Variation of installed industrial capacity has been found to follow a cyclic pattern. This paper discusses the application of control theory to the problem of the timely acquisition of extra production capacity. The control system based model presented here is compared with a System Dynamics model proposed by Sterman. Key differences are the method of implementing rational decisions about deployment of extra capacity and the use of a nonlinear APVIOBPCS inventory model. Benefits of this new model are a more measurable process and the ability to select parameter values to optimise capacity deployment. Simulation of the model indicates that the results found by Sterman underestimate the production backlog and time taken to reach equilibrium. The use of a Proportional, Integral, and Derivative (PID) controller in the capacity control loop model illustrates that it is possible not only to alter the backlog levels but at the same time to reduce the sales force and improve the revenue. The model also shows clearly that the impact of not increasing capacity promptly results in catastrophic failure of sales as a structural, rather than a business, problem. This model is simple enough to be implemented as a spreadsheet for use as a guide by managers.

Today’s consumer market is dominated by two main factors: one is the need for rapid development of new products and the other is the need for an equally rapid response to market led demand. Business cycles have long dominated economic analysis but most researchers have concentrated on examining the average effects on the economy discussing the long term expansion and decline of the whole system (Sterman and Mosekilde, [

Sectors of the market growth model (after Morecroft [

Morecroft’s [

Although the model described here was intended for high-tech companies the challenges involved in capacity planning are more generic. Bakke and Hellberg [

In view of the risks to the large amount of capital invested and the time factors involved manufacturers normally take a very cautious approach to building up capacity. It is of particular importance in supply chain design that the decisions regarding supply chain capacity and the policy of capacity increases or decreases are as efficient as possible. The decisions susceptibilities can be classified into three broad areas:

capacity levels that do not meet the full actual demand leading to nonavailability of products, loss of revenue, and market share;

delays in acquiring new capacity that involve considerable risk and may result in loss of both a market opportunity and invested capital;

excess capacity that results in low plant utilisation and ties up capital leading to low return on investment.

Akkermans et al. [

Karrabuk and Wu [

The factors underlying financial reasons for investment are not well behaved continuous linear functions and the key financial decisions in a company are normally made only when the strategic case for investment or disinvestment is very clear cut (Ceryan and Koren [

The correct timing of an indicated capacity expansion is therefore vital. This paper describes an implementation of Sterman’s [

The purpose of this paper is to investigate an improved model of capacity acquisition developed by incorporating rate of change of demand effects and simplifying the criteria for decisions thus enabling an earlier intervention. This approach derives from Sterman’s observation that managers are unable to effectively detect rates of change of demand for long lead times.

We will show that a linear control based model of a production system combined with a nonlinear inventory subsystem is able to represent the average behaviour of the system capacity and using Proportional, Integral, and Derivative (PID) control improves the model’s overall performance, reducing the oscillatory behaviour of the Sterman model.

We now review relevant work on capacity management by other researchers, especially concentrating on System Dynamics (SD) models outlying their principal conclusions. Modelling the behaviour of systems including supply chain effects has traditionally used SD methods based on the work of Forrester [

Capacity modelling results reported in the literature, such as Suryani [

As Sterman et al. [

The second class of problems pursued in the current literature (Duffie et al. [

Orcun et al. [

Investigations were undertaken by Georgiadis and colleagues at the University of Thessaloniki using SD to model a range of closed loop supply chains which included a remanufacturing component (Georgiadis et al. [

Georgiadis and Athanasiou [

Cannella et al. [

All the above cited papers indicated that the observed responses of the models developed showed the presence of cyclical capacity variation.

This section will introduce the model created by Sterman and its limitations and explain the basic equations used in the model. The MATLAB™/Simulink™ version of that model used here is described together with sample responses for comparison.

Sterman’s [

Sterman’s SD model for high-tech company growth.

This model based on the analysis of Forrester and Morecroft assumes that the firm manufactures high-tech products as a build-to-order system. It was not based on one company but it included most of the representational features of such companies while being as simple as possible. However it includes data derived from averaging a number of responses to critical questions. The system model shown in Figure

The Simulink representation of Sterman’s model presented here uses the relationships and equations with the variables expressed as continuous functions of time, for example,

The rate of change of the backlog LEVEL (BL) is the difference between the order rate (OR) and the shipping rate (SR):

The mathematical performance of the system is dictated by three differential equations. In the Simulink representation of Figure

Simulink version of Sterman’s Model.

