An upper bound for the competitive location and capacity choice problem with multiple demand scenarios

A new mathematical model is considered related to competitive location problems where two competing parties, the Leader and the Follower, successively open their facilities and try to win customers. In the model, we consider a situation of several alternative demand scenarios which differ by the composition of customers and their preferences.We assume that the costs of opening a facility depend on its capacity; therefore, the Leader, making decisions on the placement of facilities, must determine their capacities taking into account all possible demand scenarios and the response of the Follower. For the bilevel model suggested, a problem of finding an optimistic optimal solution is formulated. We show that this problem can be represented as a problem of maximizing a pseudo- Boolean function with the number of variables equal to the number of possible locations of the Leader’s facilities.We propose a novel systemof estimating the subsets that allows us to supplement the estimating problems, used to calculate the upper bounds for the constructed pseudo-Boolean function, with additional constraints which improve the upper bounds.     

A Simple Model for Computing the Trade-Offs between Plant Capacity and Operating Strategies

A method is presented for determining the optimal capacity of a production system which encounters a strong seasonal demand for its output. A model is developed for analysing the trade-offs between plant capacity and different production strategies. The production strategies are computed by linear programming for a range of plant capacities and a given demand pattern. The annual capital costs of the plant are calculated as a function of plant capacity and the two costs are added in order to determine the plant capacity corresponding to the least cost.     

Price setting policies for service systems in case of uncertain demand and service time

Building anM/M/1 queueing model, we determine price and capacity of a service system within monopolistic and competitive frameworks.The existence of a solution for a monopolist is shown and optimal capacity is determined, if demand is elastic.We also determine, within a competitive framework, the number of firms, outputs of each per unit of time, and the relation between price and the number of firms for elastic demands (which has a maximum).Finally, we show that competition supplies a much larger output than monopoly at the price set by the monopolist, if elasticity equals 2.     

3.1. OR in banking

In this paper, we describe a deterministic multiperiod capacity expansion model in which a single facility serves the demand for many products. Potential applications for the model can be found in the capacity expansion planning of communication systems as well as in the production planning of heavy process industries. The model assumes that each capacity unit simultaneously serves a prespecified (though not necessarily integer) number of demand units of each product. Costs considered include capacity expansion costs, idle capacity holding costs, and capacity shortage costs. All cost functions are assumed to be nondecreasing and concave. Given the demand for each product over the planning horizon, the objective is to find the capacity expansion policy that minimizes the total cost incurred. We develop a dynamic programming algorithm that finds optimal policies. The required computational effort is a polynomial function of the number of products and the number of time periods. When the number of products equals one, the algorithm reduces to the well-known algorithm for the classical dynamic lot size problem.     

Global Solution Approach for a Nonconvex MINLP Problem in Product Portfolio Optimization

The rigorous and efficient determination of the global solution of a nonconvex MINLP problem arising from product portfolio optimization introduced by Kallrath (2003) is addressed. The objective of the optimization problem is to determine the optimal number and capacity of reactors satisfying the demand and leading to a minimal total cost. Based on the model developed by Kallrath (2003), an improved formulation is proposed, which consists of a concave objective function and linear constraints with binary and continuous variables. A variety of techniques are developed to tighten the model and accelerate the convergence to the optimal solution. A customized branch and bound approach that exploits the special mathematical structure is proposed to solve the model to global optimality. Computational results for two case studies are presented. In both case studies, the global solutions are obtained and proved optimal very efficiently in contrast to available commercial MINLP solvers.     

Optimization models for assessing the peak capacity utilization of intelligent transportation systems

With limited economic and physical resources, it is not feasible to continually expand transportation infrastructure to adequately support the rapid growth in its usage. This is especially true for traffic coordination systems where the expansion of road infrastructure has not been able to keep pace with the increasing number of vehicles, thereby resulting in congestion and delays. Hence, in addition to striving for the construction of new roads, it is imperative to develop new intelligent transportation management and coordination systems. The effectiveness of a new technique can be evaluated by comparing it with the optimal capacity utilization. If this comparison indicates that substantial improvements are possible, then the cost of developing and deploying an intelligent traffic system can be justified. Moreover, developing an optimization model can also help in capacity planning. For instance, at a given level of demand, if the optimal solution worsens significantly, this implies that no amount of intelligent strategies can handle this demand, and expanding the infrastructure would be the only alternative. In this paper, we demonstrate these concepts through a case study of scheduling vehicles on a grid of intersecting roads. We develop two optimization models namely, the mixed integer programming model and the space-time network flow model, and show that the latter model is substantially more effective. Moreover, we prove that the problem is strongly NP-hard and develop two polynomial-time heuristics. The heuristic solutions are then compared with the optimal capacity utilization obtained using the space-time network model. We also present important managerial implications.     

