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Introduction Part 2 Introduction Chapter 1

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Case of United Airlines Kinds of problems : Develop master flight schedule Forecast demand for its routes Determine wich planes to purchase or to lease Assign planes and crews to the route Set fares

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Case of United Airlines (2) Kinds of problems : Accept reservations in various fare categories Purchase fuel Schedule airport ticket agents and service personnel Schedule maintenance crews

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Case of United Airlines (3) Kinds of problems : Maintain service facilities Lease airport gates Design and monitor its frequent flyer program

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Case of United Airlines (4) Factors impacting these decisions : Budget, equipment and personnel restrictions Union agreements FAA maintenance guidelines Safe distance and aircraft turnaround requirements Flexibility to react to complications due to weather, congestion, defects,….

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Mathematical Modeling –A process that translates observed phenomena into mathematical expressions. –Classification of constraints Nonnegativity constraintX>=0. Lower bound cconstraintX>=L(>=0) Upper bound constraintX<=U Integer constraintX=integer Binary ConstraintX=0 or 1. 1.3 Mathematical Modeling and the Management Science Process

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Mathematical Modeling -Classification of models –Optimization Models - find the best solution under a set of restrictions and scarce resources. –Prediction models - usually provide important input to optimization models. –Classification by type of data available: Deterministic models, Stochastic models. 1.3 Mathematical Modeling and the Management Science Process

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The Management Science Process –By and large, the Management Science process can be described by the following four step procedure. Problem Definition Mathematical Modeling Solution of the model Communication -Marketing of the results

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The Team Concept –For large scale projects an inter--disciplinarian team work is a necessity. Basic Steps of the Management Science Process –Defining the Problem. –Building a Mathematical Model. –Solving a Mathematical Model.

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2.2 Defining the Problem Management Science is Applied When - – Designing and implementing new operations – Evaluating ongoing Operations and Procedures. – Determining and recommending corrective actions. How do we get started ? Can we do better ? Help !!!

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2.2 Defining the Problem “Problems” in Applying Management Science G“Fuzzy”, incomplete, conflicting data. G Conflicting goals. G Differing opinions among and between workers and management. G Limited budgets for analyse G Narrow time frame. G Political “turf war”. GNo exact idea of what is wanted

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How to Start and How to Proceed – Identify the problem. – Observe the problem from various points of view. – Keep things simple. – Identify constraints. –Recognize political realities. –Work with management, get feedback.

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Delta Hardware Stores : Problem Statement DHS has 3 warehouses in California in San Jose, Fresno and Azusa. Each month DHS restocks his warehouses with its own brand of paint. DHS has his own paint manufacturing in Phoenix.

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Delta Hardware Stores : Problem Statement The capacity of the manufacturing is sometimes insufficient to cover the demand. It’s not cost effective to expand the production capacity. DHS has decided to subcontract to produce paints under his label

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Delta Hardware Stores : Problem Statement The problem is : Determine a least cost distribution scheme of paint produced at its manufacturing plant and shipments from the subcontractor to meet the demand of the warehouses.

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Problem Definition Mathematical Modeling Solution of the model Communication -Marketing of the results

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2.3 Building a Mathematical Model Identify Decision Variables –Which factors are controllable? Quantify the Objective and Constraints – Formulate the function to be optimized (profit, cost). – Formulate the requirements and/or restrictions.

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Delta Hardware Stores : The decision variables The questions are : How much paint should be shipped this month from phoenix to the 3 stores? How much extra paint should be shipped from the subcontractor to the 3 stores?

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Delta Hardware Stores : The decision variables X1 : amount shipped from Phoenix to San Jose X2 : amount shipped from Phoenix to Fresno X3 : amount shipped from Phoenix to Azusa X4 : amount shipped from subcontractor to San Jose X5 : amount shipped from subcontractor to Fresno X6 : amount shipped from subcontractor to Azusa

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2.3 Building a Mathematical Model Construct a Model Shell – Help focus on the exact data required. Gather Data – Consider time / cost issues for collecting, organizing and sorting relevant data generating a solution approach using the model

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Delta Hardware Stores : The model shell The production capacity in Phoenix is finite. The requirements of the 3 stores are different. The delivery trucks carry 1000 gallons at a time. The cost of shipping is function of the distance. The cost of the subcontractor is higher than the cost in DHS plant.

