Capacity planning in networks of queues with manufacturing applications

This paper addresses capacity planning in systems that can be modeled as a network of queues. More specifically, we present an optimization model and solution methods for the minimum cost selection of capacity at each node in the network such that a set of system performance constraints is satisfied. Capacity is controlled through the mean service rate at each node. To illustrate the approach and how queueing theory can be used to measure system performance, we discuss a manufacturing model that includes upper limits on product throughput times and work-in-process in the system. Methods for solving capacity planning problems with continuous and discrete capacity options are discussed. We focus primarily on the discrete case with a concave cost function, allowing fixed charges and costs exhibiting economies of scale with respect to capacity to be handled.     

Capacity planning in manufacturing and computer networks

This paper addresses capacity planning in manufacturing and computer networks. More specifically, given a manufacturing or computer system modeled as a network of queues, we consider the minimum cost selection of capacity levels from a discrete set of choices such that a single system performance constraint is satisfied. We focus on settings where the cost of obtaining capacity is a concave function, allowing fixed charges and economies of scale to be handled. To solve this class of capacity planning problems, we present a branch and bound algorithm that globally minimizes a concave cost function over a single convex nonlinear performance constraint and lower and upper bounds on the discrete capacity variables. We also present reoptimization procedures that allow the subproblems to be solved more efficiently. Computational results with the algorithm are reported.     

An extended enterprise planning methodology for the discrete manufacturing industry

Extended factories consisting of geographically dispersed independent production facilities are already a reality in the global economy. Production facilities concentrate on core technologies and create partner networks for the manufacturing of their products, a trend initially visible in semiconductor manufacturing but quickly spreading to other industries. A methodology, more flexible and efficient than the traditional time-bucket-based techniques and dynamic dispatching heuristics, to plan the Extended Semiconductor Enterprise and schedule work at the different production entities is presented in this paper. The generic approach also opens opportunities for applications in other discrete manufacturing industries. The methodology uses stepwise search procedures to improve plans and make-or-buy decision processes to solve resource constraints. Focus of the paper is principally on resource scheduling and less on logistics and distribution topics.     

Price-based planning and scheduling of multiproduct orders in symbiotic manufacturing networks

This paper addresses the problem of planning and scheduling operations when processing a multiproduct order in a Symbiotic Manufacturing Network (SMN). The order is characterized for each of its products, by linear routings of operations which can be performed in parallel by different contracting firms of the network. We first present a characterization of the decision process for planning and scheduling operations in SMN. Next, we propose a bidding scheme expressing pricetime trade-off and a multi-commodity network model that can be used to plan and schedule many types of order-based customized production. Finally, an illustrative example is detailed to permit full comprehension of the approach.     

Capacity planning for phased implementation of flexible manufacturing systems under budget restrictions

We consider a problem of gradually replacing conventional dedicated machines with flexible manufacturing modules (FMMs) under budget restrictions over a finite planning horizon assuming that dedicated machines cannot be purchased during the planning horizon and acquired FMMs are kept until the end of the horizon. In the problem, a replacement schedule is to be determined and operations are to be assigned to the FMMs or the dedicated machines with the objective of minimizing the sum of discounted costs of acquisition and operation of FMMs and operation costs of conventional dedicated machines. In this research, the problem is formulated as a mixed integer linear program and solved by a Lagrangean relaxation approach. A subgradient optimization method is employed to obtain lower bounds of solutions and a multiplier adjustment method is devised to improve the lower bounds. We develop a linear programming-based Lagrangean heuristic algorithm to find a good feasible solution of the original problem in a reasonable amount of computation time. The algorithm is tested on randomly generated test problems and the results are reported.     

Capacity planning with congestion effects

Important operational performance measures for a successful firm include not only price and quality, but also fast and on time delivery of customer orders. Capacity is a key issue in determining the lead time from customer order to delivery. However, capacity planning models seldom consider the impact of capacity levels on lead time performance. An important characteristic of this paper is the incorporation of congestion effects and their impact on lead time in making capacity acquisition decisions. It is especially important in a make-to-order environment, where customer orders arrive randomly and lead to high variability and congestion. This work was motivated by our observations of such tradeoffs at firms in several industries. We present a model to make equipment choice decisions in a multi-product, multi-machine, and single-stage production environment with congestion effects. The model is a nonlinear integer program. We present a heuristic solution procedure for this problem, which is based on a lower bound for the formulation that can be solved efficiently. The computational study shows that the solution procedure is quite effective in solving industry size problems. We illustrate the application of the model using data from a chemical-testing laboratory. We also discuss various extensions of the model.     

Hedging of game options in discrete markets with transaction costs

Game options introduced in [10] in 2000 were studied, by now, mostly in frictionless both complete and incomplete markets. In complete markets the fair price of a game option coincides with the value of an appropriate Dynkin's game, whereas in markets with friction and in incomplete ones there is a range of arbitrage free prices and superhedging comes into the picture. Here we consider game options in general discrete time markets with transaction costs and construct backward and forward induction algorithms for the computation of their prices and superhedging strategies from both seller's (upper arbitrage free price) and buyer's (lower arbitrage free price) points of view extending to the game options case most of the results from [12].     