Flexible capacity strategy with multiple market periods under demand uncertainty and investment constraint

We establish a flexible capacity strategy model with multiple market periods under demand uncertainty and investment constraints. In the model, a firm makes its capacity decision under a financial budget constraint at the beginning of the planning horizon which embraces n market periods. In each market period, the firm goes through three decision-making stages: the safety production stage, the additional production stage and the optimal sales stage. We formulate the problem and obtain the optimal capacity, the optimal safety production, the optimal additional production and the optimal sales of each market period under different situations. We find that there are two thresholds for the unit capacity cost. When the capacity cost is very low, the optimal capacity is determined by its financial budget; when the capacity cost is very high, the firm keeps its optimal capacity at its safety production level; and when the cost is in between of the two thresholds, the optimal capacity is determined by the capacity cost, the number of market periods and the unit cost of additional production. Further, we explore the endogenous safety production level. We verify the conditions under which the firm has different optimal safety production levels. Finally, we prove that the firm can benefit from the investment only when the designed planning horizon is longer than a threshold. Moreover, we also derive the formulae for the above three thresholds.     

Wireless network capacity management: A real options approach

This paper applies financial option valuation methods to new wireless network capacity investment decision timing. In particular, we consider the case of network capacity for cellular telephone service. Given a cluster of base stations (with a certain traffic capacity per base station), we determine when it is optimal to increase capacity for each of the base stations contained in the cluster. We express this in terms of the fraction of total cluster capacity in use, i.e. we calculate the optimal time to upgrade in terms of the ratio of observed usage to existing capacity. We study the optimal decision problem of adding new capacity in the presence of stochastic wireless demand for services. A four factor algorithm is developed, based on a real options formulation. Numerical examples are provided to illustrate various aspects of the model.     

Robust supply chain network design with multi-products for a company in the food sector

The paper aims to solve a problem faced by a company competing in the snacks market in Turkey. In line with the growth in this market, the company needs to make important decisions over the next few years about the timing and location of a new plant, its initial capacity, the timing and amount of additional capacity to be installed at the new and existing plants, the assignment of demand points to plants and the amount of raw materials to be shipped from suppliers to the plants in each period. The objective is to minimize the total cost of various components. The problem is formulated as a multi-period supply chain network design model with multi products. The resulting mixed-integer linear programming model is solved by the commercial solver CPLEX. This model enables us to carry out all analyses requested by the company in an efficient way. After this deterministic model is solved on the basis of a 9% annual increase in demand, it is extended to a minimax regret model to deal with uncertainty in demand quantities. The results suggest that opening the new plant in the city of İzmir is indeed a robust solution that is unaffected in different scenarios that are based on three distinct demand increase rates. Even though the location of the new plant remains unchanged with respect to scenarios, the optimal robust solution differs from the optimal solution of each scenario in terms of the capacity expansion decisions. After all obtained results had been communicated to the company managers and executives, the new plant construction was started in 2016 very close to the city that the mathematical model had determined. The new plant is expected to start operating in 2018.     

Integrating effective flexibility measures into a strategic supply chain planning model

This paper develops models for capacity, product mix, distribution and input supply flexibility and integrates them in a strategic level, mixed integer supply chain (SC) planning model as a way of addressing demand and supply uncertainty, as well as improving market responsiveness. Capacity flexibility is modeled via the SC’s production capacity planning to address budgeted demand and ensure the fulfillment of prospective demand increases when considering various market scenarios. This model selects an optimal number of products from fast moving and extended product range options—based on the product mix flexibility. The model confirms a quick response to a changing marketplace by considering elements like transportation and supply lead time along with the probabilities of stock out options when addressing input supply and distribution flexibility. This paper proposes a solution procedure to solve the model for real world problems, and investigates the sensitivity of the model outputs with respect to changes in flexibility measures.     

