This example consists in the optimal design of a chemical process by reducing a superstructure, an its most interesting feature are that both real and binary parameters are needed to solve it.

In this example, we want to produce a certain chemical C with a process which uses B as raw material. B can be bought, or produced from A with processes I and II. These processes cannot be built simultaneously. Additional data:

Conversions:                  

                                            Process II: B = 0,92 · A

                                            (A,B and C are in ton/h)

Maximum demand of C:  1 ton/h

Raw materials costs:        A: 1800 $/ton

                                            B: 7000 $/ton

                                            C: 13000 $/ton

Construction costs: 

To solve this problem, we must optimize the following superstructure:

This problem can be written as follows:

Mass Balances:

            B₁ = 0,75·A₁

            B₂ = 0,92·A₂

            C  =  0,9(B₁ + B₂ + B₃)

Maximum Production of C:

            C <=1· y₁

Logic Constraints:

            y₁ + y₂ <= 1

            y₁ <= y3

            y₂ <= y3

Construction Costs:

            Cost1 = (1000  + 1000 · A₁) · y₁

            Cost2 = (1500+ 1200 · A₂) · y₂ 

            Cost3 = (3500 + 2000 · (B₁ + B₂ + B₃)) · y₃ 

Objective Function:

            Profit = 13000·C · y₃  - 1800·(A₁

            Objective = maximize  (Profit)

This problem can be implemented in GOAL as follows:

PARAMETERS {{ REAL: A1[0;2] REAL: A2[0;2] REAL: B3[0;2] BINARY: y1[0;1] BINARY: y2[0;1] BINARY: y3[0;1] }} MODEL {{ 'Optimum design with mixed integer optimization '*************************************************************** '***** DECLARATIONS ******************************************** '*************************************************************** 'TODO: Declare your global variables here using a Dim sentence '*************************************************************** '***** PREPROCESSING ******************************************* '*************************************************************** Sub PreProcessing() 'TODO: Add the code of the preprocessing function here 'This code will be executed only once at the beginning of the execution End Sub '*************************************************************** '***** OBJECTIVE FUNCTION ************************************** '*************************************************************** Function ObjectiveFunction() B1 = 0.75*A1 B2 = 0.92*A2 C = 0.9*(B1 + B2 + B3) Cost1 = (1000 + 1000*A1)*y1 Cost2 = (1500 + 1200*A2)*y2 Cost3 = (3500 + 2000 * (B1 + B2 + B3))*y3 Profit = 13000*C*y3 - 1800*(A1*y1 + A2*y2) - 7000*B3*y3 - Cost1 - Cost2 - Cost3 ObjectiveFunction = Profit End Function '*************************************************************** '***** POSTPROCESSING ****************************************** '*************************************************************** Sub PostProcessing() 'TODO: Add the code of the postprocessing function here 'This code will be executed only once at the end of the execution End Sub }} CONSTRAINTS {{ #r1: Function Constr_1() Constr_1 = C <= y3 End Function #r2: Function Constr_2() Constr_2 = y1 + y2 <= 1 End Function #r3: Function Constr_3() Constr_3 = y1 <= y3 End Function #r4: Function Constr_4() Constr_4 = y2 <= y3 End Function }} ALGORITHM {{ POPULATION: 70 GENERATIONS: 100 REPRODUCTION_TYPE: 1 REPRODUCTION_PROBABILITY: 0,85 SELECTION_TYPE: 2 SELECTION_PROBABILITY: 0,85 MUTATION: 0,005 ELITISM: 1 REFRESH_PERIOD: 10 REFRESH_PERCENT: 10 }}

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Last modification: 01/27/03