Ch5_Example_Problems

Ch5_Example_Problems - 5-1 Machine Shop Example Machines...

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Unformatted text preview: 5-1 Machine Shop Example Machines Profit per machine Resource constraints purchase price ($) floor space (sq. ft.) Purchases Presses = Lathes = Profit = 2.22 5.56 1055.56 Press 100 8000 15 Lathe 150 4000 30 Usage 40000 200 Resources Available <= 40000 <= 200 maximize Z = 100x1 + 150x2 subject to 8000x1 + 4000x2 <= 40000 15x1 + 30x2 <= 200 x1, x2 >= 0 Page 1 5-1 es Left over 0 0 + 150x2 00x2 <= 40000 0x2 <= 200 , x2 >= 0 1 Page 2 5-2 Recreational Facilities (0 - 1) Example Swimming Pool 300 35000 4 1 Tennis Center 90 10000 2 1 Athletic Field 400 25000 7 0 Projects: Daily usage Resource Constraints cost ($/facility) space (acres/facility) contingency Projects selected: Swimming pool = Tennis center = Athletic field = Gymnasium = Total daily usage = Gymnasium 150 90000 3 0 Usage 60000 7 1 Resources Constraints Available <= 120000 <= 12 <= 1 1 0 1 0 700 maximize Z = 300x1 + 90x2 + 400x3 + 150x4 subject to 35000x1 + 10000x2 + 25000x3 + 90000x4 <= 120000 4x1 + 2x2 + 7x3 + 3x4 <= 12 x1 + x2 <= 1 x1, x2, x3, x4 = 0 or 1 Page 3 5-2 ces Left over 60000 5 0 150x4 x3 + 90000x4 <= 120000 + 7x3 + 3x4 <= 12 2 x1 + x2 <= 1 x1, x2, x3, x4 = 0 or 1 Page 4 5-8 Machine Shop Example Machines Profit per machine Resource constraints purchase price ($) floor space (sq. ft.) Purchases Presses = Lathes = Profit = 1 6 1000 Press 100 8000 15 Lathe 150 4000 30 Usage 32000 195 Resources Available <= 40000 <= 200 maximize Z = 100x1 + 150x2 subject to 8000x1 + 4000x2 <= 40000 15x1 + 30x2 <= 200 x1, x2 >= 0 Page 5 5-8 es Left over 8000 5 + 150x2 00x2 <= 40000 0x2 <= 200 , x2 >= 0 1 Page 6 5-12 Investments Example Investments Profit per Investment Cost per investment Investment decisions Condos = 4 Land = 4.167 Bonds = 0 Profit = 42250 Condos 9000 50000 Land 1500 12000 Bonds 1000 8000 Invested 250000 Constraint <= Budget 250000 maximize Z = 9000x1 + 1500x2 + 1000x3 subject to 50000x1 + 12000x2 + 8000x3 <= 250000 x1 <= 4 x2 <= 15 x3 <= 20 x2 >= 0 x1, x3 >= 0 and integer Page 7 5-12 0x3 00x3 <= 250000 x1 <= 4 x2 <= 15 x3 <= 20 x2 >= 0 x1, x3 >= 0 and integer Page 8 A Capital Budgeting Example NPV returns Xi Project ($1000s) 1 website 1 120 2. warehouse 0 85 3. clothing department 0 105 4. computer department 1 140 5. ATMs 1 70 Funds spent per year Constraints Available funds per year Departmental space restriction Total return = 330 Project costs/year ($1000s) 1 2 3 55 40 25 45 35 20 60 25 50 35 30 30 30 135 105 55 <= <= <= 150 110 60 1 maximize Z = 120x1 + 85x2 + 105x3 + 140x4 + 70x5 subject to 55x1 + 45x2 + 60x3 + 50x4 + 30x5 <= 1 40x1 + 35x2 + 25x3 + 35x4 + 30x5 <= 1 25x1 + 20x2 + 30x4 <= 6 x3 + x4 <= 1 xi = 0 x1 + 85x2 + 105x3 + 140x4 + 70x5 + 45x2 + 60x3 + 50x4 + 30x5 <= 150 + 35x2 + 25x3 + 35x4 + 30x5 <= 110 25x1 + 20x2 + 30x4 <= 60 