The smooth function in (

This section outlines the basis of the new control system (CS) model used to improve the behaviour of a variable capacity firm. The changes from the Sterman model in order for the manager to regain control over the capacity increase process are outlined. In the Sterman model the production capacity is outside the control of the order fulfillment organisation. In the new CS model this parameter is brought under the direct control of management as a key decision factor. The fundamental differences between the two models are indicated in Figure

Comparison of the Sterman and new CS Models.

Linear control model with inventory.

The purpose of the new model variant reported here is to address two common issues reported in the literature:

Firstly, the capacity acquisition process was not timely enough to prevent companies failing. So the aim was to devise a new decision process that would prevent the oscillatory problems seen in the SD model. This was a clear conclusion from the work of Forrester, Morecroft, and Sterman. At a process level it was also the conclusion reached by Deif and ElMaraghy [

Secondly, the Sterman model does not include the inventory control procedures normally used by managers and hence does not include all the associated delays which would be significant to the operation of the whole system. Based on this factor alone it would be expected that a real system would respond more slowly than the Sterman model.

These SD lookup tables were then replaced with a smoothed function constant and the formulation of the extra capacity defined by relationships for the error in (i.e., difference between) capacity (ECAP) and the desired capacity:

The delays in the inventory production and forecasting process were not modelled in Sterman’s work. To include this factor a subsystem model of an automatic pipeline and variable inventory order based production control system (APVIOBPCS) was added to the simulation. In this implementation the desired production could exceed the capacity so a switch was added (see subsystem model Figure

Capacity and inventory subsystems.

The smoothing functions and delays were then replaced by simple first-order delay functions (blocks). In the Simulink models information transmission is represented by the arrows. The overall model is shown in Figures

Another modification made to Sterman’s original SD form was the addition of PID control for the system gains (Figure

The justification for including PID functions follows from Diehl’s [

The CS model can be compared to the very much simpler models of White and Censlive [

Order rate performance of the two models.

Shipment rate.

Delivery delay.

Capacity utilisation.

Sales force.

Backlog.

Capacity.

Expected revenue.

Net stock.

Exogenous step input shipment rate.

Exogenous input step capacity response.

Step input sales force.

Step input delivery delay.

Step input effect on backlog response.

Step input on capacity utilisation.

Net stock responses for step input in demand.

Order rate gain variation.

In this section the results from the two models, Sterman’s model and the new CS model, will be compared. Two sets of results are presented here; the first set of results is for the case where there is no exogenous or external input; here sales are generated by the in-house sales force.

Figure

It is clear that the structure of the SD model and hence the company upon which it is based has serious structural flaws if operated as the model predicts. Since the decision processes are fairly simple we can see what the effect of the nonlinear functions is since these introduce higher order dynamics. From experiments conducted by Sterman it is clear that managers cannot readily appreciate these higher order dynamics. The key to controlling the growth of this company would be to increase capacity in a timely manner. Using the CS model it is easier to see when to implement capacity changes.

The control system model for different values of proportional gain KE provides a similar range of results to those of the Sterman model, but without the large oscillations. The sales rate (Figure

Figure

The variation in capacity will be a severe problem to deal with as an investment issue. The system responses show a lower average and peak backlog. These excessive backlog and delivery delays will cause loss of customers and may result in them moving to a rival supplier. Shipment rates also vary greatly in the Sterman model creating logistic problems not present in the control system data.

The final curve shown in Figure

A step exogenous demand is made for the second set of results. This represents a sudden surge in customers, say from an advertising campaign. The picture (Figures

These results show that the CS model allows better use of the existing capacity and allows extra capacity to be scheduled faster than the SD model.

The implications of the results of the simulations are now discussed with the possible implications for managers outlined.

