Approximating the solution of a dynamic, stochastic multiple knapsack problem

The model is analysed of an environment where orders arrive probabilistically over time, with their revenues and capacity requirements becoming known upon arrival. The decision is whether to accept an order, receiving a reward and reserving capacity, or reject an order, freeing capacity for possible future arrivals. The dynamic, stochastic multiple knapsack problem is modelled with stochastic dynamic programming (SDP). Multiple knapsacks are used as orders may stay in the system for multiple periods. As the state space grows exponentially in the number of knapsacks and the number of possible orders per period, linear programming and duality are utilised to quickly approximate the end-of-horizon values for the SDP. This helps mitigate end-of-study effects when solving the SDP directly, allowing for the solution of larger problems and leading to increased quality in solutions.