Two cities use incinerators to. reduce waste before placing it in. landfills. Citie Incinerators Landfills. Cities Incinerators Landfills

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1 It costs $3 per mile to transport a ton of material (either debris or waste). Distances (in miles) between locations are: Two cities use incinerators to reduce waste before placing it in landfills. City# City # Incinerator Incinerator # # Incin # Incin # Landfill Landfill # # Formulate an LP that can be used to minimize the total cost of disposing of the waste of both cities. Dennis Bricer Dept of Mechanical & Industrial Engineering The University of Iowa Incinerators 9/0/00 page of 0 Incinerators 9/0/00 page 3 of 0 City produces 500 tons of waste per day, and City produces 400 tons of waste per day. Cities Incinerators Landfills Waste must be incinerated at incinerator or incinerator. Each incinerator can process up to 500 tons of waste per day. The cost to incinerate waste is $40/ton at incinerator and $30/ton at incinerator. Incineration reduces each ton of waste to 0. tons of debris, which must be dumped at one of two landfills. Each landfill can receive at most 00 tons of debris per day. Definition of variables Xi = tons/day of City #i waste that is sent to Incinerator # i=,; =, Y = tons/day of debris sent from Incinerator # to Landfill # =,; =, (There are a total of 8 variables.) Citie Incinerators Landfills X Y X X Y Y X Y Incinerators 9/0/00 page of 0 Incinerators 9/0/00 page 4 of 0

2 Cities Incinerators Landfills X Y X X Y Y X Y LP Model formulation MIN 30 X + 45 X + 48 X + 56 X + 5 Y + 4 Y + 7 Y + 8 Y SUBJECT TO X + X = 500 X + X = 400 X + X <= 500 X + X <= (X + X ) = Y + Y 0.(X + X ) = Y + Y Y + Y <= 00 Y + Y <= 00 X i >=0, Y i >= 0 For all i=, & =, Thus, City # sends its waste to Incinerator #, & City # sends its waste to Incinerator #. Incinerator # sends all its debris to Landfill # & Incinerator # sends all its debris to Landfill #. Cities Incinerators Landfills Incinerators 9/0/00 page 5 of 0 Incinerators 9/0/00 page 7 of 0 LINDO output OBJECTIVE FUNCTION VALUE ) Use a modeling language (e.g. LINGO or MPL) to formulate this model! VARIABLE VALUE REDUCED COST X X Y Y ROW SLACK OR SURPLUS DUAL PRICES ) ) ) ) ) ) ) ) Incinerators 9/0/00 page 6 of 0 Incinerators 9/0/00 page 8 of 0

3 MODEL:! WASTE INCINERATION & DISPOSAL IN LANDFILLS; SETS:! DECISION VARIABLES ARE X, Y, & X = TRANSPORT FROM CITIES TO INCINERATORS = WASTE BURNED IN INCINERATOR Y = TRANSPORT FROM INCINERATORS TO LANDFILLS; CITIES/../: OUTPUT; INCINERATORS/../: ICOST, ICAP, ; LANDFILLS/../: LCAP; PICKUP(CITIES,INCINERATORS): DX, X; DISPOSE(INCINERATORS,LANDFILLS): DY, Y; ENDSETS Notes: be sure to end comments with a semicolon! Variable is defined for convenience only The mathematical statement of the problem Obective function Minimize C D X + D Y x y d i i i City Incinerator Incinerator Landfill b + C Incinerator MIN 3[(30X +5X +36X +4X )+(5Y +8Y +9Y +6Y )] DY*Y) BURNCOST Incinerators 9/0/00 page 9 of 0 Incinerators 9/0/00 page of 0 DATA: OUTPUT= ; ICOST = 40 30; ICAP = ; LCAP = 00 00; DX = 30 5! DISTANCES BETWEEN CITIES ; 36 4;! & INCINERATORS; DY = 5 8! DISTANCES BETWEEN INCINERATORS; 9 6;! & LANDFILLS; COSTPERMI = 3.0 ;! TRANSPORTATION COST PER MILE; REDUCTION = 0. ;! REDUCTION FACTOR:WASTE TO DEBRIS; ENDDATA Computing travel distances i City Incinerator D X to loop through the deliveries of trash to incinerators: with or without DX(I,J)*X(I,J) ) or DX*X ) so DY*Y) BURNCOST x i i Incinerators 9/0/00 page 0 of 0 Incinerators 9/0/00 page of 0

4 constraints subect to Incinerator = I i City X = OUTPUT for all i City X i i i for all Incinerator 0. = Y for all Incinerator Incinerator Landfill Y L for all Landfill X 0, Y 0, & 0 for all i,,& i = i City I X = X + X <= 500 = X + X <= 500 i for all Incinerator (J)=@SUM(CITIES(I): X(I,J) (J) <= ICAP(J for Incinerator for Incinerator Incinerators 9/0/00 page 3 of 0 Incinerators 9/0/00 page 5 of 0 constraints Incinerator X = OUTPUT for all i City i i X + X = 500 for City X + X = 400 for City to generate a set of constraints for X(I,J)) = OUTPUT(I)! be sure to close the parenthesis ; 0. = Y for all Incinerator Landfill 0. = Y + Y for Incinerator 0. = Y + Y for Incinerator REDUCTION* (J) Y(J,K) Incinerators 9/0/00 page 4 of 0 Incinerators 9/0/00 page 6 of 0

5 Incinerator Y L for all Landfill Y + Y <= 00 for Landfill Y + Y <= 00 Y(J,K) ) <= LCAP(K) Global optimal solution found at step: 4 Obective value: Variable Value Reduced Cost TRANSPORTCOST BURNCOST ( ) ( ) X(, ) X(, ) X(, ) X(, ) Y(, ) Y(, ) Y(, ) Y(, ) Incinerators 9/0/00 page 7 of 0 Incinerators 9/0/00 page 9 of 0 BURNCOST @SUM(INCINERATORS(J): X(I,J)) = OUTPUT(I) (J)=@SUM(CITIES(I): X(I,J) (J) <= ICAP(J REDUCTION* (J) Y(J,K) Y(J,K) ) <= LCAP(K) END Row Slac or Surplus Dual Price Incinerators 9/0/00 page 8 of 0 Incinerators 9/0/00 page 0 of 0