/* The variables can have any name, but they must start with an alphabetic character and can be followed by alphanumeric characters. Variable names are not case-insensitive, me- aning that "x3" and "X3" represent the same variable.*/ min: 3Y +2x2 +4x3 +7x4 +8X5 5Y + 2x2 >= 9 -3X4 3Y + X2 + X3 +5X5 = 12 6Y + 3x2 + 4X3 <= 124 -5X4 y + 3x2 +6X5 <= 854 -3X4

/* This is a formulation of a linear programming problem in JSON format. */ { "objective": { "type": "min", "coefficients": { "Y": 3, "X2": 2, "X3": 4, "X4": 7, "X5": 8 } }, "constraints": [ { "coefficients": { "Y": 5, "X2": 2, "X4":-3 }, "relation": "ge", "rhs": 9, "name":"VINCOLO1" }, { "coefficients": { "Y": 3, "X2": 1, "X3": 1, "X5": 5 }, "relation": "eq", "rhs": 12, "name":"VINCOLO2" }, { "coefficients": { "Y": 6, "X2": 3, "X3": 4, "X4":-5 }, "relation": "le", "rhs": 124, "name":"VINCOLO3" } ], "bounds": { "Y": { "lower": -1, "upper": 4 }, "X2": { "lower": null, "upper": 5 } } }

min: 3Y +2x2 +4Z +7x4 +8X5 5Y +2x2 +3X4 >= 9 3Y + X2 + Z +5X5 = 12 6Y +3.0x2 +4Z +5X4 <= 124 Y +3x2 + 3X4 +6X5 <= 854 /* To make a variable free is necessary to set a lower bound to -∞ (both +∞ and -∞ are repre- sented with '.' in the text format) */ -1<= x2 <= 6 . <= z <= .

min: 3x1 +X2 +4x3 +7x4 +8X5 5x1 +2x2 +3X4 >= 9 3x1 + X2 +X3 +5X5 >= 12.5 6X1+3.0x2 +4X3 +5X4 <= 124 X1 + 3x2 +3X4 +6X5 <= 854 int x2, X3

min: 3x1 +X2 +4x3 +7x4 +8X5 /* Constraints can be named using the syntax "constraint_name: ....". Names must not contain spaces. */ constraint1: 5x1 +2x2 +3X4 >= 9 constraint2: 3x1 + X2 +X3 +5X5 >= 12.5 row3: 6X1+3.0x2 +4X3 +5X4 <= 124 row4: X1 + 3x2 +3X4 +6X5 <= 854 /*To declare all variables as integers, you can use the notation "int all", or use the notation that with the wildcard '*', which indicates that all variables that start with a certain prefix are integers.*/ int x*

min: 3x1 +X2 +4x3 +7x4 +8X5 5x1 +2x2 +3X4 >= 9 3x1 + X2 +X3 +5X5 >= 12.5 6X1+3.0x2 +4X3 +5X4 <= 124 X1 + 3x2 +3X4 +6X5 <= 854 1<= X2 <=3 /*A set of SOS1 variables limits the values of these so that only one variable can be non-zero, while all others must be zero.*/ sos1 x1,X3,x4,x5

/* All variables are non-negative by default (Xi >=0). The coefficients of the variables can be either or numbers or mathematical expressions enclosed in square brackets '[]' */ /* Objective function: to maximize */ max: [10/3]Y + 20.3Z /* Constraints of the problem */ 5.5Y + 2Z >= 9 3Y + Z + X3 + 3X4 + X5 >= 8 6Y + 3.7Z + 3X3 + 5X4 <= 124 9.3Y + 3Z + 3X4 + 6X5 <= 54 /* It is possible to specify lower and upper bounds for variables using the syntax "l <= x <= u" or "x >= l", or "x <= u". If "l" or "u" are nega- tive, the variable can take negative values in the range. */ /* INCORRECT SINTAX : X1, X2, X3 >=0 */ /* CORRECT SINTAX : X1>=0, X2>=0, X3>=0 */ Z >= 6.4 , X5 >=5 /* I declare Y within the range [-∞,0] */ . <= Y <= 0 /* Declaration of integer variables. */ int Z, Y