Mixing network and LP

 

It is possible to mix LP and network sections. This is of interest mostly when assembling a model from pieces defined independently, but can be used in other contexts, for instance, to save space.

 

The problem in this section was inspired by work done by Wallace [1]. It concerns a hypothetical fish processing company with several processing plants and a fleet that can fish in different locations. The aim is to expand capacity of both the fleet and the production facilities, subject to a joint budget constraint, to send the fleet to the locations, land the ensuing catch, and finally to process the catch into a number of products. The objective is to minimize net cost, which is subject to uncertainty on both the supply side (availability of fish) and the demand side (price to customers).

 

This can be formulated mathematically as follows.


 

 

where

 

F = {1,…,F}   is the set of factories

M = {0,…,M}is the set of resources; 0 represents the fishing fleet

G = {1,…,G}  is the set of fishing grounds

Q = {1,…,Q}   is the set of products

N = {0,…,N}  is the set of nodes; 0 is the root node, {1,…,N1} are the second stage nodes, the remainder are third stage nodes

im                     is the existing capacity of resource m

em                     is the cost of adding one unit of capacity of resource m

xm                    is the capacity of resource m added in the first stage

cfg                    is the cost of sending one unit of fishing capacity from plant f to fishing ground g

xfg                    is the amount of fishing capacity sent from plant f to fishing ground g         

tgj                     is the cost of relocating one unit of fishing capacity from fishing ground g to fishing ground j

ygjn                   is the amount of fishing capacity relocated from fishing ground g to fishing ground j in node n

rgf                     is the cost of returning one unit of fishing capacity from fishing ground g to plant f

vgfn                   is the amount of fishing capacity returned from fishing ground g to plant f in node n

hfqn                   is the net profit obtained from one unit of product q produced at plant f in node n

afqm                  is the amount of resource m needed to produce one unit of product q at plant f

zfqn                   is the amount of product q produced at plant f in node n

B                      is the available budget

sgn                    is the amount of fish available at fishing ground g in node n

pn                    is the (path) probability of reaching node n

p(n)                  is the predecessor of node n in the event tree

 

 

 

Time file

Core file

Stoch file

 

 

Reference

1. S.W. Wallace, “Solving stochastic programs with network recourse”, Networks 16 (1986) 295–317.