INEN 420 Final Project. Rhoda Daniel Javier

Transcription

1 INEN 420 Final Project Rhoda Daniel Javier

2 Grummins Engine Company Facts Produces 2 types of diesel trucks (<300/year) Selling prices and manufacturing costs Government regulation: Pollution emission Cost $2, to hold a truck Maximum Demand for Trucks: Year Type 1 Type

3 Grummins Engine Company Assumptions: Zero trucks in stock at the end of the 3 rd year Should not keep more trucks in inventory than demand predicts, so production is regulated by the amount of trucks in inventory and the amount of trucks that can be sold Trucks are only produced and sold, not acquired by any other method such as auctions, trading, etc.

4 Grummins Engine Company Formulation: Decision variables: Pij= Number of trucks (each type i) produced for each year j. Sij= Number of trucks (each type i) sold for each year j. Rij= Number of trucks (each type i) that remain in stock at the end of each year j. For i=1, 2; j=1, 2, 3.

5 Grummins Engine Company Formulation: Objective Function: Max Z= 20 (S11+S12+S13) + 17 (S21+S22+S23) - 15 (P11+P12+P13) - 14 (P21+P22+P23) - 2 (R11+R12+R21+R22) (in $ thousands)

6 Grummins Engine Company Formulation: Constraints: P11+P21 <=320 (Production) P12+P22 <=320 P13+P23 <=320 S11 <=100 (Sale) S12 <=200 S13 <=300 S21 <=200 S22 <=100 S23 <=150

7 Grummins Engine Company Formulation: Constraints: R11-P11+S11 =0 (Remain in stock) R21-P21+S21 =0 R12-P12+S12-R11 =0 R22-P22+S22-R21 =0 P13+R12-S13 =0 P23+R22-S23 =0 5P11+5P12+5P13-5P21-5P22-5P23<=0 (Emissions requirement) Pij, Sij, Rij >= 0

8 Grummins Engine Company Optimal Solution (using LINDO/Excel Solver): Z = 3, (in $ thousands) S11 = 100 S12 = 200 S13 = 150 S21 = 200 S22 = 100 S23 = 150

9 Grummins Engine Company Optimal Solution: P11 = 100 P12 = 200 P13 = 150 P21 = 200 P22 = 100 P23 = 150 R11 = 0 R12 = 0 R21 = 0 R22 = 0

10 Grummins Engine Company Optimal Solution: 10 iterations Company should sell every truck they make each year

11 Grummins Engine Company Sensitivity Analysis: Decision Variables Current Sale & Production ($ thousands) Range of Increase/Decrease Objective Function Coefficient Range of Decision Variables S <= <= + 20<=C11<= + S <= <= + 20<=C12<= + S <= <= 0 15<=C13<= 20 S <= <= + 9<=C21<= + S <= <= + 9<=C22<= + S <= <= + 9<=C23<= + P <= <=0 13<=C14<=15 P <= <=0 13<=C15<=15

12 Grummins Engine Company Sensitivity Analysis: Decision Variables Current Sale & Production ($ thousands) Range of Increase/ Decrease Objective Function Coefficient Range of Decision Variables P <= <=2 15<=C16<=17 P <= <=8 12<=C24<=22 P <= <=2 12<=C25<=16 P <= <= 2 - <=C26<= 16 R <= <=+ 0 <=C1<=+ R <= <=+ 0 <=C2<=+ R <= <=+ 0 <=C3<=+ R <= <=+ 0 <=C4<=+

13 Grummins Engine Company Sensitivity Analysis: Decision Variable Current Right- Hand Side Range of Increase/Decrease Ranges of Right- Hand Sides b <= 300 <= b 1 b <= 300 <= b 2 b <= 300 <= b 3 b <= <= <= b 4 <= 120 b <= <= <= b 5 <= 220 b <= 150 <= b 6 b <= <= <= b 7 <= 220

14 Grummins Engine Company Sensitivity Analysis: Decision Variable Current Right- Hand Side Range of Increase/Decrease Ranges of Right- Hand Sides b <= <= 20 0 <= b 8 <= 120 b <= <= 10 0 <= b 9 <= 160 b <= <= <= b 10 <= 20 b <= <= <= b 11 <= 150 b <= <= <= b 12 <= 20 b <= <= <= b 13 <= 100 b <= <= <= b 14 <= 150 b <= <= <= b 15 <= 10 b <= <= <= b 16 <= 100

15 Wheat Warehouse Facts: Capacity = 20,000 bushels Month # 1: 6,000 bushels Sell up to the initial stock at the current month s selling price. Buy as much wheat as wanted (Up to 20K) Month Selling Price ($) Purchase Price ($)

