Tan C. Miller

Seasonal exponential smoothing with damped trends: An application for production planning

Abstract This paper reviews a spreadsheet-based forecasting approach which a process industry manufacturer developed and implemented to link annual corporate forecasts with its manufacturing/distribution operations. First, we consider how this forecasting system supports overall production planning and why it must be compatible with corporate forecasts. We then review the results of substantial testing of variations on the Winters three-parameter exponential smoothing model on 28 actual product family time series. In particular, we evaluate whether the use of damping parameters improves forecast accuracy. The paper concludes that a Winters four-parameter model (i.e. the standard Winters three-parameter model augmented by a fourth parameter to damp the trend) provides the most accurate forecasts of the models evaluated. Our application confirms the fact that there are situations where the use of damped trend parameters in short-run exponential smoothing based forecasting models is beneficial.

NETWORK FACILITY-LOCATION MODELS IN STACKELBERG-NASH-COURNOT SPATIAL COMPETITION

A hierarchical mathematical programming approach is combined with sensitivity analysis (of variational inequalities) to formulate a facility-location model for a firm competing on a discrete network. It is assumed that the locating firm will act as the leader firm in an industry characterized by Stackelberg leader-follower(s) oligopolistic competition. The othern competitors in this industry are assumed to act as Cournot firms that each operate under the Cournot assumption of zero conjectural variation with respect to theirn — 1 Cournot competitors. It is further assumed that then Cournot firms will react to the location/production/shipping activities of the Stackelberg firm. Therefore, when the Stackelberg firm makes its location, production, and shipping decisions it takes into account the reaction of then Cournot firms to its (the Stackelberg firms) integrated location and distribution decisions. Specifically, a Cournot reaction function is developed and imbedded in the Stackelberg firms profit-maximizing objective function to project the anticipated reaction of the Cournot firms to the Stackelberg firms location decision.

Spatial Market Equilibria on Networks

Within the framework of bilevel facility location models in spatial economic competition, there are several different submodels that one can employ to describe economic competition and determine a market equilibrium among firms on the network. Such economic models include the classic spatial price equilibrium (SPE) model, first introduced by Samuelson (1952), and enhanced for over three decades since then (e.g., Takayama and Judge, 1964) as well as models of oligopolistic competition (e.g., Cournot-Nash). In this chapter, we will review the classic SPE model and the oligopolistic spatial Cournot-Nash model. Additionally, we will review non-extremal versions of these models (i.e., variational inequality formulations) and consider their existence and uniqueness properties.

The Importance of Including Reaction Functions and Analysis of Economic Equilibria in Facility Location Models: An Example

Throughout this book, we have reviewed the motivation for developing equilibrium facility location models, the individual components which comprise such models, as well as techniques for generating algorithms to solve these models. In this chapter, by means of several linked numerical examples, we further illustrate the potential insights which an equilibrium facility location modeling approach can identify. Specifically, we will compare the results of five different scenarios for making location decisions all based on the same data set to depict how the inclusion or exclusion of reaction functions can affect the quality of the solutions obtained.

Classical Plant Location on Networks

The purpose of this chapter is to succinctly review certain classical models for locating production activities on networks. By classical model, we mean one for which the locating firm is either a price taker or a monopolist, so that the analysis of competition is especially simple. Although these classical models are non-equilibrium models, an understanding of their assumptions and mathematical formulations is essential to understanding the equilibrium network facility location models presented in subsequent chapters.

Aspatial Stackelberg Nash Cournot Equilibria

In this chapter, we review a Stackelberg-Nash-Cournot equilibrium problem applied to one market. This will help set a foundation for the review of multi-market spatial equilibria later in this book. Further, by focusing on a one market, non-locational problem, we can develop insights into the properties of market equilibria problems such as the curvature of the revenue function in a Stackelberg-Nash-Cournot model and issues regarding the uniqueness of solutions to this problem. In a multi-market model with locational decisions included, these properties become more difficult to illustrate. Additionally, the one market model serves to illustrate how market equilibria problems are amenable to formulation and solution by nonlinear complementarity techniques as well as variational inequalities. Finally, this chapter provides an important first glimpse of the role which sensitivity analysis can play in market equilibria models.

Stackelberg Equilibria on Networks

In this chapter, we formulate and discuss a sensitivity based approach to solving Stackelberg-Nash-Cournot (SNC) equilibrium problems on a discrete network. The Cournot-Nash model detailed in Chapter 4 represents the first critical component required to develop a SNC equilibrium model. As we will explore in this chapter, performing sensitivity analysis on the variational inequality formulation of the Cournot-Nash model facilitates the development of Cournot reaction functions. We can then use these functions to predict how oligopolistic Cournot competitors will react to the production and distribution decisions of an oligopolistic Stackelberg competitor. This chapter also includes a discussion of the existence properties of sensitivity based SNC equilibrium problems.

Dynamic Models: Equilibrium and Disequilibrium Approaches

In this chapter, we consider two different approaches to formulating multi-period, dynamic facility location models which are based on the construct of an economic equilibrium (or disequilibrium). Our dynamic models build on the single period location models presented in previous chapters, and represent a natural extension of those models. Plant facility location, almost by definition, implies a significant commitment on the part of a firm to a location for a period of years. Thus, the ability to explicitly evaluate the impact of a location decision over a multi-period (multi-year) planning horizon represents an important capability and enhancement. In particular, the multi-period planning horizon facilitates the evaluation of the proper timing of location decisions, in addition to the determination of the best location(s). Further, this allows the firm’s location decision to better meet forecast growth and/or decrease in market demand over time.

A Facility Sizing and Location in Stackelberg Nash Cournot Equilibrium Model

In this chapter, we consider models for facility sizing and location under oligopolistic market equilibrium conditions on a spatial network. As in Chapter 6, we employ the paradigm of a large firm locating facilities on a defined network. To solve facility sizing/location models in oligopolistic competition, sensitivity based bilevel algorithms are developed. We will observe that there exist both significant similarities and differences between the algorithms developed in this chapter and those presented in Chapter 6 where a spatial price equilibrium characterized the economic conditions under which the firm of interest made its locational decisions.

A Facility Sizing and Location in Spatial Price Equilibrium Model

As reviewed in Chapter 4, within the framework of bilevel facility location models in spatial equilibrium, there are several different submodels that one can employ to describe economic competition amongst firms on the network. These economic models include the classic spatial price equilibrium (SPE) model, first introduced by Samuelson (1952) and enhanced for over three decades since then (e.g., Takayama and Judge, 1964) as well as models of oligopolistic competition (e.g., Cournot-Nash). In this chapter, we present a facility sizing and location in spatial price equilibrium model. We review the modifications required to the SPE submodel and to the facility location submodel which facilitate their linkage in an integrated model. Additionally, we consider how the sensitivity analysis results of Chapter 5 are used to generate a linear approximation of the implicit function relating changes in production and shipping levels to changes in market prices. This represents the “linking” component of the facility sizing and location in spatial price equilibrium model. This chapter also presents existence results for the integrated model, as well as a solution algorithm and numerical results.