Order fulfilment systems are forced to manage a volatile customer demand while meeting customer-required short order deadlines. To handle these challenges, we introduce the Strategy of Levelled Order Release (LOR) for workload balancing over time. The contributions of this work are (1) the workload balancing concept LOR, (2) a discrete-time Markov chain for performance analysis, and (3) an algorithm for capacity planning under performance constraints in order fulfilment systems with LOR.

In this thesis, we study a series of closely related multi-stage stochastic programming models in production planning, from both a modeling and an algorithmic point of view. We first consider a very simple multi-stage stochastic lot-sizing problem, involving a single item with no fixed charge and capacity constraint. Although a multi-stage stochastic integer program, this problem can be shown to have a totally unimodular constraint matrix. We develop primal and dual algorithms by exploiting the problem structure. Both algorithms are strongly polynomial, and therefore much more efficient than the Simplex method. Next, motivated by applications in semiconductor tool planning, we develop a general capacity planning problem under uncertainty. Using a scenario tree to model the evolution of the uncertainties, we present a multi-stage stochastic integer programming formulation for the problem. In contrast to earlier two-stage approaches, the multi-stage model allows for revision of the capacity expansion plan as more information regarding the uncertainties is revealed. We provide analytical bounds for the value of multi-stage stochastic programming over the two-stage approach. By exploiting the special simple stochastic lot-sizing substructure inherent in the problem, we design an efficient approximation scheme and show that the proposed scheme is asymptotically optimal. We conduct a computational study with respect to a semiconductor-tool-planning problem. Numerical results indicate that even an approximate solution to the multi-stage model is far superior to any optimal solution to the two-stage model. These results show that the value of multi-stage stochastic programming for this class of problem is extremely high. Next, we extend the simple stochastic lot-sizing model to an infinite horizon problem to study the planning horizon of this problem. We show that an optimal solution of the infinite horizon problem can be approximated by optimal solutions of a series of finite horizon problems, which implies the existence of a planning horizon. We also provide a useful upper bound for the planning horizon.

Manufacturing and service firms use capacity planning to match capacity with demand. It is usually subject to stochastic and time-dependent variability in demand and capacity. This dissertation investigates how to set capacity in stochastic and time-dependent operations systems and analyzes the benefit of incorporating future demand changes in the decision making. The book contains the following three scientific essays that constitute the dissertation of Jannik Vogel. [1] Quality-speed trade-offs in dynamic service systems: Do future demand changes matter? [2] Planning capacity in stochastic MTO production systems with time-dependent demand: The clear the queue effect [3] The optimal time to adapt the processing rate in a make-to-order production system

In batch production systems with zero fulfillment time, inventory on all items has to be maintained to respond to stochastic customer demand. Meeting a desired service level would require high safety stock levels. Cyclical inventory cost is charged to each make-to-stock (MTS) item. These two types of costs lead to an expensive inventory holding policy. However if customer contracts can be revised with positive fulfillment time, there is an opportunity to reduce both types of inventory costs. The positive order fulfillment time allows the manufacturer to reduce safety stock by scheduling production in response to arriving orders and to reduce cyclical inventory cost by converting some items to make-to-order (MTO). This research considers a formal approach to determine a policy to minimize the inventory cost with desired service level and positive order fulfillment time. Our production system is a single server multi-item system with constant order fulfillment time and homogeneous stochastic processing time, hence, it can be modeled as an M/G/1 queue. In addition, the queue operates on a first-come-first-serve (FCFS) scheduling rule. For this type of system, we derive the conditions under which a pure MTO policy can be applied. We also use a mixed-integer nonlinear programming model to determine a combined MTS/MTO policy when pure MTO is not feasible. In terms of this combined MTO/MTS policy, through numerical study we show the percentage of items that will be MTO is dependent on initial system load and the order fulfilment time. We also find out that we expect to save more, from this combined MTO/MTS, facing a system with a small initial load and a long order fulfillment time. Moreover, an item with small arrival rate, large order size mean and large unit time holding cost is more likely to be MTO. This should result in insights for a manufacturer using combined MTS/MTO policy with positive order fulfillment time.

With supply chains becoming increasingly extended, the uncertainties in the upstream production process can greatly affect the material flows that aim toward meeting the uncertain demands at the downstream. We analyze a two-location system in which the upstream production facility experiences random capacities and the downstream store faces random demands. Different from the widely used approach that seeks the decomposition of the profit function based on the echelon inventories, our approach builds on the notions of stochastic functions, in particular, the stochastic linearity in midpoint and the directionally concave order. With these notions, we establish the concavity and submodularity of the profit functions in the transformed decision variables. In general, it is optimal to follow a two-level state-dependent threshold policy such that an order is issued at a location if and only if the inventory position of that location is below the corresponding threshold. In the special case where the salvage values are linear in the ending inventories, the profit function becomes separable in the inventory positions, and the optimal policy reduces to the echelon base-stock policy. The effect of the uncertain capacity and demand depends critically on whether the production capacity is limited or ample in relation to the demand. Only when the capacity and the demand do not differ much, the upstream facility carries positive inventory; otherwise, all units produced are shipped immediately toward the downstream. We further extend our analysis to systems with general stochastic production functions and with multiple locations.

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