Formulation of a General Problem
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1 Formulation of a General Problem Pei Huang 1 1 Department of Agricultural Economics Texas A&M University Based on materials written by Gillig & McCarl and improved upon by many previous lab instructors Special thanks to Mario Andres Fernandez Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
2 Outline 1 Review of a Simple Problem 2 Model Formulation Set Denition Data Entry Variable Specication Equation Specication 3 Solution Reports Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
3 Review Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
4 Review Variable specications Objective function Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
5 Review Variable specications Objective function Considered variables Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
6 Review Variable specications Objective function Considered variables Equation specications Declaration Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
7 Review Variable specications Objective function Considered variables Equation specications Declaration Algebraic structure specication Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
8 Review Variable specications Objective function Considered variables Equation specications Declaration Algebraic structure specication Model statement Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
9 Review Variable specications Objective function Considered variables Equation specications Declaration Algebraic structure specication Model statement Solve statement Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
10 What if we have much more data? A simple problem max 109X corn + 90X wheat + 115X cotton s.t. X corn + X wheat + X cotton 100 6X corn + 4X wheat + 8X cotton 60 X bread, X meat, X wine 0 Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
11 What if we have much more data? A simple problem A general problem max 109X corn + 90X wheat + 115X cotton s.t. X corn + X wheat + X cotton 100 6X corn + 4X wheat + 8X cotton 60 X bread, X meat, X wine 0 max s.t. c i X i i a ij X i b j i X i 0 j Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
12 What if we have much more data? A simple problem A general problem max 109X corn + 90X wheat + 115X cotton s.t. X corn + X wheat + X cotton 100 6X corn + 4X wheat + 8X cotton 60 X bread, X meat, X wine 0 max s.t. c i X i i a ij X i b j i X i 0 j It impossible to enter variables and equations item by item for the problem on the right with substantially large i and j. Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
13 Handling Summation in GAMS Algebra i x i ij x ij GAMS Syntax z=sum(i,x(i)) z=sum((i,j),x(i,j)) or z=sum(j,sum(i,x(i,j)) i,j: a set; z: a scalar or variable; x(i),x(i,j): a parameter or variable dened over i or (i,j); SUM: the summation function in GAMS Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
14 Handling Summation in GAMS Algebra i x i ij x ij GAMS Syntax z=sum(i,x(i)) z=sum((i,j),x(i,j)) or z=sum(j,sum(i,x(i,j)) i,j: a set; z: a scalar or variable; x(i),x(i,j): a parameter or variable dened over i or (i,j); SUM: the summation function in GAMS i ( xi + j y ij ) = i x i + ij y ij z=sum(i,x(i)+sum(j,y(i,j)) or z=sum(i,x(i))+sum((i,j),y(i,j)) Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
15 Model Formulation in GAMS Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
16 Model Formulation in GAMS Variable specications Objective function Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
17 Model Formulation in GAMS Variable specications Objective function Considered variables Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
18 Model Formulation in GAMS Variable specications Objective function Considered variables Equation specications Declaration Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
19 Model Formulation in GAMS Variable specications Objective function Considered variables Equation specications Declaration Algebraic structure specication Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
20 Model Formulation in GAMS Variable specications Objective function Considered variables Equation specications Declaration Algebraic structure specication Model statement Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
21 Model Formulation in GAMS Variable specications Objective function Considered variables Equation specications Declaration Algebraic structure specication Model statement Solve statement Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
22 Model Formulation in GAMS Set denition Data entry Variable specications Objective function Considered variables Equation specications Declaration Algebraic structure specication Model statement Solve statement Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
23 Set Denition Subscripts (algebra) Sets (GAMS) Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
24 Set Denition Subscripts (algebra) Sets (GAMS) Elements in a set are included in / /, and separated by Enter. Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
