The stock size problem

Consider a number of jobs that have to be completed within some fixed period of time. Each job consumes or supplies a certain, job-dependent quantity of a resource thus changing the contents of a stock. Natural examples for this scenario arise in the case of trucks delivering goods to a trade house and taking other goods away from it, or in the case where processes change their environment. The goal is to find an ordering of the jobs such that all jobs can be completed and such that the maximum stock size needed over the time period becomes minimum. Since this problem can be shown to be NP-hard, we present three polynomial time approximation algorithms which yield job sequences with maximum stock size at most 2, respectively 8/5 times and 3/2 times the optimum stock size. These approximation algorithms are also compared by a numerical simulation. Moreoever, we consider a variation of the problem where consecutive processes are allowed to directly exchange resources without using the stock.