Lecture1 - revised - PR601 Operations Research Decision...

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PR601: Operations Research & Decision Sciences Chanaka Senanayake, PhD(NUS, Singapore) Department of Production Engineering Faculty of Engineering University of Peradeniya email: [email protected] 1
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Course Overview Content-Part I 1 Linear Programming 2 Mixed Integer Programming 3 Decision Theory 4 Simulation for Decision Making (Monte Carlo Simulation) Schedule/Assessments Lectures (Part I) on 26 th April, 10 th , 17 th , 24 th , 31 st of May. No lecture on 3 rd May. Three assignments - end of 2,3,4 lectures Mid-Semester (20% of Grade) - Dates TBA 2
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Course Overview Lectures-Part I Session 1: 9.00am-10.30am, 15 mt break Session 2: 10.45am-11.30am, 15 mt break Session 3: 11.45am-12.30pm Lectures, Discussions, Examples, Take-home practice problems Presentation slides - uploaded 2 days in advance. 3
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Course Overview Textbooks/References Any book on Operations Research - including Integer Programming LP: Factory Physics - Hopp and Spearman LP: Operations Management - Russell and Taylor 4
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Introduction Linear Programming (LP) is a powerful mathematical tool used for solving constrained optimization problems. was first used to find optimal schedules or ’programs’ of resource allocation. is used to determine an optimal level of operational activity in order to achieve an objective , subject to restrictions called constraints . 5
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Lecture 01: Learning Outcomes At the end of today’s lecture, you should be able to formulate a simple optimization problem as a LP problem use the graphical solution method to solve LP problems use sensitivity analysis to a given LP problem 6
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Formulation The first step in using Linear programming is to formulate a practical problem in mathematical terms Components of a Linear Programming model Decision variables - quantities under our control (e.g. number of workers to hire, level of inventory to hold). Objective function - what we want to maximize or minimize specified in terms of the decision variables (e.g. maximize profit, minimize cost). Constraints - restrictions on our choices of the decision variables (e.g. due to capacity constraints, raw material limitations). 7
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Formulation Example #1 Suppose an electrical manufacturing firm wants to manufacture radios, toasters and clocks. Decision variables - The number of each item to produce, represented by x 1 , x 2 , x 3 (radio,toaster, clock, respectively). Now, suppose the profit from a radio is $6, the profit from a toaster is $4, and the profit from a clock is $2. Objective function - Maximize total profit (Z). i.e. Maximize Z = 6 x 1 + 4 x 2 + 2 x 3 If it requires 2 hours of labour to produce a radio, 1 hour to produce a toaster, and 1.5 hours to produce a clock and only 40 hours of labour are available. Constraint - For restriction on total labour hours: 2 x 1 + 1 x 2 + 1 . 5 x 3 40 8
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Formulation General structure of a Linear Programming model Maximize (or minimize) Z = c 1 x 1 + c 2 x 2 + ... + c n x n Subject to a 11 x 1 + a 12 x 2 + ... + a 1 n x n b 1 a 21 x 1 + a 22 x 2 + ... + a 2 n x n b 2 : a n 1 x 1 + a n 2 x 2 + ... + a nn x n b n x i 0 ( i = { 1 , 2 ,..., n } ) where x i = decision variables, b i = constraint levels, c j = objective function coefficients, a ij = constraint coefficients.
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