Mathematical Modeling Tran, Van Hoai Faculty of Computer Science & Engineering HCMC University of Technology Tran Van Hoai

What is it ? MAXIMIZE50D+ 30C+ 6M SUBJECT TO 7D+ 3C+ 1.5M≤ 2000 D≥ 100 C≤ 500 D,C,M≥ 0 D,C integers Tran Van Hoai2 Mathematical Modeling = process to translate observed or desired phenomena into mathematical expressions (Total profit) (Raw steel) (Contract) (Cushions) (Nonnegativity) (Discrete)

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 Tran Van Hoai3

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 Tran Van Hoai4

Modeling variable constraints Limited number of cushions (lót nệm) C ≤ 500 Contract commitments C ≥ 100 Trivial constraints D, C, M ≥ 0 D, C integers Tran Van Hoai5

Solving the model is quite simple Tran Van Hoai6 MAXIMIZE50D+ 30C+ 6M SUBJECT TO 7D+ 3C+ 1.5M≤ 2000 D≥ 100 C≤ 500 D,C,M≥ 0 D,C integers MAXIMIZE50D+ 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) D = 100 (desks) C = 433 (chairs) M = 2/3 (pound)

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 Tran Van Hoai7

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 Tran Van Hoai8 Good principle wrong answer to right question is not fatal Right question to wrong answer is disastrous (thảm khốc) Good principle wrong answer to right question is not fatal Right question to wrong answer is disastrous (thảm khốc)

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) Tran Van Hoai9

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 – Making company be sure of its objective 5.Identify constraints 6.Seek continuous feedback Tran Van Hoai10 Relate closely to models

Delta Hardware Store Problem statement Tran Van Hoai11 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 To find least cost distribution scheme (from its plant, shipments from subcontractor) To meet demands its warehouses

MS process – step 2: Building mathematical model “Put scattered thoughts, ideas, conflicting objectives/constraints into logical coherent decision framework” “Mathematical modeling is an art” Tran Van Hoai12

Suggested approach 1.Identify decision variables 2.Quantify the objectives/constraints 3.Construct a model shell 4.Gather data – Consider time/cost issues Tran Van Hoai13

Decision variables & decision makers “Controllable” or “uncontrollable” depend on who has control Tran Van Hoai14 PRODUCTION PROCESS Inputs Manager Owner $

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 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 Tran Van Hoai15 Controllable input = decision variable Uncontrollable input = parameter Hardest part to build mathematical model

Delta Hardware Store Variable definition X1X1 Amount of paint shipped from Phoenix to San Jose X2X2 Amount of paint shipped from Phoenix to Fresno X3X3 Amount of paint shipped from Phoenix to Azusa X4X4 Amount of paint subcontracted for San Jose X5X5 Amount of paint subcontracted for Fresno X6X6 Amount of paint subcontracted for Azusa Tran Van Hoai16 Decision maker has no control over demand, production capacities, unit costs

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 Tran Van Hoai17 Total profit = Total revenues – Total cost

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) Tran Van Hoai18

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 Tran Van Hoai19

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 Tran Van Hoai20 Delta Hardware Store Informal model

Objective function MManufacturing cost at Phoenix plant T 1, T 2 T 3 Shipping cost from Phoenix to San Jose, Fresno, Azusa CFixed cost per 1000 gallons from subcontractor S 1, S 2 S 3 Shipping charge by subcontractor to San Jose, Fresno, Azusa Tran Van Hoai21 MINIMIZE(M+T 1 )X 1 + (M+T 2 )X 2 + (M+T 3 )X 3 + (C+S 1 )X 4 + (C+S 2 )X 5 + (C+S 3 )X 6

Constraints (1) Tran Van Hoai22 Q1Q1 Capacity of the Phoenix plant Q2Q2 Maximum number of gallons available from subcontractor R 1 R 2 R 3 Respective orders at warehouses in San Jose, Fresno, Azusa 1. Number of truckloads shipped out from Phoenix cannot exceed plant capacity X 1 + X 2 + X 3 ≤ Q 1 2. Number of gallons ordered from subcontractor cannot exceed order limit X 4 + X 5 + X 6 ≤ Q 2 1. Number of truckloads shipped out from Phoenix cannot exceed plant capacity X 1 + X 2 + X 3 ≤ Q 1 2. Number of gallons ordered from subcontractor cannot exceed order limit X 4 + X 5 + X 6 ≤ Q 2