16 Wheat Warehouse Assumptions: Wheat can only be sold or purchased (at the given rates) Ending inventory: Ending inventory = beginning inventory amount sold + amount purchased

17 Wheat Warehouse Formulation: Decision variables: si = amount of wheat (in thousands) sold during month i, i = 1,,10 pi = amount of wheat (in thousands) purchased during month i, i=1,,10 ej = # of bushels (in thousands) left at the end of month j, j=1,,9

18 Wheat Warehouse Formulation: Objective Function: Max Z = 3s1 8p1 + 6s2 8p2 + 7s3 2p3 + s4 3p4 + 4s5 4p5 + 5s6 3p6 + 5s7 3p7 + s8 2p8 + 3s9 5p9 + 2s10 5p10 (in $ thousands)

19 Wheat Warehouse Formulation: Constraints: s1 <= 6 (selling restrictions) s2 e1 <= 0 s3 e2 <= 0 s4 e3 <= 0 s5 e4 <= 0 s6 e5 <= 0 s7 e6 <= 0 s8 e7 <= 0 s9 e8 <= 0 s10 e9 <= 0

20 Wheat Warehouse Formulation: Constraints: p1 s1 p2 s2 + e1 <= 20 p3 s3 + e2 <= 20 p4 s4 + e3 <= 20 p5 s5 + e4 <= 20 p6 s6 + e5 <= 20 p7 s7 + e6 <= 20 p8 s8 + e7 <= 20 p9 s9 + e8 <= 20 p10 s10 + e9 <= 20 <= 14 (purchasing restrictions)

21 Formulation: Constraints: Wheat Warehouse e1 + s1 p1 = 6 (ending inventory ) e2 + s2 p2 e1 = 0 e3 + s3 p3 e2 = 0 e4 + s4 p4 e3 = 0 e5 + s5 p5 e4 = 0 e6 + s6 p6 e5 = 0 e7 + s7 p7 e6 = 0 e8 + s8 p8 e7 = 0 e9 + s9 p9 e8 = 0 si, pi >= 0, i = 1,,10; ei >= 0, i = 1,,9

22 Wheat Warehouse Optimal Solution (using LINDO/Excel Solver after 20 iterations): Z = 162 (in $ thousands) s3 = 6,000 p3 = 20,000 s5 = 0 p5 = 0 s6 = 20,000 p6 = 20,000 s7 = 20,000 p7 = 0 p8 = 20,000 s9 = 20,000 s10 = 0

23 Wheat Warehouse Sensitivity Analysis: Decision Variable Current Selling/Purchase Price Range of Increase/Decrease Objective Function Coefficient Ranges of Decision Variables s 1 $3 - <= <= 4 - <= c 1 <=7 p 1 $8-1 <= 7 <= c 11 s 2 $6 - <= <= 1 - <= c 2 <= 7 p 2 $8-1 <= 7 <= c 21 s 3 $7-1 <= <= 1 6 <= c 3 <= 8 p 3 $2-1 <= <= 1 1 <= c 31 <= 3 s 4 $1 - <= <=1 - <= c 4 <= 2 p 4 $3-1 <= 2 <= c 41 s 5 $4-2 <= <=0 2 <= c 5 <= 4 p 5 $4 0 <= 4 <= c 51

24 Wheat Warehouse Sensitivity Analysis: Decision Variable Current Selling/Purchase Price Range of Increase/Decrease Objective Function Coefficient Ranges of Decision Variables s 6 $5-1 <= 4 <= c 6 p 6 $3 - <= <= 2 - <= c 61 <= 5 s 7 $5-2 <= 3 <= c 7 p 7 $3-1 <= <= 2 2 <= c 71 <= 5 s 8 $1 - <= <= 1 - <= c 8 <= 2 p 8 $2-1 <= <= 1 1 <= c 81 <= 3 s 9 $3-1 <= <= 2 2 <= c 9 <= 5 p 9 $5-2 <= 3 <= c 91 s 10 $2-2 <= <= 1 0 <= c 10 <= 3 p 10 $5-5 <= 0 <= c 11

25 Wheat Warehouse Sensitivity Analysis: Decision Variable Current Right- Hand Side Range of Increase/Decrease Ranges of Right-Hand Sides b <= 0 <= b 1 b <= -6 <= b 2 b <= -6 <= b 3 b <= -20 <= b 4 b <= -20 <= b 5 b <= -20 <= b 6 b <= 0 <= b 7 b <= 0 <= b 8 b <= 0 <= b 9