Optimal use of hub facilities: A two-hub model with fixed arc costs

Summary We introduce a model of a communication network design problem involving the utilization of hub facilities. That is, for a problem with two sets of customers and no intraset demand we seek to determine how the hub node associated with each set should be utilized. We assume that the only costs are the fixed costs associated with creating each of the three types of connecting arcs. A key parameter is the “group” size which is the number of communication circuits which can be bundled together in an arc. The optimal design depends strongly on how closely the arcs can be filled to capacity. The general demand problem is shown to be NP-Hard. However, for unit demand, we derive an almost “all or nothing” result which specifies that all flow should be direct node-to-node or, on the other hand, all or almost all flow should go via the hubs. Research supported in part by Grant SAB-94-0115 from the Spanish Interministerial Commission of Science and Technology while this author was on sabbatical leave at the Polytechnic University of Catalonia in Barcelona.     

Stockpiling under price uncertainty and storage capacity constraints

A model is presented for the operation of an agent whose responsibility is to purchase and perhaps stockpile sufficient quantities of a certain commodity in order to satisfy an exogenous constant demand per time period. This is the situation faced by a state agency which is responsible for the purchasing of oil products to satisfy the demand of a country in which demand for energy has stabilized at a certain level.An important feature of the model is that the price of the commodity is described by a stochastic process. This reflects the volatility of prices of oil products. Furthermore, the model takes storage capacity constraints explicitly into account, and thus can help to assess the optimal level of storage capacity expansion.The relevant stochastic dynamic programming equations are derived and solved for the least cost function, which turns out to be piecewise linear in the inventory level. The storage capacity enters only in the computation of the constant term of the value function. The solution of the dynamic programming equation leads also to the optimal purchasing strategy of agencies with different levels of flexibility in their policy: in the model, an agency can be allowed or not to resell from the stock and it can have finite or infinite storage capacity.     

An efficiency frontier approach for the design of cellular manufacturing systems in a lumpy demand environment

A new multi-objective approach for the cell formation problem in a lumpy demand environment is presented. The objectives addressed in this paper are grouping efficiency and capacity requirements. In lumpy demand the required capacity is affected by demand variability and the correlation between the part types assigned to the cells. We claim that since the required capacity is determined by part types grouping, part type demands variability and their correlation should be taken into consideration as part of the cell formation. This new approach is discussed and formulated as a mixed integer programming model and illustrated by a wide range of typical examples. These examples demonstrate that when using traditional approaches designers do not obtain optimal solutions and may make decisions on the basis of wrong results. The proposed approach helps designers eliminate these problems and produce a reasonable cell design. A genetic algorithm is proposed and examined for designing large-scale systems.     

The acquisition of new technology and its impact on a firm's competitive position

The strategic decision concerning the optimal and dynamic acquisition of new technology is examined. The model focuses on a profit maximizing firm that optimally derives its price, level of output, and its level and composition of productive capacity over time. The acquisition of new technology and reduction of existing capacity may occur simultaneously, so that the composition of the firm's productive resources may be upgraded over time. It is assumed that the acquisition of new technology causes a reduction in production costs and a direct increase in the firm's demand. The demand experienced by the firm may be directly increased as a result of acquiring new technology due to benefits such as expanded product-mix or volume capabilities, improved quality of output, or improved customer service (shorter production lead time). In addition, it is shown that demand is indirectly increased due to the reduced production costs that enable the firm to charge a lower price. Therefore, the strategic impact of acquiring new technology is captured, since its effect on future demand and the firm's ability to meet the demand are considered. The importance of capturing the increased demand potential offered by the new technology is demonstrated through the analysis of numerical examples. In addition, the effect on the optimal solution caused by a variety of environmental conditions is examined. For example, the impact of technological innovation is observed by defining (i) the cost of acquiring technology as a decreasing function of time, and (ii) the effectiveness of new technology on reducing operating costs as an increasing function of time.     

Optimal diffusion of a new technology when both demand and supply are nonstatic

The investment problem of a monopolized sector selling an innovated product is explored. Learning by doing is supposed to occur on the supply side, while learning by using is introduced to explain demand growth. Pontryagin's maximum principle is applied to the resulting optimal control problem, which includes supply capacity and cumulative output as state variables. The optimal investment policy turns out to be of a very simple form: all profit is retained and invested until capacity achieves its optimal size. In spite of this, the new technology price displays a variety of time patterns that heavily depend on the actual demand and cost conditions, as one would expect in the real world.The authors express their gratitude to Professor Sergio Rinaldi for helpful comments. This work was partially supported by Centro Teoria dei Sistemi, CNR, Milano, Italy.     