Constraints (2) Tran Van Hoai23 3. Number of gallons received at each warehouse equals to its total order X 1 + X 4 = R 1 X 2 + X 5 = R 2 X 3 + X 6 = R 3 4. All shipments are nonnegative and integers X 1, X 2, X 3, X 4, X 5, X 6 ≥ 0 X 1, X 2, X 3, X 4, X 5, X 6 integer 3. Number of gallons received at each warehouse equals to its total order X 1 + X 4 = R 1 X 2 + X 5 = R 2 X 3 + X 6 = R 3 4. All shipments are nonnegative and integers X 1, X 2, X 3, X 4, X 5, X 6 ≥ 0 X 1, X 2, X 3, X 4, X 5, X 6 integer Need gathering (or approximating) data for parameters

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” RULE OF THUMB “Pareto principle” or “80/20 rule” Data gathering- time/cost issues Tran Van Hoai24

Simplify the problem – Transportation problem with only cost for manufacturing, ordering, transportation – Partial truckload, wholesale pricing, time- dependent cost,…are ignored Tran Van Hoai25 Delta Hardware Store Data gathering R1R1 4S1S1 $1200 R2R2 2S2S2 $1400 R3R3 5S3S3 $1100 Q2Q2 5C$5000

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) Tran Van Hoai26 Q 1 = AVG(production) past months = 7.9 (~8)

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) $ Tran Van Hoai27 M = $3.00 * 1000 = $3000 Q 1 = $100 + $150 + $800 = $1050 Q 2 = $100 + $100 + $555 = $750 Q 3 = $100 + $120 + $430 = $650 M = $3.00 * 1000 = $3000 Q 1 = $100 + $150 + $800 = $1050 Q 2 = $100 + $100 + $555 = $750 Q 3 = $100 + $120 + $430 = $650

Final model Minimize 4050X X X X X X 6 S.t. X 1 + X 2 + X 3 ≤ 8 X 4 + X 5 + X 6 ≤ 5 X 1 + X 4 = 4 X 2 + X 5 = 2 X 3 + X 6 = 4 X i ≥ 0, integer  i=1,…, Tran Van Hoai28

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 Tran Van Hoai29 Cost/time must be considered Large classes of problems have efficient solution techniques Cost/time must be considered Large classes of problems have efficient solution techniques

How to choose solution techniques? Can apply observation of experts Tran Van Hoai30 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 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

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 – Do the results make sense ? Intuitive ? – Can solution be integrated in current conditions ? Changes needed ? – Does solution modify plans of the organization ? Tran Van Hoai31 Testing/Validating is time-consuming process Historical/Simulated (hypothetical) data can be used Testing/Validating is time-consuming process Historical/Simulated (hypothetical) data can be used

Iterative development If one team not successful, other team comes with fresh mind Tran Van Hoai32 MODEL – SOLVE – VERIFY ManagerAnalysist

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 Tran Van Hoai33

Report Adjustable Cells FinalReducedObjectiveAllowable CellNameValueCost Coefficie ntIncreaseDecrease $B$13 PHOENIX PLANT SAN JOSE $C$13 PHOENIX PLANT FRESNO E+30 $D$13 PHOENIX PLANT AZUSA E+30 $B$14 SUBCONTRACTOR SAN JOSE $C$14 SUBCONTRACTOR FRESNO E $D$14 SUBCONTRACTOR AZUSA E Tran Van Hoai34

MS process – step 4: Communicating/Implementing results Prepare a business report/presentation Monitor the progress of the implementation Tran Van Hoai35 HOMEWORK Read textbook Writing business report/memos Using speadsheets in management science models Using Excel Solver to find an optimal solution and analyze results HOMEWORK Read textbook Writing business report/memos Using speadsheets in management science models Using Excel Solver to find an optimal solution and analyze results

Next Linear Programming Models Integer Linear Programming Models Tran Van Hoai36