Optimization Modeling

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24 Chapter 2. Formulating <strong>Optimization</strong> Models Figure 2.8: Unbounded objective, unbounded feasible region References Readers who want to know more about the theoretical aspects of linear programming, or the simplex method, can refer to [Ch83] or [Lu73], and there are many other excellent works on the subject. Figure 2.9: Bounded objective, unbounded feasible region 2.2 Formulating mixed integer programming models This section In this section some basic characteristics of mixed integer programming models and their solution will be discussed. Need for integer solutions ... Recall the example of the chips-producing company in the previous section where the optimal solution was (X p ,X m ) = (57.5, 50). If chips are packed in bags of 1 kilogram, this optimal solution cannot be translated into a practical
2.2. Formulating mixed integer programming models 25 solution. One approach is to round down the solution. In this example, rounding yields the feasible solution (X p ,X m ) = (57, 50), but the profit is reduced from $190 to $189. There are many real-world problems that require their solutions to have integer values. Examples are problems that involve equipment utilization, setup costs, batch sizes, and “yes-no” decisions. Fractional solutions of these problems do not make real-world sense; just imagine constructing half a building. ... illustrated Rounding is a straightforward way to obtain integer values. There might be doubt, however, whether such a solution is optimal or even feasible. In practice, rounding is a satisfactory approach when: Pros and cons of rounding ... the figures to be rounded are so large that the error is negligible, the input data are not known with certainty, so that a fractional solution is not that accurate anyway, the rounding procedure is certain to give a feasible solution, and algorithms developed especially to search for an integer solution are too expensive in terms of computer resources and time. The alternative to rounding has already been mentioned—making use of an integer programming algorithm. Next, the potato chips example will be used to present some aspects of integer programming. When an integer programming algorithm is used in the example above, the solution (X p ,X m ) = (49, 58) yields a higher profit, $189.5. ... an alternative Consider again the chips-producing company. Their special chips are becoming so popular that a large supermarket chain wants to sell these potato chips in their stores. The additional restriction is that the chips must be delivered in batches of 30 kg, for efficiency reasons. As a result, the optimal production plan, determined above, is no longer adequate. The optimal amounts of 57.5 kg of plain and 50 kg of Mexican chips would make up, respectively, 1.92 and 1.67 batches of potato chips. This solution does not satisfy the additional restriction imposed by the supermarket chain. An integer programming model could now be used to determine a new (and acceptable) production plan. Extending the potato chips example The linear programming model of Section 2 can be reformulated as an integer programming model by measuring the amounts of chips in batches, and by adding the requirement that the solution takes on integer values. When measuring the amounts of chips in batches of 30 kg, the production processes require the following amounts of time. Each batch of plain chips takes 60 minutes to slice, 120 minutes to fry, and 120 minutes to pack. Each batch of Mexican chips takes 120 minutes to slice, 150 minutes to fry, and 60 minutes to pack. Furthermore, the net profit on a batch of plain chips is $60, while the net profit on a batch of Mexican chips is $45. Description
Page 1: AIMMS Optimization Modeling AIMMS 3
Page 4 and 5: Copyright c○ 1993-2006 by Paragon
Page 6 and 7: vi About Aimms Stand-alone applicat
Page 8 and 9: viii Contents 3 Algebraic Represent
Page 10 and 11: x Contents Part IV Intermediate Opt
Page 12 and 13: xii Contents 22 A Facility Location
Page 14 and 15: xiv Preface Part IV—Data Managem
Page 16 and 17: xvi Preface What’s in the Optimiz
Page 18 and 19: xviii Preface tleneck capacity iden
Page 20 and 21: xx Preface
Page 23 and 24: Chapter 1 Background This chapter g
Page 25 and 26: 1.3. The role of mathematics 5 Mode
Page 27 and 28: 1.4. The modeling process 7 1.4 The
Page 29 and 30: 1.5. Application areas 9 Planning m
Page 31 and 32: 1.5. Application areas 11 The resul
Page 33 and 34: 1.6. Summary 13 A different situati
Page 35 and 36: 2.1. Formulating linear programming
Page 43: 2.1. Formulating linear programming
Page 47 and 48: 2.2. Formulating mixed integer prog
Page 49 and 50: 2.3. Formulating nonlinear programm
Page 51 and 52: 2.3. Formulating nonlinear programm
Page 53 and 54: Chapter 3 Algebraic Representation
Page 55 and 56: 3.3. Symbolic-indexed form 35 Maxim
Page 57 and 58: 3.3. Symbolic-indexed form 37 The i
Page 59 and 60: 3.4. Aimms form 39 Figure 3.1 gives
Page 61 and 62: 3.6. Summary 41 units of the variab
Page 63 and 64: 4.2. Shadow prices 43 In addition t
Page 65 and 66: 4.2. Shadow prices 45 If a binding
Page 67 and 68: 4.3. Reduced costs 47 By definition
Page 69 and 70: 4.4. Sensitivity ranges with consta
Page 71 and 72: 4.5. Sensitivity ranges with consta
Page 73 and 74: Chapter 5 Network Flow Models Here,
Page 75 and 76: 5.2. Example of a network flow mode
Page 77 and 78: 5.3. Network formulation 57 The fol
Page 79 and 80: 5.5. Pure network flow models 59 Be
Page 81 and 82: 5.6. Other network models 61 If the
Page 83: Part II General Optimization Modeli
Page 86 and 87: 66 Chapter 6. Linear Programming Tr
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74 Chapter 6. Linear Programming Tr
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76 Chapter 6. Linear Programming Tr
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78 Chapter 7. Integer Linear Progra
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Chapter 8 An Employee Training Prob
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8.2. Model formulation 93 Minimize:
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8.3. Solutions from conventional so
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8.5. Introducing probabilistic cons
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8.6. Summary 99 Minimize: Subject t
