A combinatorial approach to a class of parallel-machine, continuous-time scheduling problems

The paper analyzes a manufacturing system with N non-identical, parallel machines continuously producing one product type in response to its demand. Inventory and backlog costs are incurred when tracking the demand results in inventory surpluses and shortages respectively. In addition, the production cost of a machine is incurred when the machine is not idle. The objective is to determine machine production rates so that the inventory, backlog, and production costs are minimized. For problems with demand defined as an arbitrary function of time, numerical methods are suggested to approximate an optimal solution. The complexity of the approximation methods is polynomial, while finding an exact optimal solution requires exponential time. In a case when production is to cope with a special form of a single-mode, K-level piece-wise constant demand, we prove, with the aid of the maximum principle, that the exact optimal solution can be found as a combination of analytical and combinatorial tools in O(KN²(max{K,2N})²) time.