Pricing catastrophe options in discrete operational time

We employ a doubly-binomial process as in Gerber [Gerber, H.U., 1988. Mathematical fun with the compound binomial process. ASTIN Bull. 18, 161-168] to discretize and generalize the continuous “randomized operational time” model of Chang et al. ([Chang, C.W., Chang, J.S.K., Yu, M.T., 1996. Pricing catastrophe insurance futures call spreads: A randomized operational time approach. J. Risk Insurance 63, 599-616] and CCY hereafter) from a complete-market continuous-time setting to an incomplete-market discrete-time setting, so as to price a richer set of catastrophe (CAT) options. For futures options, we derive the equivalent martingale probability measures by benchmarking to the shadow price of a bond to span arrival uncertainty, and the underlying futures price to span price uncertainty. With a time change from calendar time to the operational transaction-time dimension, we derive CCY as a limiting case under risk-neutrality when both calendar-time and transaction-time intervals shrink to zero. For a cash option with non-traded underlying loss index, we benchmark to the market reinsurance premiums to span claim uncertainty, and with a time change to claim time, we derive the cash option price as a binomial sum of claim-time binomial Asian option prices under the martingale measures.     

Pricing catastrophe options in discrete operational time

We employ a doubly-binomial process as in Gerber [Gerber, H.U., 1988. Mathematical fun with the compound binomial process. ASTIN Bull. 18, 161–168] to discretize and generalize the continuous “randomized operational time” model of Chang et al. ([Chang, C.W., Chang, J.S.K., Yu, M.T., 1996. Pricing catastrophe insurance futures call spreads: A randomized operational time approach. J. Risk Insurance 63, 599–616] and CCY hereafter) from a complete-market continuous-time setting to an incomplete-market discrete-time setting, so as to price a richer set of catastrophe (CAT) options. For futures options, we derive the equivalent martingale probability measures by benchmarking to the shadow price of a bond to span arrival uncertainty, and the underlying futures price to span price uncertainty. With a time change from calendar time to the operational transaction-time dimension, we derive CCY as a limiting case under risk-neutrality when both calendar-time and transaction-time intervals shrink to zero. For a cash option with non-traded underlying loss index, we benchmark to the market reinsurance premiums to span claim uncertainty, and with a time change to claim time, we derive the cash option price as a binomial sum of claim-time binomial Asian option prices under the martingale measures.     

Continuity correction for discrete barrier options with two barriers

Discrete barrier options are the options whose payoffs are determined by underlying prices at a finite set of times. We consider the discrete barrier option with two barriers. Broadie et al. (1997) [16] proposed a continuity correction for the discretely monitored barrier option. We extend this idea to barrier option with two barriers. The proof for discrete chained barrier option is provided and numerical results show the continuity correction approximation is remarkably accurate.     

Efficient pricing of discrete Asian options

Asian options are popular path-dependent financial derivatives. This paper uses lattices to price fixed-strike European-style Asian options that are discretely monitored. The algorithm proposed can also be applied to floating-strike Asian options as well because fixed-strike and floating-strike Asian options are related through an equation. The discretely monitored version is usually found in practice instead of the continuously monitored version usually encountered in the literature. This paper presents the first provably quadratic-time convergent lattice algorithm for pricing fixed-strike European-style discretely monitored Asian options. It is the most efficient lattice algorithm with convergence guarantees. The algorithm relies on the Lagrange multipliers to choose the number of states for each node of the lattice. Extensive numerical experiments and comparisons with many existing numerical methods confirm the performance claims and the competitiveness of our algorithm. This result places fixed-strike European-style discretely monitored Asian options in the same complexity class as vanilla options.     

Capacity planning with working time accounts in services

This paper presents, formalizes and classifies working time accounts (WTAs), a means of achieving flexible production capacity through the flexible organization of working time. We propose linear programming models to plan working time, using WTA, at companies in the service industry. A computational experiment shows that the models can be solved for any capacity planning problems of reasonable size.     

Numerical valuation of discrete double barrier options

In the present paper we explore the problem for pricing discrete barrier options utilizing the Black-Scholes model for the random movement of the asset price. We postulate the problem as a path integral calculation by choosing approach that is similar to the quadrature method. Thus, the problem is reduced to the estimation of a multi-dimensional integral whose dimension corresponds to the number of the monitoring dates.We propose a fast and accurate numerical algorithm for its valuation. Our results for pricing discretely monitored one and double barrier options are in agreement with those obtained by other numerical and analytical methods in Finance and literature. A desired level of accuracy is very fast achieved for values of the underlying asset close to the strike price or the barriers.The method has a simple computer implementation and it permits observing the entire life of the option.     