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Delta Hardware Stores : The model shell To obtain the best prices, orders must be in 1000-gallon increments. The subcontractor charges : –a fixed fee for each 1000 gallons ordered, –a delivery charge to each of the 3 warehouses.

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Delta Hardware Stores : The model shell The objective is to minimize the total monthly costs of manufacturing, transporting and subcontracting subject to : –the Phoenix plant cannot produce more than his capacity –the amount ordered from the subcontractant cannot exceed a maximum limit –the orders of the warehouses are to be fulfilled

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If : M is the manufacturing cost for 1000 gallons in Phoenix T1, T2 ant T3 are the truckloads shipping cost from Phoenix to the 3 stores C is the fixed purchase cost per 1000 gallons from the subcontractor S1, S2 and S3 are the shipping charges per 1000 gallons from the subcontractor to the 3 stores Then we have : Min (M+T1)X1 + (M+T2)X2 + (M+T3)X3 + (C+S1)X4 + (C+S2)X5 + (C+S3)X6 Delta Hardware Stores : The model shell

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The constraints are : number of truckloads from Phoenix cannot exceed the capacity : X1 + X2 + X3 Q1 number of 1000-gallons ordered from the subcontractor cannot exceed a limit : X4 + X5 + X6 Q2 number of 1000-gallons received in the stores equals their orders : X1 + X4 = R1 X2 + X5 = R2 X3 + X6 = R6 all shipments must be nonnegative and integer Delta Hardware Stores : The model shell

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After quantification of the parameters, we have : Minimize 4050 X1 + 3750 X2 + 3650 X3 + 6200X4 + 6400X5 + 6100 X6 Subject to : X1 + X2 + X3 8 X4 + X5 + X6 5 X1 + X4 = 4 X2 + X5 = 2 X3 + X6 = 5 Xi 0 and integer Delta Hardware Stores : The model shell

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Problem Definition Mathematical Modeling Solution of the model Communication -Marketing of the results

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2.4 Solving a Mathematical Model Choose an Appropriate Solution Technique –An optimization algorithm. –A heuristic algorithm. 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 solutions; they simply want a better one….. –If the solution technique costs more than you will save, don’t use it…..

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2.4 Solving a Mathematical Model Generate Model Solutions Test / Validate Model Results –Is the solution reasonable? –Are radical changes needed? –Does it fit present and future plans? –Unacceptable results? Return to modeling. Perform “What--If” Analyses

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Delta Hardware Stores : Model Solution

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Problem Definition Mathematical Modeling Solution of the model Communication - Marketing of the results

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2.5 Communicating the results Prepare a business report or presentation. –Be concise –Use common everyday language –Make liberal use of graphics Monitor the progress of the implementation

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2.5 Communicating the results Introduction - Problem statement Assumptions/approximations made Solution approach - computer program used Results - presentation/analysis What-if analyses Overall recommandation Appendices Parts of a Business Presentation

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The Management Science Process Problem Definition Mathematical Modeling Solution of the model Communication -Marketing of the results

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Advantages of using the management science approach Helps the DM to focus on the true goals of the problem Helps deal with a problem in its entirety Helps sort out data that are relevant to the problem Describes the problemin concise mathematical relationships

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Advantages of using the management science approach Helps reveal cause-and-effect relations in the problem Can be used to solve complex problems with large amounts data Yields an optimal (or at least a good) solution

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Disadvantages of using the management science approach Uses idealized models that may be oversimplified May not be cost effective Can be misused by untrained personel Requires quantification of all model input

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Disadvantages of using the management science approach May create models requiring excessive computer ressources May create models that are to difficult to explain to users Can yield suspect or insatisfactory results due to a rapidly changing environment

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