x3 + x4 <= 1 xi = 0 or 1 A Fixed Charge and Facility Location Example Shipping Costs: Farms 1 2 3 4 5 6 Production Capacity Potatoes Shipped Plants B 0 8.1 0 1.9 0 0 10 10 Annual Fixed Cost 405 390 450 368 520 465 Projected Potatoes Selection Harvest Shipped Constraints 11.2 11.2 0 10.5 10.5 0 12.8 0 0 9.3 4.7 -4.6 10.8 0 0 9.6 9.6 0 A 0 2.4 0 0 0 9.6 12 12 Selection Yi 1 1 0 1 0 1 2,082.30 C 11.2 0 0 2.8 0 0 14 14 Farm 1 2 3 4 5 6 Total Cost (Z) = minimize Z = 18x1A +15x1B + 12x1C + 13x2A + 10x 18x3C + 19x4A + 15x4B + 16x4C + 1 16x6B + 12x6C + 405y1 + 390y2 + subject to x1A + x1B + x1C - 11.2 x2A + x2B + x2C - 10.5 x3A + x3B + x3C - 12.8 x4A + x4B + x4C - 9.3 x5A + x5B +x5C - 10.8 x6A + x6B + x6C - 9.6 x1A + x2A + x3A + x4A + x5A + x1B + x2B + x3B + x4B + x5B + x1C + x2C + x3C + x4C + x5C + x 8x1A +15x1B + 12x1C + 13x2A + 10x2B +17x2C + 16x3A + 14x3B + 18x3C + 19x4A + 15x4B + 16x4C + 17x5A + 19x5B + 12x5C + 14x6A + 16x6B + 12x6C + 405y1 + 390y2 + 450y3 + 368y4 + 520y5 + 465y6 x1A + x1B + x1C - 11.2y1 <= 0 x2A + x2B + x2C - 10.5y2 <= 0 x3A + x3B + x3C - 12.8y3 <= 0 x4A + x4B + x4C - 9.3y4 <= 0 x5A + x5B +x5C - 10.8y5 <= 0 x6A + x6B + x6C - 9.6y6 <= 0 x1A + x2A + x3A + x4A + x5A + x6A = 12 x1B + x2B + x3B + x4B + x5B + x6B = 10 x1C + x2C + x3C + x4C + x5C + x6C = 14 xij >= 0 yi = 0 or 1 A Set Covering Example Hub Sites 1. Atlanta 2. Boston 3. Charlotte 4. Cincinnati 5. Detroit 6. Indiannapolis 7. Milwaukee 8. Nashville 9. New York 10. Pittsburgh 11. Richmond 12. St. Louis Hub Selections Total Hubs 1 Atlanta 1 0 1 0 0 0 0 1 0 0 0 0 0 4 2 Boston 0 1 0 0 0 0 0 0 1 0 0 0 1 3 4 Charlotte Cincinnati 1 0 0 0 1 0 0 1 0 1 0 1 0 0 0 1 0 0 0 1 1 0 0 0 1 0 Cities Within 300 Miles 5 6 7 8 Detroit Indiannapolis Milwaukee Nashville 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 1 1 1 1 1 1 1 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 minimize Z = x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x1 subject to x1 + x3 + x2 + x1 + x3 + x1 x4 + x5 + x8 + x1 x4 + x5 + x6 + x7 + x8 + x12 > x4 + x5 + x6 + x7 + x8 + x12 > x5 + x6 + x1 + x4 + x6 + x8 + x12 x2 + x9 + x1 x4 + x5 + x10 + x11 x3 + x9 + x10 + x11 x6 + x8 + x1 s 9 10 New York Pittsburgh 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0 1 1 1 0 0 0 0 11 Richmond 0 0 1 0 0 0 0 0 1 1 1 0 0 12 St. Louis 0 0 0 0 0 1 0 1 0 0 0 1 1 Sum 1 1 1 1 1 2 1 1 1 1 1 1 5 + x6 + x7 + x8 + x9 + x10 + x11 + x12 x1 + x3 + x8 >= 1 x2 + x9 > = 1 x1 + x3 + x11 > = 1 x4 + x5 + x8 + x10 >= 1 x5 + x6 + x7 + x8 + x12 >= 1 x5 + x6 + x7 + x8 + x12 >= 1 x5 + x6 + x7 >= 1 x1 + x4 + x6 + x8 + x12 >= 1 x2 + x9 + x11 >= 1 x4 + x5 + x10 + x11 >= 1 x3 + x9 + x10 + x11 >= 1 x6 + x8 + x12 >= 1 xi = 0 or 1 ...
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This note was uploaded on 06/17/2010 for the course MAT 540 taught by Professor Azad during the Spring '10 term at Strayer.

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