# Production Planning Problems: Models and
Software

## Laurence Wolsey, University of Utrecht, on leave from CORE, Universite
Catholique de Louvain, Belgium

Download slides here (gzipped PostScript file).

This talk is on modeling and solving the capacitated economic lot-sizing
problem that arises in production planning.

The problem is to find a production plan that will meet the demand
*d*_{t} > 0 of a product over *NT* time periods. There
exists a capacity limit *c*_{t} on the amount of production in
period *t*. If we produce *x*_{t} units in period
*t*, a fixed set-up cost *f*_{t} and a variable
production cost *p*_{t} x_{t} is charged. In addition,
if *s*_{t} units are kept in stock from period *t* to
*t+1*, an inventory holding cost of *h*_{t} s_{t}
is incurred. We would like find a production plan that minimizes the total
production and inventory costs.

A fixed charge network flow formulation of the problem is given in
Slide 4. The network has a source node for total production and a sink
node for each demand period.

Different versions of the problem are classified and a generic formulation
is given in Slides 6 and 7. The generic model handles multiple
products and backlogging of a product (negative inventory).

The motivation for developing a modeling and branch-cut system specifically
for the lot-sizing problem is to make use of the knowledge created by
research over the last 15 years. EMOSL (Extended Modelling and Subroutine
Library) of XPRESS enables the user to create a model in mps format by
using a language based on lot-sizing terminology. Routines to generate
valid inequalities, seperation routines and specialized heuristics are
included in the optimizer module.

The system has been tested on a testbed of various types of lot-sizing
problems. An order of 10 speedup in run time is achieved in some of the
test problems.