A stochastic periodic review integrated inventory model involving defective items, backorder price discount, and variable lead time

The purpose of this article is to investigate a stochastic integrated supplier-retailer inventory problem. The model analyzed in this article explores the problem of the protection interval, the backorder price discount, the lead time, and the numbers of shipments from the supplier to the retailer in one production run as control variables to widen applications for an integrated periodic review inventory model. We consider the situation in which the supplier and the retailer establish a long-term strategic partnership and contract to jointly determine the best strategy. We assume that the protection interval demand follows a normal distribution. Our objective is to determine the optimal review period, the optimal backorder price discount, the optimal lead time, and the optimal number of shipments from the supplier to the retailer in one production run, so that the joint expected annual total cost incurred has the minimum value. Furthermore, an algorithm of finding the optimal solution is developed. Also, the sensitivity analysis included and a numerical example is given to illustrate the results of the proposed model.     

Stochastic multi-site capacity planning of TFT-LCD manufacturing using expected shadow-price based decomposition

This paper presents a stochastic optimization model and efficient decomposition algorithm for multi-site capacity planning under the uncertainty of the TFT-LCD industry. The objective of the stochastic capacity planning is to determine a robust capacity allocation and expansion policy hedged against demand uncertainties because the demand forecasts faced by TFT-LCD manufacturers are usually inaccurate and vary rapidly over time. A two-stage scenario-based stochastic mixed integer programming model that extends the deterministic multi-site capacity planning model proposed by Chen et al. (2010) [1] is developed to discuss the multi-site capacity planning problem in the face of uncertain demands. In addition a three-step methodology is proposed to generate discrete demand scenarios within the stochastic optimization model by approximating the stochastic continuous demand process fitted from the historical data. An expected shadow-price based decomposition, a novel algorithm for the stage decomposition approach, is developed to obtain a near-optimal solution efficiently through iterative procedures and parallel computing. Preliminary computational study shows that the proposed decomposition algorithm successfully addresses the large-scale stochastic capacity planning model in terms of solution quality and computation time. The proposed algorithm also outperforms the plain use of the CPLEX MIP solver as the problem size becomes larger and the number of demand scenarios increases.     

A non-stationary periodic review production–inventory model with uncertain production capacity and uncertain demand

In this paper we consider a nonstationary periodic review dynamic production–inventory model with uncertain production capacity and uncertain demand. The maximum production capacity varies stochastically. It is known that order up-to (or base-stock, critical number) policies are optimal for both finite horizon problems and infinite horizon problems. We obtain upper and lower bounds of the optimal order up-to levels, and show that for an infinite horizon problem the upper and the lower bounds of the optimal order up-to levels for the finite horizon counterparts converge as the planning horizons considered get longer. Furthermore, under mild conditions the differences between the upper and the lower bounds converge exponentially to zero.     

产能分享对设备制造商和用户企业的影响研究

本文建立Stackelberg博弈模型比较了传统制造模式与产能分享模式下设备制造商和设备用户企业的利润,发现设备用户企业总是从产能分享模式中受益,但是订单需求增量不同的企业受益大小不同;平台制定的设备租赁价格对设备制造商的利润和设备用户企业的购买决策都将产生影响,对于任意固定价格的设备,平台的制定的最优租赁价格是唯一的,并且平台的最优利润是设备价格的倒U型函数;产能分享业务的出现对设备制造商产生了损益影响,当平台采取最优的定价策略时,价格相对比较高的设备的购买需求增加,设备制造商从产能分享业务中受益,价格相对比较低的设备的购买需求降低,设备制造商在产能分享业务中利益受损。最后通过算例分析对上述结果进行了验证。     

Multi-resource investment strategies: Operational hedging under demand uncertainty

Consider a firm that markets multiple products, each manufactured using several resources representing various types of capital and labor, and a linear production technology. The firm faces uncertain product demand and has the option to dynamically readjust its resource investment levels, thereby changing the capacities of its linear manufacturing process. The cost to adjust a resource level either up or down is assumed to be linear. The model developed here explicitly incorporates both capacity investment decisions and production decisions, and is general enough to include reversible and irreversible investment. The product demand vectors for successive periods are assumed to be independent and identically distributed. The optimal investment strategy is determined with a multi-dimensional newsvendor model using demand distributions, a technology matrix, prices (product contribution margins), and marginal investment costs. Our analysis highlights an important conceptual distinction between deterministic and stochastic environments: the optimal investment strategy in our stochastic model typically involves some degree of capacity imbalance which can never be optimal when demand is known.