# Common Lisp Linear Programming

This is a Common Lisp library for solving linear programming problems. It’s designed to provide a high-level and ergonomic API for specifying linear programming problems as lisp expressions.

The core library is implemented purely in Common Lisp with only a few community-standard libraries as dependencies (ASDF, Alexandria, Iterate). However, the solver is designed to support alternative backends without any change to the user’s code. Currently, there is a backend for the GNU Linear Programming Kit (GLPK).

## Installation

The linear-programming library is avalible in both the main Quicklisp distribution and Ultralisp, so it can loaded with with `(ql:quickload :linear-programming)`

.
You can check that it works by running `(asdf:test-system :linear-programming)`

.

If you are not using Quicklisp, place this repository, Alexandria, and Iterate somewhere where ASDF can find them.
Then, it can be loaded with `(asdf:load-system :linear-programming)`

and tested as above.

## Usage

See neil-lindquist.github.io/linear-programming/ for further documentation.

Consider the following linear programming problem.

maximize x + 4y + 3z

such that

- 2x + y ≤ 8
- y + z ≤ 7
- x, y, z ≥ 0

First, the problem needs to be specified. Problems are specified with a simple DSL, as described in the syntax reference.

```
(use-package :linear-programming)
(defvar problem (parse-linear-problem '(max (= w (+ x (* 4 y) (* 3 z))))
'((<= (+ (* 2 x) y) 8)
(<= (+ y z) 7))))
```

Once the problem is created, it can be solved with the simplex method.

```
(defvar solution (solve-problem problem))
```

Finally, the optimal tableau can be inspected to get the resulting objective function, decision variables, and reduced-costs (i.e. the shadow prices for the variable’s lower bounds).

```
(format t "Objective value solution: ~A~%" (solution-variable solution 'w))
(format t "x = ~A (reduced cost: ~A)~%" (solution-variable solution 'x) (solution-reduced-cost solution 'x))
(format t "y = ~A (reduced cost: ~A)~%" (solution-variable solution 'y) (solution-reduced-cost solution 'y))
(format t "z = ~A (reduced cost: ~A)~%" (solution-variable solution 'z) (solution-reduced-cost solution 'z))
;; ==>
;; Objective value solution: 57/2
;; x = 1/2 (reduced cost: 0)
;; y = 7 (reduced cost: 0)
;; z = 0 (reduced cost: 1/2)
```

Alternatively, the `with-solution-variables`

and `with-solved-problem`

macros simplify some steps and binds the solution variables in their bodies.

```
(with-solution-variables (w x y z) solution
(format t "Objective value solution: ~A~%" w)
(format t "x = ~A (reduced cost: ~A)~%" x (reduced-cost x))
(format t "y = ~A (reduced cost: ~A)~%" y (reduced-cost y))
(format t "z = ~A (reduced cost: ~A)~%" z (reduced-cost z)))
;; ==>
;; Objective value solution: 57/2
;; x = 1/2 (reduced cost: 0)
;; y = 7 (reduced cost: 0)
;; z = 0 (reduced cost: 1/2)
(with-solved-problem ((max (= w (+ x (* 4 y) (* 3 z))))
(<= (+ (* 2 x) y) 8)
(<= (+ y z) 7))
(format t "Objective value solution: ~A~%" w)
(format t "x = ~A (reduced cost: ~A)~%" x (reduced-cost x))
(format t "y = ~A (reduced cost: ~A)~%" y (reduced-cost y))
(format t "z = ~A (reduced cost: ~A)~%" z (reduced-cost z)))
;; ==>
;; Objective value solution: 57/2
;; x = 1/2 (reduced cost: 0)
;; y = 7 (reduced cost: 0)
;; z = 0 (reduced cost: 1/2)
```