Lecture 20110128

# Lecture 20110128 - IMSE2008 Operational Research Techniques...

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IMSE2008 Operational Research Techniques Lecture 01/28 LP Formulation eometry of Linear Programs Geometry of Linear Programs Miao Song Dept of Industrial & Manufacturing Systems Engineering

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eview of Lecture 01/21 Review of Lecture 01/21 What is Linear Programming? ormulation of LP Formulation of LP Decision variables Objective function Constraints A production problem scheduling problem (to be continued) A scheduling problem (to be continued) 1/28/2011 2
cheduling Postal Workers Scheduling Postal Workers Each postal worker works for 5 consecutive days, followed by 2 days off, repeated weekly. Day Mon Tue Wed Thu Fri Sat Sun Demand 17 13 15 19 14 16 11 Minimize the number of postal workers (for the time being, we will permit fractional workers on each day.) 1/28/2011 3

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P for the Scheduling Problem LP for the Scheduling Problem 2 34 56 7 min xx x x x x x  12 14 7 s.t. 17 x x x 7 2 36 7 13 15 x x x x x x 7 19 x x x 5 3 45 6 14 16 x x x x  23 7 11 x 1/28/2011 4 0 for 1 to 7 j x j
ome Modifications of the Model Some Modifications of the Model Suppose that we need to ensure that at least 30% of the workers have Sunday off. 1/28/2011 5

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genda Agenda Formulation of Linear Programming Piecewise Linear Objective Functions j Piecewise Linear Constraints Geometry of Linear Programming 1/28/2011 6
Motivating Example A Motivating Example Suppose that the desirable number of workers on day j is d j , but it is not required. et e the “excess” number of workers day Let s j be the excess number of workers day j . s j > 0 if there are more workers on day j than d ; otherwise s 0 . j j What is the minimum cost schedule, where the “cost” of having too many workers on day j is f j ( s j ) , which is a non-linear function? What are the new decision variables? What is the resulting nonlinear model? 1/28/2011 7

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n Nonlinear Functions On Nonlinear Functions Occasionally a nonlinear program can be transformed into a linear program Rare, but useful when it occurs In general, non-linear programming solvers can work well on a minimization problem when the objective function is onvex convex 1/28/2011 8
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## This note was uploaded on 12/09/2011 for the course IMSE 0301 taught by Professor Song during the Spring '11 term at HKU.

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Lecture 20110128 - IMSE2008 Operational Research Techniques...

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