25 Set Denition Subscripts (algebra) Sets (GAMS) Elements in a set are included in / /, and separated by Enter. Explanatory text and set name, Explanatory text and elements are separated by Space. Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
26 Set Denition Subscripts (algebra) Sets (GAMS) Elements in a set are included in / /, and separated by Enter. Explanatory text and set name, Explanatory text and elements are separated by Space. Don't forget the semi-colon at the end of set denition! Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
27 Set Denition Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
28 Set Denition Elements in a set are included in / /, and separated by commas,. Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
29 Set Denition Elements in a set are included in / /, and separated by commas,. Explanatory text and Explanatory text are identied with Space after the element or set. Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
30 Subset To check model feasibility, we can dene a small subset over the whole set, and test the model. Example SET Crops The whole set for crops /Corn Soybeans Wheat Cotton Hay/ Crop(Crops) /Corn Soybeans Wheat/; A small set for crops Crop(Crops)=YES; OBJT.. Profits =e= SUM(Crop, Price(Crop) * Production(Crop)); Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
31 Subset To check model feasibility, we can dene a small subset over the whole set, and test the model. Example SET Crops The whole set for crops /Corn Soybeans Wheat Cotton Hay/ If the last line is activated, it will run the model with the whole set. Crop(Crops) /Corn Soybeans Wheat/; A small set for crops Crop(Crops)=YES; OBJT.. Profits =e= SUM(Crop, Price(Crop) * Production(Crop)); Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
32 Subset To check model feasibility, we can dene a small subset over the whole set, and test the model. Example SET Crops The whole set for crops /Corn Soybeans Wheat Cotton Hay/ If the last line is activated, it will run the model with the whole set. If the last line is commented out, then it will run the model with the small set. Crop(Crops) /Corn Soybeans Wheat/; A small set for crops Crop(Crops)=YES; OBJT.. Profits =e= SUM(Crop, Price(Crop) * Production(Crop)); Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
33 Subset To check model feasibility, we can dene a small subset over the whole set, and test the model. Example SET Crops The whole set for crops /Corn Soybeans Wheat Cotton Hay/ Crop(Crops) /Corn Soybeans Wheat/; A small set for crops If the last line is activated, it will run the model with the whole set. If the last line is commented out, then it will run the model with the small set. The subset Crop is used for all set-dependency calculations in subsequent code instead of Crops. Crop(Crops)=YES; OBJT.. Profits =e= SUM(Crop, Price(Crop) * Production(Crop)); Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
34 Data Entry Input data representing c i, a ij, and b j Scalar Parameter Table Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
35 Scalar For items that are not set dependent Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
36 Scalar For items that are not set dependent Scalar Format SCALAR ItemName explanatory text /value/; Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
37 Scalar For items that are not set dependent Scalar Format SCALAR ItemName explanatory text /value/; Example SCALARS LandAvail Land availability /100/ Pricecorn 1992 price per bushel /2.2/; Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
38 Scalar For items that are not set dependent Scalar Format SCALAR ItemName explanatory text /value/; Example SCALARS LandAvail Land availability /100/ Pricecorn 1992 price per bushel /2.2/; One SCALAR (or SCALARS) denition can include multiple scalars. Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
39 Parameter For items that are vectors Set-dependent Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
40 Parameter For items that are vectors Set-dependent Parameter Format PARAMETER ItemName(set1) explanatory text /element1 value element2 value/; Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
41 Parameter For items that are vectors Set-dependent Parameter Format PARAMETER ItemName(set1) explanatory text /element1 value element2 value/; Example PARAMETER revenue(crop) /corn 109 wheat 90 cotton 115/; the return to an acre of crop Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
42 Table For items with two or more dimensions Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
43 Table For items with two or more dimensions Table Format TABLE ItemName(set1, set2) explanatory text set2_elem1 set2_elem2 set1elem1 value11 value12 set1elem2 value21 value22 ; Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