Managing manufacturing risks by using capacity options

In this study, we investigate the strategy of increasing production capacity temporarily through contingent contractual agreements with short-cycle manufacturers to manage the risks associated with demand volatility. We view all these agreements as capacity options. More specifically, we consider a manufacturing company that produces a replenishment product that is sold at a retailer. The demand for the product switches randomly between a high level and a low level. The production system has enough capacity to meet the demand in the long run. However, when the demand is high, it does not have enough capacity to meet the instantaneous demand and thus has to produce to stock in advance. Alternatively, a contractual agreement with a short-cycle manufacturer can be made. This option gives the right to receive additional production capacity when needed. There is a fixed cost to purchase this option for a period of time and, if the option is exercised, there is an additional per unit exercise price which corresponds to the cost of the goods produced at the short-cycle manufacturer. We formulate the problem as a stochastic optimal control problem and analyse it analytically. By comparing the costs between two cases where the contract with the short-cycle manufacturer is used or not, the value of this option is evaluated. Furthermore, the effect of demand variability on this contract is investigated.     

A stochastic production planning problem in hybrid manufacturing and remanufacturing systems with resource capacity planning

DIRECT is derivative-free global-search algorithm has been found to perform robustly across a wide variety of low-dimensional test problems. The reason DIRECT’s robustness is its lack of algorithmic parameters that need be “tuned” to make the algorithm perform well. In particular, there is no parameter that determines the relative emphasis on global versus local search. Unfortunately, the same algorithmic features that enable DIRECT to perform so robustly have a negative side effect. In particular, DIRECT is usually quick to get close to the global minimum, but very slow to refine the solution to high accuracy. This is what we call DIRECT’s “eventually inefficient behavior.” In this paper, we outline two root causes for this undesirable behavior and propose modifications to eliminate it. The paper builds upon our previously published “MrDIRECT” algorithm, which we can now show only addressed the first root cause of the “eventually inefficient behavior.” The key contribution of the current paper is a further enhancement that allows MrDIRECT to address the second root cause as well. To demonstrate the effectiveness of the enhanced MrDIRECT, we have identified a set of test functions that highlight different situations in which DIRECT has convergence issues. Extensive numerical work with this test suite demonstrates that the enhanced version of MrDIRECT does indeed improve the convergence rate of DIRECT.     

Production planning under uncertainty in textile manufacturing

Textile manufacturing consists of yarn production, fabric formation, and finishing and dyeing stages. The subject of this paper is the yarn production planning problem, although the approach is directly applicable to the fabric production planning problem due to similarities in the respective models. Our experience at an international textile manufacturer indicates that demand uncertainty is a major challenge in developing yarn production plans. We develop a stochastic programming model that explicitly includes uncertainty in the form of discrete demand scenarios. This results in a large-scale mixed integer model that is difficult to solve with off-the-shelf commercial solvers. We develop a two-step preprocessing algorithm that improves the linear programming relaxation of the model and reduces its size, consequently improving the computational requirements. We illustrate the benefits of a stochastic programming approach over a deterministic model and share our initial application experience.     

Minimax production planning in failure-prone manufacturing systems

In this paper, we consider a minimax production planning model of a flexible manufacturing system with machines that are subject to random breakdown and repair. The objective is to choose the rate of production that minimizes the related minimax cost of production and inventory/shortage. The value function is shown to be the unique viscosity solution to the associated Hamilton-Jacobi-Isaacs equation. Under certain conditions, it is shown that the value function is continuously differentiable. A verification theorem is given to provide a sufficient condition for optimal control. Finally, two examples are solved explicitly.This research was supported by the Natural Sciences and Engineering Research Council of Canada under Grants OGP0036444 and A4169.     

Pricing of general European options on discrete dividend-paying assets with jump-diffusion dynamics

Stocks regularly pay dividends at discrete intervals of time while statistical evidence indicates the existence of small “jumps” in the stock price dynamics. In this paper, we find closed-form solutions for the valuation of European options when the underlying asset is modeled by a jump-diffusion process and pays discrete or continuous dividends. The formula is very general and can be used with any specification on the distribution of the jump. Moreover, the formula is written in terms of the Black–Scholes formula with no jumps or dividends and thus indicates the effect of the jumps and the effect of the inclusion of discrete (or continuous) dividends on the price of the option.     

Robust production planning in a manufacturing environment with random yield: A case in sawmill production planning

This paper addresses a multi-period, multi-product sawmill production planning problem where the yields of processes are random variables due to non-homogeneous quality of raw materials (logs). In order to determine the production plans with robust customer service level, robust optimization approach is applied. Two robust optimization models with different variability measures are proposed, which can be selected based on the tradeoff between the expected backorder/inventory cost and the decision maker risk aversion level about the variability of customer service level. The implementation results of the proposed approach for a realistic-scale sawmill example highlights the significance of using robust optimization in generating more robust production plans in the uncertain environments compared with stochastic programming.     

Capacity switching options under rivalry and uncertainty

Deregulated infrastructure industries exhibit stiff competition for market share. Firms may be able to limit the effects of competition by launching new projects in stages. Using a two-stage real options model, we explore the value of such flexibility. We first demonstrate that the value of investing in a sequential manner for a monopolist is positive but decreases with uncertainty. Next, we find that a typical duopoly firm’s value relative to a monopolist’s decreases with uncertainty as long as the loss in market share is high. Intriguingly, this result is reversed for a low loss in market share. We finally show that this loss in value is reduced if a firm invests in a sequential manner and specify the conditions under which sequential capacity expansion is more valuable for a duopolist firm than for a monopolist.