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Transcript of Mathematical Modeling Tran, Van Hoai Faculty of Computer Science & Engineering HCMC University of...
Tran Van Hoai 1
Mathematical Modeling
Tran, Van HoaiFaculty of Computer Science & Engineering
HCMC University of Technology
2012-2013
Tran Van Hoai 2
What is it ?
MAXIMIZE 50D + 30C + 6MSUBJECT TO 7D + 3C + 1.5M ≤ 2000
D ≥ 100 C ≤ 500
D, C, M ≥ 0D, C integers
2012-2013
Mathematical Modeling = process to translate observed or desired phenomena into mathematical
expressions
(Total profit)(Raw steel)(Contract)(Cushions)(Nonnegativity)(Discrete)
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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
2012-2013
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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 ≤ 2000NewOffice only has 2000 pounds of raw steel
2012-2013
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Modeling variable constraints
• Limited number of cushions (lót nệm)C ≤ 500
• Contract commitmentsC ≥ 100
• Trivial constraintsD, C, M ≥ 0
D, C integers
2012-2013
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Solving the model is quite simple
2012-2013
MAXIMIZE 50D + 30C + 6MSUBJECT TO 7D + 3C + 1.5M ≤ 2000
D ≥ 100 C ≤ 500
D, C, M ≥ 0D, 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
2012-2013 Tran Van Hoai
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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
2012-2013
Good principlewrong 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)
2012-2013
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Suggested approach
1. Observe operations– Understanding at least as well as those
directly involved
2. Ease into complexity3. Recognize political realities4. Decide what is really wanted
– Making company be sure of its objective
5. Identify constraints6. Seek continuous feedback
2012-2013 Tran Van Hoai
Relate closely to models
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Delta Hardware StoreProblem statement
2012-2013
Google.com
• 3 warehouses• 1 production
plant– Do not expand
production capacity
– Subcontract other manufacturer (label product s by Delta)
To find least cost distribution scheme (from its plant, shipments from subcontractor)
To meet demands its warehouses
12
MS process – step 2:Building mathematical model
• “Put scattered thoughts, ideas, conflicting objectives/constraints into logical coherent decision framework”
• “Mathematical modeling is an art”
2012-2013 Tran Van Hoai
13
Suggested approach
1. Identify decision variables2. Quantify the objectives/constraints3. Construct a model shell4. Gather data – Consider time/cost issues
2012-2013 Tran Van Hoai
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Decision variables & decision makers
• “Controllable” or “uncontrollable” depend on who has control
2012-2013
PRODUCTION PROCESS
Inputs Manager
Owner$
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Quick guide
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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• Controllable input = decision variable• Uncontrollable input = parameter
Hardest part to build mathematical model
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Delta Hardware StoreVariable definition
X1 Amount of paint shipped from Phoenix to San JoseX2 Amount of paint shipped from Phoenix to FresnoX3 Amount of paint shipped from Phoenix to AzusaX4 Amount of paint subcontracted for San JoseX5 Amount of paint subcontracted for FresnoX6 Amount of paint subcontracted for Azusa
2012-2013
Decision maker has no control over demand, production capacities, unit costs
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Quantify objective/constraints
• 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
2012-2013
Total profit = Total revenues – Total cost
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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)
2012-2013
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Delta Hardware StoreAdditional 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
2012-2013
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• Create a model in wordsMinimize overall monthly cost (manufacturing,
transporting, subcontracting)Subject to
1. Phoenix plant cannot operate beyond its capacity2. Amount order to subcontractor is not over a
maximum limit3. Orders at each warehouse will be fulfilled
2012-2013
Delta Hardware StoreInformal model
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Objective function
M Manufacturing cost at Phoenix plantT1, T2 T3 Shipping cost from Phoenix to San Jose, Fresno, AzusaC Fixed cost per 1000 gallons from subcontractorS1, S2 S3 Shipping charge by subcontractor to San Jose, Fresno,
Azusa
2012-2013
MINIMIZE (M+T1)X1+ (M+T2)X2+ (M+T3)X3+(C+S1)X4+ (C+S2)X5+ (C+S3)X6
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Constraints (1)
2012-2013
Q1 Capacity of the Phoenix plantQ2 Maximum number of gallons available from
subcontractorR1 R2 R3 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)
2012-2013
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 integersX1, X2, X3, X4, X5, X6 ≥ 0
X1, X2, X3, X4, X5, X6 integer
Need gathering (or approximating) data for parameters
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• Time/cost of collecting, organizing, sorting relevant– “Hard” data >< “soft” data– Harder the data, more costly/time consuming to obtaint
• Time/cost of generating solution approach– Simplifying solution technique can lead to unrealistic
• Time/cost of using the model– Management must respond rapidly to dynamic
business → impact on model selected A business client settles for 80% of optimal solution
at 20% of cost to obtain it
RULE OF THUMB“Pareto principle” or “80/20 rule”
Data gathering- time/cost issues
2012-2013 Tran Van Hoai
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• Simplify the problem– Transportation problem with only cost for
manufacturing, ordering, transportation– Partial truckload, wholesale pricing, time-
dependent cost,…are ignored
2012-2013
Delta Hardware StoreData gathering
R1 4 S1 $1200R2 2 S2 $1400R3 5 S3 $1100Q2 5 C $5000
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Production limit
• No plant runs continuously at full capacity – due to machine failure, partial staffing, limited
resource• Two possibilities
– Theoretical production limit * reduction factor– Ask plant manager “what is best estimations?”– Make a forecast
• E.g., compute an average production (except outlier)
2012-2013
Q1 = AVG(production)past months = 7.9 (~8)
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Plant product/transportation costs
• Production cost– Direct: $2.25– Indirect: $6000/8000
• Transportation cost– Loading (at Phoenex): $100– Unloading: (San Jose) $150, (Fresno) $100, (Azusa)
$120– Mileage: (to San Jose) $800, (to Fresno) $550, (to
Azusa) $430
2012-2013
M = $3.00 * 1000 = $3000Q1 = $100 + $150 + $800 = $1050Q2 = $100 + $100 + $555 = $750Q3 = $100 + $120 + $430 = $650
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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
2012-2013
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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
2012-2013
Cost/time must be consideredLarge classes of problems have efficient solution techniques
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How to choose solution techniques?
• Can apply observation of experts
2012-2013
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 solutionsGood solutions are not for real-life situation
• We need test/validate to answer– Do the results make sense ? Intuitive ?– Can solution be integrated in current conditions ?
Changes needed ?– Does solution modify plans of the organization ?
2012-2013
Testing/Validating is time-consuming processHistorical/Simulated (hypothetical) data can be used
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Iterative development
• If one team not successful, other team comes with fresh mind
2012-2013
MODEL – SOLVE – VERIFY
ManagerAnalysist
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What-if
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
2012-2013
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ReportAdjustable Cells
Final Reduced Objective Allowable Allowable
Cell Name Value CostCoefficie
nt Increase Decrease
$B$13PHOENIX PLANT SAN JOSE 1 0 4050 2150 300
$C$13PHOENIX PLANT FRESNO 2 0 3750 500 1E+30
$D$13PHOENIX PLANT AZUSA 5 0 3650 300 1E+30
$B$14SUBCONTRACTOR SAN JOSE 3 0 6200 300 2150
$C$14SUBCONTRACTOR FRESNO 0 500 6400 1E+30 500
$D$14SUBCONTRACTOR AZUSA 0 300 6100 1E+30 300
2012-2013
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MS process – step 4:Communicating/Implementing results
• Prepare a business report/presentation• Monitor the progress of the implementation
2012-2013
HOMEWORKRead 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