lecture2_3

lecture2_3 - LINEAR PROGRAMMING 1.224J/ESD.204J...

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LINEAR PROGRAMMING 1.224J/ESD.204J TRANSPORTATION OPERATIONS, PLANNING AND CONTROL: CARRIER SYSTEMS Professor Cynthia Barnhart Professor Nigel H.M. Wilson Fall 2003
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Announcements –Reader – Problem set #1 –December 1 recitation & December 5 class –OPL Studio examples 12/31/2003 Barnhart 1.224J 2
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LINEAR PROGRAMMING Sources: -Introduction to linear optimization (Bertsimas, Tsitsiklis) -Nathaniel Grier’s paper -1.224 previous material
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Outline 1. Modeling problems as linear programs 2. Solving linear programs 12/31/2003 Barnhart 1.224J 4
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Outline 1. Modeling problems as linear programs What is a linear Program Formulation Set Notation Review Example: Transit Ridership Standard Form of an LP Linearity –E x a m p l e s 12/31/2003 Barnhart 1.224J 5
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What is a Linear Program (LP)? 1. Objective Function summarizes objective of the problem (MAX, MIN) 2. Constraints of problem: limitations placed on the problem; control allowable solutions Problem statement: ‘given….’, ‘must ensure…’, ‘subject to’ Equations or inequalities 3. Decision Variables quantities, decisions to be determined multiple types (real numbers, non-negative, integer, binary) In an LP, the decision variables are real numbers Choice of decision variables will determine difficulty in formulating and solving the problem 12/31/2003 Barnhart 1.224J 6
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Set Notation Review Set:collection of distinct objects R: set of real numbers Z: set of integers 0: empty set Superscript +: non-negative elements of a set : ‘is an element of’ {} : ‘the set containing’ (members of the set are between brackets) : or | : ‘such that ’ example: { x S : x 0 } :’there exists’ : ‘for all’ 12/31/2003 Barnhart 1.224J 7
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Example: Transit Ridership A transit agency is performing a review of the service it provides. It has decided to measure its overall effectiveness in terms of the total number of riders it serves. The agency operates a number of modes of transport. The table shows the average number of riders generated by each trip (by mode) and the cost of each trip (by mode) Mode Heavy Rail Light Rail BRT Bus Ave. Ridership per trip (r i ) 400 125 60 40 Ave. Cost per trip (c i ) 200 80 40 30 Give a formulation of the problem to maximize the total number of riders the agency services given a fixed daily budget of $5,000. 12/31/2003 Barnhart 1.224J 8
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Transit Ridership Formulation 1. Decision Variables? –X 1 = number of trips made using heavy rail 2 = number of trips made using light rail 3 = number of trips made using bus rapid transit (BRT) 4 = number of trips made using bus 2. Objective Function? MAX (Total Ridership) Ridership= 400*X 1 +125*X 2 +60*X 3 +40*X 4 3. Constraints? Cost budget Cost=200*X 1 +80*X 2 +40*X 3 +30*X 4 12/31/2003 Barnhart 1.224J 9
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Transit Ridership Model MAX ( X * 400 + X * 125 2 + X * 60 3 + X * 40 4 ) 1 s.t.
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lecture2_3 - LINEAR PROGRAMMING 1.224J/ESD.204J...

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