix to Azusa
Amount of paint subcontracted for San Jose
Amount of paint subcontracted for Fresno
Amount of paint subcontracted for Azusa
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Quantify objective/constraints
Total profit = Total revenues – Total cost
• Often, there is single objective function
≥2 objective functions → multicriteria decision
problem
• Constraints can be definitional in nature
– Artificial constraints can be added to strengthen
model
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Quick guide
• Create limiting condition in words as follows
(amount of resource required)
(Has some relation to)
(Availability of the resource)
• Translate to math expressions, using known,
parameters, and variables
• Move variables to left side, constants to right side
• Construct model shell
– Use generic symbols for parameters (until actual data
determined)
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Delta Hardware Store
Additional observation
• Additional information
– Finite production capacity at Phoenix plant
– Limited amount of paint available from
subcontractor
– Different requirements for 3 warehouses
– Orders in unit of 1000 gallons of paints (=a truck
delivery), cost = f( time, distance )
– Subcontractor charges fixed fee for a 1000-gallon
order, a delivery charge for each city
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Delta Hardware Store
Informal model
• Create a model in words
Minimize overall monthly cost (manufacturing,
transporting, subcontracting)
Subject to
1. Phoenix plant cannot operate beyond its capacity
2. Amount order to subcontractor is not over a
maximum limit
3. Orders at each warehouse will be fulfilled
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Objective function
M
T1, T2 T3
C
S1, S2 S3
Manufacturing cost at Phoenix plant
Shipping cost from Phoenix to San Jose, Fresno, Azusa
Fixed cost per 1000 gallons from subcontractor
Shipping charge by subcontractor to San Jose, Fresno,
Azusa
MINIMIZE
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(M+T1)X1+ (M+T2)X2+ (M+T3)X3+
(C+S1)X4+ (C+S2)X5+ (C+S3)X6
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Constraints (1)
Q1
Q2
R1 R 2 R3
Capacity of the Phoenix plant
Maximum number of gallons available from
subcontractor
Respective orders at warehouses in San Jose, Fresno,
Azusa
1. Number of truckloads shipped out from Phoenix
cannot exceed plant capacity
X1 + X2 + X3 ≤ Q1
2. Number of gallons ordered from subcontractor
cannot exceed order limit
X4 + X5 + X6 ≤ Q2
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Constraints (2)
3. Number of gallons received at each warehouse
equals to its total order
X1 + X4 = R1
X2 + X5 = R2
X3 + X6 = R3
4. All shipments are nonnegative and integers
X1, X2, X3, X4, X5, X6 ≥ 0
X1, X2, X3, X4, X5, X6 integer
Need gathering (or approximating)
data for parameters
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Data gathering- time/cost issues
• Time/cost of collecting, organizing, sorting
relevant
– “Hard” data >< “soft” data
– Harder the data,
moreOF
costly/time
RULE
THUMBconsuming to
obtaint
“Pareto principle” or “80/20 rule”
• Time/cost of generating solution approach
– Simplifying solution technique can lead to unrealistic
business client
settles
80% of optimal solution
• ATime/cost
of using
thefor
model
at 20%
cost torapidly
obtaintoitdynamic
– Management
mustofrespond
business → impact on model selected
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Delta Hardware Store
Data gathering
• Simplify the problem
– Transportation problem with only cost for
manufacturing, ordering, transportation
– Partial truckload, wholesale pricing, timedependent cost,…are ignored
R1
R2
R3
Q2
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4
2
5
5
S1
S2
S3
C
Tran Van Hoai
$1200
$1400
$1100
$5000
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Production limit
• No plant runs continuously at full capacity
– due to machine failure, partial staffing, limited
resource
• Two possibilities
Q1 = AVG(production)past months = 7.9 (~8)
– Theoretical production limit * reduction factor
– Ask plant manager “what is best estimations?”
– Make a forecast
• E.g., compute an average production (except outlier)
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Plant product/transportation costs
• Production cost
– Direct: $2.25
M = $3.00 * 1000 = $3000
– Indirect: $6000/8000
Q1 = $100 + $150 + $800 = $1050
• Transportation
cost
Q2 = $100 + $100 + $555 = $750
– LoadingQ(at=Phoenex):
$100
$100
+
$120
+ $430 = $650
3
– Unloading: (San Jose) $150, (Fresno) $100, (Azusa)
$120
– Mileage: (to San Jose) $800, (to Fresno) $550, (to
Azusa) $430
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Final model
Minimize 4050X1 + 3750X2 + 3650X3 + 6200X4 + 6400X5 + 6100X6
S.t.
X1 +
X2 +
X3
≤ 8
X4 +
X5 +
X6 ≤ 5
X1 +
X4
= 4
X2 +
X5
= 2
X3 +
X6 = 4
Xi ≥ 0, integer i=1,…,6
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MS process – step 3:
Solving mathematical model
•
•
•
•
•
Choose an appropriate solution techniques
Generate model solutions
Test/Validate model results
Return to modeling step if unacceptable results
Perform “what-if” analyses
Cost/time must be considered
Large classes of problems have efficient solution techniques
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How to choose solution techniques?
• Can apply observation of experts
Woolsey’s Laws
- Managers would rather live with a problem they can’t
solve than use a technique they don’t trust
- Managers don’t want the best solution, they simply
want a better one
- If the solution technique will cost you more than you
will save, don’t use it
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Test/Validate model results
• Due to simplification, optimal/heuristical,
simulated solutions
Good solutions are not for real-life situation
• We need test/validate to answer
Testing/Validating
time-consuming
– Do
the results makeissense
? Intuitive ? process
Historical/Simulated (hypothetical) data can be used
– Can solution be integrated in current conditions ?
Changes needed ?
– Does solution modify plans of the organization ?
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Iterative development
Analysist
Manager
MODEL – SOLVE – VERIFY
• If one team not successful, other team comes
with fresh mind
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What-if analyses
• Computer solution to a model is “an answer”
for the model
• Managers need anticipating more
– Management concerns
– Potential new opportunities
– Possible changes
What-if
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Report
Adjustable Cells
Final
Cell
$B$13
$C$13
$D$13
$B$14
$C$14
$D$14
Name
PHOENIX PLANT
SAN JOSE
PHOENIX PLANT
FRESNO
PHOENIX PLANT
AZUSA
SUBCONTRACTOR
SAN JOSE
SUBCONTRACTOR
FRESNO
SUBCONTRACTOR
AZUSA
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Value
Reduced Objective Allowable Allowable
Coefficie
Cost
nt
Increase Decrease
1
0
4050
2150
300
2
0
3750
500
1E+30
5
0
3650
300
1E+30
3
0
6200
300
2150
0
500
6400
1E+30
500
0
300
6100
1E+30
300
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MS process – step 4:
Communicating/Implementing results
• Prepare a business report/presentation
• Monitor the progress of the implementation
HOMEWORK
Read textbook
-1.5. Writing business report/memos
-1.6 . Using speadsheets in management science
models
-2.5. Using Excel Solver to find an optimal solution and
analyze results
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Next
• Linear Programming Models
• Integer Linear Programming Models
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