26 Wheat Warehouse Sensitivity Analysis: Decision Variable Current Right- Hand Side Range of Increase/Decrease Ranges of Right- Hand Sides b <= 0 <= b 10 b <= 0 <= b 11 b <= 6 <= b 12 b <= 20 <= b 13 b <= <= 0 20 <= b 14 <= 20 b <= <= 0 0 <= b 15 <= 20 b <= 0 <= b 16 b <= 0 <= b 17 b <= 0 <= b 18

27 Wheat Warehouse Sensitivity Analysis: Decision Variable Current Right- Hand Side Range of Increase/Decrease Ranges of Right- Hand Sides b <= 0 <= b 19 b <= 0 <= b 20 b <= <= 14 0 <= b 21 <= 20 b <= -6 <= b 22 b <= <= 20 0 <= b 23 <= 20 b <= 0 <= b 24 b <= -20 <= b 25 b <= -20 <= b 26 b <= <= 0-20 <= b 27 <= 0 b <= -20 <= b 28 b <= <= 0-20 <= b 29 <= 0

28 Power Generation Facts Puerto Rico Electric Power Authority (PREPA) accounts for a majority of net electricity generation (5 plants). ASE-PR and Eco Electrica will provide at least 20% of the power demand during high peak demand (0100 PM). Plant 2 will supply only 50% of its maximum output. Plant 4 will supply only 70% of the maximun output. Power Grid: Total Seven Plants beginning Tropical Storm Jeanne: Cause damage to Puerto Rico s power grid ($60 million).

29 Power Generation EcoElectrica ASE-PR New Power Plant Substations

30 Power Generation Cost of Shipping to Substation # (As of 30 SEP 2004) FROM (Plant) Supply (MW) San Juan Palo Seco Aguirre Costa Sur Arecibo ASE-PR Eco Electrica Expected Demand in MW (2005)

31 Power Generation Assumptions: Plants operate at 90% of their maximum capacity. Power supply to the substation is only being used by the intended sources. In the event of any bad weather, at most two plants will be disconnected.

32 Power Generation Formulation: Two different Conditions: Minimize the cost of meeting each substation s peak power demand for next year Minimize the cost of meeting each substation s peak power demand if Plant 2 and 4 are disconnected due to bad weather

33 Formulation: Power Generation Decision variables (both conditions): Xij= number of megawatts produced at plant i and sent to substations j (Power is sent to each substation during high peak hour (0100 PM)). We define seven plants (i = 1, 2,., 7). Plant 1 is San Juan, Plant 2 is Palo Seco, Plant 3 is Aguirre, Plant 4 is Costa Sur, Plant 5 is Arecibo and Plants 6 and 7 are the two new facilities (ASE-PR and Eco Electrica). The twelve substations are defined: j = 1,2,3,4,,12.

34 Power Generation Formulation: Objective Function (Both Conditions) Min Z = 6X11 + 5X12 + 6X13 + 7X14 + 9X X16 + 9X X X X X X X21 + 6X22 + 5X23 + 8X X X X X X29 +16X X X X X X X X X36 + 9X37 + 6X38 + 7X39 + 9X X X X X X X X X X47 + 9X48 + 5X49 + 8X X x X51 + 8X52 + 9X X X X X X58 + 9X59 + 9X X X X61 + 8X62 + 9X63 + 9X64 + 9X65 + 9X66 + 7X67 + 5X68 + 6X69 + 7X X X X X X X X X76+ 14X77 + 7X78 + 5X79 + 8X X X712

35 Power Generation Formulation: Constraints (Condition 1): (Supply Constraints 90%) X11+X12+X13+X14+X15+X16+X17+X18+X19+X110+X111+X112 <= 360 X21+X22+X23+X24+X25+X26+X27+X28+X29+X210+X211+X212 <= 301 X31+X32+X33+X34+X35+X36+X37+X38+X39+X310+X311+X312 <= 810 X41+X42+X43+X44+X45+X46+X47+X48+X49+X410+X411+X412 <= 743 X51+X52+X53+X54+X55+X56+X57+X58+X59+X510+X511+X512 <= 224 X61+X62+X63+X64+X65+X66+X67+X68+X69+X610+X611+X612 <= 409 X71+X72+X73+X74+X75+X76+X77+X78+X79+X710+X711+X712 <= 451

