M670_11 (Linear Programming)

M670_11 (Linear Programming) - Linear Programming k MGMT...

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1 Linear Programming k MGMT 670: Business Analytics Krannert Graduate School of Management Purdue University
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2 Categories of Mathematical Models Descriptive : Understand the system Simulation, Regression Analysis, Time Series Analysis,… Prescriptive : Find the “optimal” solution Critical fractile, LP, Networks, IP, CPM, EOQ, NLP, … Model Quantitative Category Techniques
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3 Prescriptive Mathematical Models § INPUTS Relate decision variables (controllable inputs) with fixed or variable parameters (uncontrollable inputs) § MODEL Frequently seek to maximize or minimize some objective function subject to constraints § OUTPUTS Optimal solution is the value of the decision variables that results in the most desirable objective value
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4 Transforming Model Inputs into Output Uncontrollable Inputs (Environmental Factors) Controllable Inputs (Decision Variables) Output (Projected Results) Mathematical Model Unit price = $100, Unit cost = $30, Capacity = 50 Produce Profit = Revenue – Expenses Produce  50 Produce*=50 Profit* = 50×(100-30)=$3500 * denotes optimal solution and optimal objective value
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5 Steps in Formulating Linear Programming Models 1. Define decision variables 2. Express objective as a function of decision variables and known parameters 3. Formulate constraints to meet requirements and not exceed available capacity (using decision variables and known parameters)
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6 An Example LP Problem Blue Ridge Hot Tubs produces two types of hot tubs: A and B. Processing requirements and contribution margins are: There are 200 pumps , 1566 hours of labor , and 2880 feet of tubing available for production. What production plan will maximize profit ? Product A Product B Pumps 1 1 Labor 9 hrs 6 hrs Tubing 12 feet 16 feet Unit Profit $350 $300
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7 Formulating an LP Model 1. Define decision variables . 2. The objective Maximize Profit MAX
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8 Formulating an LP Model (continued) 3. Formulate constraints (Pump) pumps required pumps available (Labor) (Tubing)
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9 Feasibility versus Optimality § The constraints of an LP problem defines its feasible region . Any production plan for Product A and B that satisfies Pump, Labor and Tubing constraints is a feasible plan § The best point in the feasible region is the optimal solution to the problem. Among all feasible plans, the production plan that maximizes profit is the optimal plan.
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10 XB XA 250 200 150 100 50 0 0 50 100 150 200 250 tubing constraint 12XA + 16XB ≤ 2880 Feasible Region Constraints: Feasible Plans labor constraint 9XA + 6XB ≤ 1566 pump constraint XA + XB ≤ 200 (0, 200) (200, 0)
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11 Objective: Profit § Objective function: MAX 350XA + 300XB § Some plans yield the same profit: (6,0) or (0,7) each yield a profit of 2100 § Plans that yield the same profit P .
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This note was uploaded on 10/17/2011 for the course MGMT 670 taught by Professor Tawarmalani during the Spring '11 term at Purdue University.

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M670_11 (Linear Programming) - Linear Programming k MGMT...

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