lp_solve_online_dev

You can try out the web app deployed on Heroku here:

• https://lpsolveweb.herokuapp.com/

This project aims to build an accessible, online linear programming optimization interface. The web app works with a complementary backend server that processes algebraic optimization models and finds the optimal solution, if possible. The Github repository for the server can be found in this GitHub repository.

Syntax and Semantics

The syntax is largely based on that used in lpsolve library and IDE, and it additionally implements 'for' and 'summation' statements to make linear modeling more succint.

Objective Statement

• Declare whether the objective is max or min. Alternatively, the terms maximize and minimize can be used respectively.
• Include a colon (:) to distinguish the objective from the expression.
• Add the linear expression to be optimized.
• End the objective statement with a semi-colon (;).
• Only 1 objective statement can be declared.
• Examples:
  • max: 2x + y;
  • maximize: 2x + y;
  • min: 3x_1 + 2x_2 - 4x_3;
  • minimize: 3x_1 + 2x_2 - 4x_3;

Functional Constraints

Constraints apply a restriction on variables based on expressions and values. The server recognizes two types of constraints named relation constraints and range constraints. Only 1 of the 2 can be declared in the same statement.

• Inequality refers to less-than (<), less-than-or-equal (<=), greater-than (>), and greater-than-or-equal (>=).
• Equality refers to equal (=).
• Every constraint declaration must end with a semi-colon (;).
• Relation Constraint:
  • Defined to be a inequality or equality, i.e. expression 1 [inequality or equality] expression 2
  • Examples:
    • 2x + 4y <= 17;
    • 3x_1 - 2x_2 + 3x_3 >= 10;
    • 2x = 5y;
• Range Constraint:
  • Defined to be a continuous inequality, i.e. value 1 [inequality] expression [inequality] value 2
  • By value, value 1 and value 2 are expressions that must evaluate to some number.
  • By continuous, a range constraint statement must use the same inequality across the statement.
  • Examples:
    • 2 <= 3x + 5y <= 17;
    • 2 >= 2x - 2y >= 0;
• An optional name for constraints can be declared at the beginning of the statement, following the pattern: [name]: [constraint];
  • Examples:
    • C1: 2x + 4y <= 12;
    • C2: 100 <= 20x + 40y <= 600;
  • Note the necessity of the colon (:) to distinguish the name from the constraint.
  • Names cannot be one of the objective types, i.e. max, min, maximize, minimize.

Type Declaration

The server recognizes three unique types: int, bin, free.

• The three types are used to constrain variables to be a certain type of value. Any variables declared in the objective statement and constraint statements are by default free.
• Type declarations are written in the following format:
  • [type]: [each variable separated by commas];
• int:
  • The variables declared as int can only be integers in the optimal solution.
• bin:
  • The variables declared as bin can only be binary values in the optimal solution, i.e. 1 or 0.
• free:
  • The variables declared as free can only be any type of rational number in the optimal solution, including integer and fractional values.
• Examples:
  • int x, y, z;
  • bin x_1, x_2, x_3;
  • free z;

For Statement

For statements can be used to declare multiple constraints or type declarations in a single statement. The following format is used to write a for statement:

• for [iterating variable] = [initial value] to [terminating value]: [constraint or type declaration];
• For each iteration, the any instance of the iterating variable in the constraint or type declaration is substituted by the current value.
• The iterating variable increments by 1.
• The terminating value is required to be greater than or equal to the inital value.
• Examples:
  • for i = 1 to 3: 2x_i <= 15;
    • This is equivalent to the following constraints:
      • 2x_1 <= 15;
      • 2x_2 <= 15;
      • 2x_3 <= 15;
  • for i = 1 to 3: int x_i;
    • This is equivalent to the following type declarations:
      • int x_1;
      • int x_2;
      • int x_3;

Summation Statement

Summations can be used express a long expression succintly. Summations are considered expressions and can be written inside objective or constraint statements. They are written in the following form:

• sum [[iterating variable] = [initial value] to [terminating value]] ([expression])
• The iterating variable, initial value, and terminating value follow the same properties as those in for statements.
• Examples:
  • max: sum [i = 1 to 4] (x_i);
    • This is equivalent to max: x_1 + x_2 + x_3 + x_4;
  • C1: sum [i = 1 to 4] (2x_i - y_i) <= 23;
    • This is equivalent to C1: 2x_1 - y_1 + 2x_2 - y_2 + 2x_3 - y_3 + 2x_4 - y_4 <= 23;