4-1 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Linear Programming: Modeling Examples Chapter 4- Part2

4-2 Chapter Topics A Marketing Example A Blend Example A Multiperiod Scheduling Example Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-3 A Marketing Example Data and Problem Definition (1 of 6) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-4 Maximize Z = 20,000x ,000x 2 + 9,000x 3 subject to: 15,000x 1 + 6,000x 2 + 4,000x 3  100,000 x 1  4 x 2  10 x 3  7 x 1 + x 2 + x 3  15 x 1, x 2, x 3  0 where x 1 = number of television commercials x 2 = number of radio commercials x 3 = number of newspaper ads A Marketing Example Model Summary (2 of 6)

4-5 Exhibit 4.10 A Marketing Example Solution with Excel (3 of 6)

4-6 Exhibit 4.11 A Marketing Example Solution with Excel Solver Window (4 of 6)

4-7 Exhibit 4.13 A Marketing Example Integer Solution with Excel (5 of 6) Exhibit 4.12

4-8 Exhibit 4.14 A Marketing Example Integer Solution with Excel (6 of 6)

4-9 A Blend Example Problem Definition and Data (1 of 8)

4-10 A Blend Example Problem Definition and Data (2 of 8)

4-11 A Blend Example Problem Definition and Data (3 of 8)

4-12 ■ Determine the optimal mix of the three components in each grade of motor oil that will maximize profit. Company wants to produce at least 3,000 barrels of each grade of motor oil. ■ Decision variables: The quantity of each of the three components used in each grade of gasoline (9 decision variables); x ij = barrels of component i used in motor oil grade j per day, where i = 1, 2, 3 and j = s (super), p (premium), and e (extra). A Blend Example Problem Statement and Variables (4 of 8)

4-13 Maximize Z = 11x 1s + 13x 2s + 9x 3s + 8x 1p + 10x 2p + 6x 3p + 6x 1e + 8x 2e + 4x 3e subject to: x 1s + x 1p + x 1e  4,500 bbl. x 2s + x 2p + x 2e  2,700 bbl. x 3s + x 3p + x 3e  3,500 bbl. 0.50x 1s x 2s x 3s  x 2s x 1s x 3s  x 1p x 2p x 3p  x 3p x 1p x 2p  x 1e x 2e x 3e  x 2e x 1e x 3e  0 x 1s + x 2s + x 3s  3,000 bbl. x 1p + x 2p + x 3p  3,000 bbl. x 1e + x 2e + x 3e  3,000 bbl. A Blend Example Model Summary (5 of 8) all x ij  0

4-14 Exhibit 4.17 A Blend Example Solution with Excel (6 of 8)

4-15 Exhibit 4.18 A Blend Example Solution with Solver Window (7 of 8)

4-16 Exhibit 4.19 A Blend Example Sensitivity Report (8 of 8)

4-17 A Multi-Period Scheduling Example Problem Definition and Data (1 of 5)

4-18 Decision Variables: r j = regular production of computers in week j (j = 1, 2, …, 6) o j = overtime production of computers in week j (j = 1, 2, …, 6) i j = extra computers carried over as inventory in week j (j = 1, 2, …, 5) A Multi-Period Scheduling Example Decision Variables (2 of 5)

4-19 Model summary: Minimize Z = $190(r 1 + r 2 + r 3 + r 4 + r 5 + r 6 ) + $260(o 1 +o 2 +o 3 +o 4 +o 5 +o 6 ) + 10(i 1 + i 2 + i 3 + i 4 + i 5 ) subject to: r j  160 computers in week j (j = 1, 2, 3, 4, 5, 6) o j  50 computers in week j (j = 1, 2, 3, 4, 5, 6) r 1 + o 1 - i 1 = 105week 1 r 2 + o 2 + i 1 - i 2 = 170week 2 r 3 + o 3 + i 2 - i 3 = 230 week 3 r 4 + o 4 + i 3 - i 4 = 180week 4 r 5 + o 5 + i 4 - i 5 = 150week 5 r 6 + o 6 + i 5 = 250week 6 r j, o j, i j  0 A Multi-Period Scheduling Example Model Summary (3 of 5)

4-20 A Multi-Period Scheduling Example Solution with Excel (4 of 5) Exhibit 4.20

4-21 Exhibit 4.21 A Multi-Period Scheduling Example Solution with Solver Window (5 of 5)

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