CPLEX Basics

Multi-criteria trade-off analysis can be accomplished with a variety of optimization tools, including Microsoft Excel (for small-scale problems) and ILOG CPLEX.

Solving Linear Optimization Problems with ILOG CPLEX

ILOG CPLEX is a tool for solving linear optimization problems, commonly referred to as Linear Programming (LP) problems, of the form (CPLEX Online Manual):

 ``` Maximize (or Minimize) c1x1 + c2x2 +...+ cnxn subject to a11x1 + a12x2 +...+ a1nxn ~ b1 a21x1 + a22x2 +...+ a2nxn ~ b2 .......... am1x1 + am2x2 +...+ amnxn ~ bm with these bounds li <= xi <= ui, ..... ln <= xn <= un ``` where "~" can be <= (less than or equal), >= (greater than or equal), or = (equal), and the upper bounds ui and lower bounds li may be positive infinity, negative infinity, or any real number.

The data you provide as input for this LP is:

 ``` Objective function coefficients: c1, ...... , cn Constraint coefficients: a11, ...... , amn Right-hand sides: b1, ........ , bm Upper and lower bounds: u1, ...... , un and l1, ...... , ln ```

CPLEX also can solve several extensions to LP:

1. Mixed Integer Programming (MIP) problems, where any or all of the LP variables are further restricted to take integer values in the optimal solution (and where MIP itself is extended to include constructs like Special Ordered Sets (SOS) and semi-continuous variables);

2. Quadratic Programming (QP) problems, where the LP objective function is expanded to include quadratic terms;

3. Network Flow problems, a special case of LP that CPLEX can solve much faster by exploiting the problem structure.

Procedure for Problem Definition and Solution

Components of problems should be entered in the following order: (1) objectives; (2) constraints; (3) bounds; (4) completion of the problem definition.

Points to note are as follows:

1. Objectives

Before entering the objective function, you must state whether the problem is a minimization or maximization. For example, you might type:

```     maximize  x1 + 2x2 + 3x3
```

In the simple example shown immediately above, the variables are named simply x1, x2, x3, but you can give your variables more meaningful names such as cars or gallons.

2. Constraints

Once you have entered the objective function, you can move on to the constraints. However, before you start entering the constraints, you must indicate that the subsequent lines are constraints by typing:

```     subject to

or

st
```

These terms can be placed alone on a line or on the same line as the first constraint if separated by at least one space. Now you can type in the constraints in the following way:

```     st
x1 + x2 + x3 <= 20
x1 - 3x2 + x3 <= 30
```

3. Bounds

Finally, you must enter the lower and upper bounds on the variables. If no bounds are specified, ILOG CPLEX will automatically set the lower bound to 0 and the upper bound to +ve infinity. You must explicitly enter bounds only when the bounds differ from the default values. In our example, the lower bound on x1 is 0, which is the same as the default. The upper bound 40, however, is not the default, so you must enter it explicitly. You must type bounds on a separate line before you enter the bound information:

```     bounds
x1 <= 40
```

Since the bounds on x2 and x3 are the same as the default bounds, there is no need to enter them.

4. Completion of the Problem Definition

You have finished entering the problem, so to indicate that the problem is complete, type:

```     end
```

on the last line.

CPLEX has a wide range of features for retrieving and displaying problem characteristics (e.g., the binary variables, the bounds, the constraints, and so forth). We refer interested readers to the Online CPLEX manual for a more complete discussion.

Optimal Solution

The optimal solution that CPLEX computes and returns is:

 ``` Variables: x1, ..... , xn ```

Simple Example

CPLEX is available on the ISR solaris (UNIX) workstations. To access CPLEX, type:

```   prompt >>  tap cplex
prompt >>  cplex
```

Now suppose that we create an input file "example1" containing a description of the linear programming problem:

```    MAX
X - 3Y
ST
A:  -X + Y  <= 3.5
B:   X + Y  <= 5.5
C:   X + 2Y <= 9.0
D:   X <= 4.5
BOUNDS
X >= 0
Y >= 0
END
```

To load "example1" into CPLEX, we simply type:

```   CPLEX>  read example1 lp
CPLEX>
```

Here, we use the argument "lp" to indicate that the file is of type "linear programming."

The following script of code shows commands for displaying the problem parameters, and finding and displaying the optimal solution:

```   CPLEX>  display problem all
Maximize
obj: X - 3 Y
Subject To
A: - X + Y <= 3.5
B: X + Y <= 5.5
C: X + 2 Y <= 9
D: X <= 4.5
Bounds
All variables are >= 0.
CPLEX>
CPLEX>  optimize
Tried aggregator 1 time.
LP Presolve eliminated 4 rows and 2 columns.
All rows and columns eliminated.
Presolve time =    0.00 sec.
Dual simplex - Optimal:  Objective =    4.5000000000e+00
Solution time =    0.00 sec.  Iterations = 0 (0)
CPLEX>
CPLEX>  display solution variables
Display values of which variable(s):  X
Variable Name           Solution Value
X                             4.500000
CPLEX>  display solution variables
Display values of which variable(s):  Y
The variable 'Y' is 0.
CPLEX>
CPLEX>  display solution variables X-Y
Variable Name           Solution Value
X                             4.500000
All other variables in the range 1-2 are zero.
CPLEX>
```

The optimal solution is at (x,y) coordinate ( 4.5, 0), where the objective function equals 4.5 - 0 = 4.5.