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9.2. Model formulation 101 each par
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9.3. Adding logical conditions 103
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9.3. Adding logical conditions 105
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9.5. Summary 107 When all elements
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Chapter 10 A Diet Problem This chap
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10.2. Model formulation 111 Paramet
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10.3. Quantities and units 113 To p
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10.3. Quantities and units 115 Mode
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Chapter 11 A Farm Planning Problem
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11.1. Problem description 119 labor
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11.2. Model formulation 121 the an
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11.2. Model formulation 123 in the
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11.3. Model results 125 In Aimms yo
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Chapter 12 A Pooling Problem In thi
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12.1. Problem description 129 When
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12.2. Model description 131 Let f(p
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12.3. A worked example 133 Due to m
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12.3. A worked example 135 In this
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Part IV Intermediate Optimization M
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140 Chapter 13. A Performance Asses
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150 Chapter 14. A Two-Level Decisio
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164 Chapter 15. A Bandwidth Allocat
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174 Chapter 16. A Power System Expa
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Chapter 17 An Inventory Control Pro
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188 Chapter 17. An Inventory Contro
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Chapter 18 A Portfolio Selection Pr
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18.1. Introduction and background 2
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18.2. A strategic investment model
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18.3. Required mathematical concept
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18.4. Properties of the strategic i
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18.4. Properties of the strategic i
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18.5. Example of strategic investme
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18.6. A tactical investment model 2
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18.7. Example of tactical investmen
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18.8. One-sided variance as portfol
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18.9. Adding logical constraints 22
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18.10. Piecewise linear approximati
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18.11. Summary 225 Note that the ab
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Chapter 19 A File Merge Problem Thi
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19.1. Problem description 229 No. o
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19.2. Mathematical formulation 231
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19.2. Mathematical formulation 233
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19.4. The simplex method 235 the mo
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19.5. Algorithmic approach 237 Assu
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19.6. Summary 239 S and x ij from i
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Chapter 20 A Cutting Stock Problem
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20.2. The initial model formulation
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20.3. Delayed cutting pattern gener
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20.3. Delayed cutting pattern gener
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20.4. Extending the original cuttin
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Chapter 21 A Telecommunication Netw
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21.2. Bottleneck identification mod
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21.2. Bottleneck identification mod
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21.3. Path generation technique 257
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21.3. Path generation technique 259
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21.4. A worked example 261 In the a
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21.5. Summary 263 21.5 Summary In t
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22.1. Problem description 265 Plant
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22.3. Solve large instances through
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22.4. Benders’ decomposition with
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22.4. Benders’ decomposition with
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22.6. Formulating dual models 273 T
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22.7. Application of Benders’ dec
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22.7. Application of Benders’ dec
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22.8. Computational considerations
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22.9. A worked example 281 For each
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22.10. Summary 283 Exercises 22.1 I
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Part VI Appendices
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Index Symbols λ-formulation, 84 A
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290 Index conditional, 81 either-or
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292 Index simplex iteration, 236 me
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294 Bibliography [Ch96] A. Charnes,
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