36 Power Generation Formulation: Constraints (Condition 2): (Supply Constraints 90%) X11+X12+X13+X14+X15+X16+X17+X18+X19+X110+X111+X112 <= 360 X21+X22+X23+X24+X25+X26+X27+X28+X29+X210+X211+X212 <= 0 X31+X32+X33+X34+X35+X36+X37+X38+X39+X310+X311+X312 <= 0 X41+X42+X43+X44+X45+X46+X47+X48+X49+X410+X411+X412 <= 743 X51+X52+X53+X54+X55+X56+X57+X58+X59+X510+X511+X512 <= 224 X61+X62+X63+X64+X65+X66+X67+X68+X69+X610+X611+X612 <= 409 X71+X72+X73+X74+X75+X76+X77+X78+X79+X710+X711+X712 <= 451

37 Power Generarion Formulation: Constraints (Both Conditions): (Supply Constraints for 2 add plants) 4X61 + 4X62+ 4X63 + 4X64 + 4X65 + 4X66 + 4X67 + 4X68 + 4X69 + 4X X X612+ 4X71 + 4X72 + 4X73 + 4X74 + 4X75 + 4X76 + 4X77 + 4X78 + 4X79 + 4X X X712 - X11 - X12 - X13 - X14 - X15 - X16 - X17 - X18 -X19 - X110 - X111 - X112 - X21 - X22 - X23 - X24 - X25 - X26 - X27 - X28 - X29 -X210 - X211 - X212 - X31 - X32 - X33 - X34 - X35 - X36 - X37 - X38 - X39 - X310 -X311 - X312 - X41 - X42 - X43 - X44 - X45 - X46 - X47 - X48 - X49 - X410 - X411 -X412 - X51 - X52 - X53 - X54 - X55 - X56 - X57 - X58 - X59 - X510 - X511 - X512 >= 0

38 Power Generarion Formulation: Constraints (Both Conditions): (Demand Constraints) X11+X21+X31+X41+X51+X61+X71 >= 190 (Substation 1) X12+X22+X32+X42+X52+X62+X72 >= 175 (Substation 2) X13+X23+X33+X43+X53+X63+X73 >= 175 (Substation 3) X14+X24+X34+X44+X54+X64+X74 >= 195 (Substation 4) X15+X25+X35+X45+X55+X65+X75 >= 165 (Substation 5) X16+X26+X36+X46+X56+X66+X76 >= 190 (Substation 6) X17+X27+X37+X47+X57+X67+X77 >= 200 (Substation 7) X18+X28+X38+X48+X58+X63+X78 >= 190 (Substation 8 X19+X29+X39+X49+X59+X69+X79 >= 200 (Substation 9) X110+X210+X310+X410+X510+X610+X710 >= 175 (Substation 10) X111+X211+X311+X411+X511+X611+X711 >= 145 (Substation 11) X112+X212+X312+X412+X512+X612+X112 >= 200 (Substation 12) Sign Restrictions: Xij >= 0, for i=1,..,7; j = 1,..,12

39 Power Generation Optimal Solution (using LINDO): Condition 1: Z = $14,912 Condition 2: Z = $ 17,237 Condition 1 (26 iterations) Plant i to Substation j Power Transmitted Condition 2 (32 iterations) Plant i to Substation j Power Transmitted X X X X X15 0 X15 0 X X X22 0 X21 0 X24 10 X36 0 X X X35 30 X X X X37 0 X51 16 X38 15 X52 0 X X55 30 X X X X X X65 44 X X X65 34 X75 91 X X X79/710 0 X78 15 X X

40 Power Generation Sensitivity Analysis: Decision Variables Cost of Shipping Power ($) Range of Increase/Decrease Objective Function Coefficient Ranges of Decision Variables X11 (NBV) 6-1 <= 5 <= C11 X12 (BV) 5-7 <= <= 0-2 <= C12 <= 5 X <= 3 <= C13 X <= 9 <= C15 X <= 8 <= C16 X <= 7 <= C17 X <= 4 <= C18 Decrease C11 to $4, X11 enter basis. Increase X11=190 C12 to $7, X12 leaves basis. Increase Enter C15 X15=101 to $10, X15 leaves basis. Z= 14,192 X <= 3 <= C19 X <= 6 <= C110 X <= 5 <= C111 X <= 8 <= C112 X <= 1 <= C21 X <= 6 <= C22 X <= 4 <= C23 X <= <= 0 8 <= C25 <= 10

41 Power Generation Sensitivity Analysis: RHS Current Value Range of Increase/Decrease Objective Function Coefficient Ranges of Decision Variables b1 0 - <= <=625 - <= b1< = 625 b <= <= <= b2 <= 370 b <= <= <= b3 <= 331 b <= 235 <= b4 b <= 375 <= b5 b <= <= <= b9 <= 291 b <= <= <= b11 <= 190 b <= <= <= b20 <= 224 Increase b3 to 190, New Z = $14,323 Increase b9 to 292, New Z = $14,472