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Class_Notes_Part_1A

# Class_Notes_Part_1A - Class Notes for DMOR Set 1 These...

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Unformatted text preview: Class Notes for DMOR: Set 1 These class notes are intended for the students of the OEM program, class of 2009. It is best to use them in concert with the notes that you take in class, and the readings from our book. 1 Goal: Identify and solve optimization problems My brewing company can develop twenty types of beer, but I have limited amounts of hops, barley, wheat, malt, and labor available. How much of each should I produce? We have a set P of projects, and a set T of teams. Each project has a specific set of teams to which it can be assigned, because some teams cannot tackle certain projects. Also, each team can only be assigned to a limited number of projects. Unfortunately, there are too many projects and not enough teams, and thus we will not complete some projects. If we know how much money we can make by completing each project, how do we assign projects to teams in order to maximize the total money earned by completing projects? Imagine a communication network as a web of nodes and links connecting different users. We want to prevent people from attacking the network by placing sensors on its nodes. The sensor can see the node on which it is placed, and it can see all of its neighbors. How do we place the minimum number of sensors on the network? (And what if one of the sensors might be lying!?) Consider a city’s transportation infrastructure, and observe that as more people use a road, the average speed on the road decreases. We wish to recommend routes for people to drive in order to reduce congestion. Should we let them each choose their fastest way home, or should we insist on certain travel routes? 2 Optimization Problems Optimization problems have... 1. A set of decision variables , which represent the solution we’re trying to find. For instance, suppose we are optimizing investment decisions, and we wish to know how much money to invest in stocks A , B , and C . Then we might define decision variables x A , x B , and x C , whose values will reveal the correct amount of money to invest in stocks A , B , and C , respectively. 2. An objective function that we either try to minimize or maximize. For instance, we might want to maximize revenue, minimize patient surgery time, minimize travel dis- tance required in a logistics system, and so on. If we were playing a game, this would be the “score.” 3. A set of constraints that reflect limitations or requirements. In the game analogy, these are the “rules.” 3 What Does a Mathematical (Linear, Integer, Mixed-Integer, Nonlinear...) Program Look Like? Minimize 5 x 1 + 2 x 2- 3 x 3 subject to 2 x 1- x 2 ≤ 5 4 x 1 + 3 x 2 + x 3 ≥ 5 3 x 2- x 3 ≥ 4 x 1 ,x 2 ,x 3 ≥ 0 (means that all variables are nonnegative) Minimize c t x subject to Ax = b x ≥ Minimize x 1 x 2 + x 2 + 4 x 3 subject to x 1 + 3 x 2 + x 3 ≥ 6 x 1- x 2- 3 x 3 ≤ 3 x 1 binary ,x 2 ,x 3 ≥ BUT NOT LIKE THIS!...
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Class_Notes_Part_1A - Class Notes for DMOR Set 1 These...

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