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Mathematical Modeling

Tran, Van HoaiFaculty of Computer Science & Engineering

HCMC University of Technology

2012-2013

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What is it ?

MAXIMIZE 50D + 30C + 6MSUBJECT TO 7D + 3C + 1.5M ≤ 2000

D ≥ 100 C ≤ 500

D, C, M ≥ 0D, C integers

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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

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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

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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

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Solving the model is quite simple

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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

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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

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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)

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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

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Relate closely to models

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Delta Hardware StoreProblem statement

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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

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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. Identify decision variables2. Quantify the objectives/constraints3. Construct a model shell4. Gather data – Consider time/cost issues

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Decision variables & decision makers

• “Controllable” or “uncontrollable” depend on who has control

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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

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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

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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)

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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

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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

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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

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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)

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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)

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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

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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

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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)

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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

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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

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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

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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

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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 ?

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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

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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

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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

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MS process – step 4:Communicating/Implementing results

• Prepare a business report/presentation• Monitor the progress of the implementation

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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

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Next

• Linear Programming Models• Integer Linear Programming Models

2012-2013