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SJSU ISE 230 - Chapter 18

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Chapter 18 Deterministic Dynamic ProgrammingDescription18.1 Two Puzzles Example18.2 A Network ProblemCharacteristics of Dynamic Programming ApplicationsSlide 718.3 An Inventory ProblemSlide 9Slide 1018.4 Resource-Allocation ProblemsSlide 12Generalized Resource Allocation ProblemSlide 14A Turnpike Theorem18.5 Equipment Replacement Problems18.6 Formulating Dynamic Programming RecursionsSlide 18A Fishery ExampleSlide 20Slide 21Slide 22Slide 23Slide 24Slide 25Incorporating the Time Value of Money into Dynamic Programming FormulationsSlide 27Computational Difficulties in Using Dynamic Programming18.7 The Wagner-Whitin Algorithm and the Silver-Meal HeuristicSlide 30Slide 31Slide 32Slide 33The Silver-Meal Heuristic18.8 Using Excel to Solve Dynamic Programming ProblemsChapter 18Deterministic Dynamic Programmingto accompanyOperations Research: Applications and Algorithms 4th editionby Wayne L. WinstonCopyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.2 DescriptionDynamic programming is a technique that can be used to solve many optimization problems.In most applications, dynamic programming obtains solutions by working backward from the end of the problem toward the beginning, thus breaking up a large, unwieldy problem into a series of smaller, more tractable problems318.1 Two Puzzles Example We show how working backward can make a seemingly difficult problem almost trivial to solve.Suppose there are 20 matches on a table. I begin by picking up 1, 2, or 3 matches. Then my opponent must pick up 1, 2, or 3 matches. We continue in this fashion until the last match is picked up. The player who picks up the last match is the loser. How can I (the first player) be sure of winning the game?4 If I can ensure that it will be opponent’s turn when 1 match remains, I will certainly win.Working backward one step, if I can ensure that it will be my opponent's turn when 5 matches remain, I will win.If I can force my opponent to play when 5, 9, 13, 17, 21, 25, or 29 matches remain, I am sure of victory.Thus I cannot lose if I pick up 1 match on my first turn.518.2 A Network ProblemMany applications of dynamic programming reduce to finding the shortest (or longest) path that joins two points in a given network.For larger networks dynamic programming is much more efficient for determining a shortest path than the explicit enumeration of all paths.6Characteristics of Dynamic Programming ApplicationsCharacteristic 1The problem can be divided into stages with a decision required at each stage.Characteristic 2Each stage has a number of states associated with it.By a state, we mean the information that is needed at any stage to make an optimal decision.Characteristic 3The decision chosen at any stage describes how the state at the current stage is transformed into the state at the next stage.7 Characteristic 4Given the current state, the optimal decision for each of the remaining stages must not depend on previously reached states or previously chosen decisions.This idea is known as the principle of optimality.Characteristic 5If the states for the problem have been classified into on of T stages, there must be a recursion that related the cost or reward earned during stages t, t+1, …., T to the cost or reward earned from stages t+1, t+2, …. T.818.3 An Inventory ProblemDynamic programming can be used to solve an inventory problem with the following characteristics:1. Time is broken up into periods, the present period being period 1, the next period 2, and the final period T. At the beginning of period 1, the demand during each period is known.2. At the beginning of each period, the firm must determine how many units should be produced. Production capacity during each period is limited.9 3. Each period’s demand must be met on time from inventory or current production. During any period in which production takes place, a fixed cost of production as well as a variable per-unit cost is incurred.4. The firm has limited storage capacity. This is reflected by a limit on end-of-period inventory. A per-unit holding cost is incurred on each period’s ending inventory.5. The firms goal is to minimize the total cost of meeting on time the demands for periods 1,2, …., T.10 In this model, the firm’s inventory position is reviewed at the end of each period, and then the production decision is made.Such a model is called a periodic review model. This model is in contrast to the continuous review model in which the firm knows its inventory position at all times and may place an order or begin production at any time.1118.4 Resource-Allocation ProblemsResource-allocation problems, in which limited resources must be allocated among several activities, are often solved by dynamic programming.To use linear programming to do resource allocation, three assumptions must be made:Assumption 1 : The amount of a resource assigned to an activity may be any non negative number.Assumption 2 : The benefit obtained from each activity is proportional to the amount of the resource assigned to the activity.12 Assumption 3: The benefit obtained from more than one activity is the sum of the benefits obtained from the individual activities.Even if assumptions 1 and 2 do not hold, dynamic programming can be used to solve resource-allocation problems efficiently when assumption 3 is valid and when the amount of the resource allocated to each activity is a member of a finite set.13Generalized Resource Allocation ProblemThe problem of determining the allocation of resources that maximizes total benefit subject to the limited resource availability may be written aswhere xt must be a member of {0,1,2,…}.Tttttxr1)(maxTttwxtgt1)(s.t.14 To solve this by dynamic programming, define ft(d) to be the maximum benefit that can be obtained from activities t, t+1,…, T if d unites of the resource may be allocated to activities t, t+1,…, T.We may generalize the recursions to this situation by writingfT+1(d) = 0 for all dwhere xt must be a non-negative integer satisfying gt(xt)≤ d.)]}([)({max)(1ttttxtxgdfxrdftt15A Turnpike TheoremTurnpike results abound in the dynamic programming literature.Why are the results referred to as a turnpike theorem? Think about taking an automobile trip in which our goal is to minimize the time needed to complete the trip.For a long trip it may be


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SJSU ISE 230 - Chapter 18

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