Algebraic Modeling Languages

A Quick Preview on  Algebraic Generation Languages.

        Algebraic, or constraint oriented, languages describe the MP matrix as a set of 
equations, a form favored by the academic and engineering communities.  The description of the 
matrix is algebraic in that a statement is an equation  or inequality comprised of summations of 
products between model activities and coefficient bearing data tables.  A single statement will 
describe a collection of similar constraints and activities.  The structure of the language 
parallels the mathematical summation notation used to formulate the model.

DICTIONARY
        CLASS V                 Set of vegetable oils
                VEG1,VEG2
        CLASS O                 Set of non-vegetable oils
                OIL1,OIL2,OIL3
        CLASS P                 Set of final product
                Y
DATA
                TABLE A         Maximum production quantities
                        VEG     OIL
        MAXM    200     250
*
                TABLE B         Cost per ton of vegetable oils
                        VEG1    VEG2
        COSTV   100     120
*
                TABLE C         Cost per ton of non-vegetable oils
                        OIL1    OIL2    OIL3
        COSTN   130     110     115
*
                TABLE D         Hardness of vegetable oils
                        VEG1    VEG2
        HARDV   8.8     6.1
*
                TABLE E         Hardness of non-vegetable oils
                        OIL1    OIL2    OIL3
        HARDN   2.0     4.2     5.0
*
                TABLE F         Selling price per ton of final product
                        PRICE
        Y               150
*

        COPY
NAME            BLEND
        FORM ROW ID
                PROF = OBJ
                VVEG(V) = MAX
                NVEG(O) = MAX
                UHAR(V,O) = MAX
                LHAR(V,O) = MIN
                CONT(V,O) = FIX
        COPY
COLUMNS
        FORM VECTOR X(V)
                PROF = TABLE B((V), COSTV)
                VVEG(V) = 1
                UHAR(V) =  TABLE D((V), HARDV)
                LHAR(V) = TABLE D((V), HARDV)
                CONT(V) = 1.0
        FORM VECTOR X(O)
                PROF = TABLE C((O), COSTN)


                NVEG(O) = 1
                UHAR(O) =  TABLE E((O), HARDN)
                LHAR(O) = TABLE E((O), HARDN)
                CONT(O) = 1.0
        FORM VECTOR X(P)
                PROF = TABLE F(PRICE, (P))
                UHAR(P) = -6.0
                LHAR(P) = -3.0
                CONT(P) = -1.0
        COPY
RHS
        FORM VECTOR RHSIDE
                VVEG(V) = TABLE A(VEG, MAXM)
                NVEG(O) = TABLE A(OIL, MAXM)
        COPY
ENDATA
        END             


Figure 2.3 The Oil-Blending Problem in MAGEN

Algebraic Generation Languages

-       are not restricted by conforming to MPS format

-       describe the mathematical programming matrix as a set of equations or inequalities

-       use index expressions such as "i OVER raw" and "t from 1 to time" to describe the sets 
over which collection of items are indexed

-       use arithmetic expressions with powerful indexed-sum such as "SIGMA" notation that 
corresponds to the algebraic use of S

        Examples:               ALPS
                                GAMS
                                LPMODEL
                                MAGIC
                                AMPL

        GAMS is the best known, most widely used and most capable of the current algebraic 
modeling systems.  It was originally developed on a CDC mainframe, has been migrated to the 
IBM mainframe, UNIX based workstations and can be run on an ordinary PC.  GAMS uses two-
dimensional data tables as its principle input medium.  GAMS has no data management capability.  
Data tables and model description both appear in a fixed order in the GAMS symbolic input file.  
GAMS automates the generation of matrix names while allowing the user to describe data with 
names of up to 10 characters.  GAMS is written in PASCAL and interfaces to multiple optimizers 
via the MPS standard files.  Thus the user is free to mix and match model managers, optimizers, 
operating systems and hardware platforms.  AMPL is a more recent modeling language with 
design and objective similar to that of GAMS.  In an AMPL representation, the data values are 
specified in a separate section and are not combined with the problem specifications as in GAMS.  
This separation of data in the AMPL model makes it more maintainable.

        GAMS and AMPL will be described in more details in PART 2.