44 Table For items with two or more dimensions Table Format TABLE ItemName(set1, set2) explanatory text set2_elem1 set2_elem2 set1elem1 value11 value12 set1elem2 value21 value22 ; Example TABLE technical(resource, crop) corn wheat cotton land labor ; the technical coefficients Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
45 Direct Assignment What if we want to post-calculate a set-dependent parameter based on model solutions? Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
46 Direct Assignment What if we want to post-calculate a set-dependent parameter based on model solutions? Directly Assigned Parameter Format PARAMETER ItemName(set1, set2,...) explanatory text; ItemName(set1, set2,...) = some mathematical expressions; Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
47 Direct Assignment What if we want to post-calculate a set-dependent parameter based on model solutions? Directly Assigned Parameter Format PARAMETER ItemName(set1, set2,...) explanatory text; ItemName(set1, set2,...) = some mathematical expressions; Example PARAMETER CalcRevenue(crop) calculate revenues by crop; CalcRevenue(crop) = Revenue(crop)*Production.L(crop); PARAMETER TotalRevenue(crop) calculate total revenue; TotalRevenue(crop) = SUM(crop, Revenue(crop)*Production.L(crop)); Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
48 Variable Specication Now set-dependent Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
49 Variable Specication Now set-dependent Variable Format VARIABLE VarName1(set1, set2,...) explanatory text VarName2(set1, set2,...) explanatory text ; Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
50 Variable Specication Now set-dependent Variable Format VARIABLE VarName1(set1, set2,...) explanatory text VarName2(set1, set2,...) explanatory text ; Example POSITIVE VARIABLE Production(crop) land allocation for crops ; Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
51 Equation Specication Equation declarations Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
52 Equation Specication Equation declarations Equation Format EQUATION EquName1(set1, set2,...) explanatory text EquName2(set1, set2,...) explanatory text ; Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
53 Equation Specication Equation declarations Equation Format EQUATION EquName1(set1, set2,...) explanatory text EquName2(set1, set2,...) explanatory text ; Example EQUATION objective maximize farm income ResourceEq(resource) resource constraints ; Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
54 Equation Specication Algebraic structure specication Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
55 Equation Specication Algebraic structure specication General Structure EquationName(set).. LHS =e= (or =l= or =g=) RHS; Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
56 Equation Specication Algebraic structure specication General Structure EquationName(set).. LHS =e= (or =l= or =g=) RHS; Example objective.. profit =e= SUM(crop, revenue(crop)*production(crop)); ResourceEq(resource).. SUM(crop, rechnical(resource, crop)*production(crop)) =l= constraint(resource); Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
57 Equation Specication Algebraic structure specication General Structure EquationName(set).. LHS =e= (or =l= or =g=) RHS; Example objective.. profit =e= SUM(crop, revenue(crop)*production(crop)); ResourceEq(resource).. SUM(crop, rechnical(resource, crop)*production(crop)) =l= constraint(resource); Sets after summation on LHS and RHS must be balanced. Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
58 Solution Reports Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
59 Solution Reports Objective function solution Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
60 Solution Reports Objective function solution Equation solutions Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
61 Solution Reports Objective function solution Equation solutions Variable solutions Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
62 Solution Reports Objective function solution Equation solutions Variable solutions Shadow price: marginal values of resources Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
63 Solution Reports Objective function solution Equation solutions Variable solutions Shadow price: marginal values of resources Reduced costs: marginal cost if a non-basic variable is forced to enter the solutions Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
64 Solution Reports Objective function solution Equation solutions Variable solutions Shadow price: marginal values of resources Reduced costs: marginal cost if a non-basic variable is forced to enter the solutions The single dot. represents zero. Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
65 Solution Reports Objective function solution Equation solutions Variable solutions Shadow price: marginal values of resources Reduced costs: marginal cost if a non-basic variable is forced to enter the solutions The single dot. represents zero. INF represents innity. Pei Huang Texas A&M University AGEC 641 Lab Session, Fall / 18
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