The control system models agree with the general trends of the variables from the Sterman SD model, but they do not exhibit large oscillations present in the SD model. Responses are adjusted by altering the error gain. The main difference between the Sterman SD model and our model is the way the errors are computed and the nonlinear gains set in the Sterman model lookup tables. The loop structure and delays are the same except for the addition of the inventory subsystem loops in the CS model. It is clear therefore that the violent oscillations in capacity response which cause major business operational problems are due to the way the decisions are implemented. If a model can be used that does not exhibit these decision trends then a growth period for the company is more likely. These results also broadly agree with those of Yuan and Ashayeri [

Sterman [

Conclusions can be drawn from the responses of the two models and the work of other researchers.

Examination of the literature about capacity provision shows that the problem of capacity planning in high-tech and low-tech firms is a serious problem, exacerbated in high-tech products by the short lifetime. Existing industry data shows cyclic variation of capacity and investment to be present. SD analysis has shown this to be largely a company management structural effect rather than a demand problem.

Sterman created an SD model to examine the critical aspects of management decision-making after his investigation of feedback decisions in supply chains. His model shows unacceptably large cyclic variations in capacity. To reduce these effects a different decision process was devised and a control engineering based (CS) model of the strategic modification of available capacity has been devised to include normal inventory control and delays to compare with the standard SD model of strategic capacity acquisition by Sterman.

This new model shows very clearly the effect the decision structure has on company performance by comparison with the model of existing companies from Sterman. It should be appreciated that nearly all SD models have embedded industry or company data in the nonlinear table functions used as they are often created for consultancy and are then specific to particular companies.

We have shown that an alteration to the decision process to make the recognition of the changes in capacity by implementing a simple difference between what is needed and what is already in place has a profound impact on the response to external demand.

The CS model does not use the nonlinear functions for decisions that the SD model uses, making it easier for managers to understand and recognise the process. When PID control is used, the performance is improved for many input conditions but a simple system using just the gain KE will have many of the overall advantages. The extreme booms and crashes predicted by the Sterman SD model are

Shipment rates occur later in the CS model for exogenous demand due to the inventory delays and the model shows that capacity utilisation is closer to 100% than that reported for the SD model. This is also true when a sudden external demand is made. Peak delivery delay and staff levels are similar to the PID controlled system model.

The control system model has

reduced variation in sales rates, capacity, delivery delay, and backlog,

more consistent shipment rates,

more consistent revenue rates.

The model is able to provide management guidance at the strategic level and ease the decision-making process, enabling a choice of system parameters to give a specific performance. The internal feedback mechanisms in the model allow managers to examine and rectify the effects on the company operations due to poor management decisions.

This model can easily be implemented as a spreadsheet for use by managers using only simple sales and other data.

The significant lesson from this work is that small changes in decisions can have large effects on the behaviour of the whole production/supply system. These changes together with making the decision at the correct time will determine the success of the strategy.

These models presented here are probably too simple to fully represent a particular company without adding extra details, but those parameters taken as constants need to be examined to determine their contribution to successful business operation. For example, the company goal for delivery delay could be made a dynamic variable.

The control system model allows the examination of the effects of adverse events such as production line machine failures or external commercial environmental factors, by simulation of random capacity disturbances. Future research will examine the effect of policy analysis on sales force productivity, the increasing use of internet marketing plus order systems, the freeing up of the concept of fixed delivery delays, and more rapid reconfiguration of production facilities. Since it is difficult to gain the trust of company managers to implement such processes without evidence of their effectiveness, this model could be developed into a product similar to the “Beer Game” as a management training aid to demonstrate the effectiveness of control of internal decision interactions.

Backlog (units)

Initial backlog = 1000

Capacity (units)

Company goal for delivery delay = 2 months

Gain of sales force

Cost per sales rep = $8000

Capacity utilisation (fraction of capacity in use)

Desired capacity (units)

Delivery delay (weeks)

Derivative gain for PID ~ 0.5

Delivery delay perceived by the company (weeks)

Desired production (units)

Extra capacity (units)

Effect of expansion pressure on desired capacity

Expected revenue

Error in capacity

Fraction of revenue to sales = 0.2

Integral gain ~ 0.001

Control system proportional gain ~ 2.5

Gain of backlog feedback

Management planning system

Market target delivery delay = 2 months

Normal delivery delay (company target) (weeks)

Normal sales effectiveness = 10

Order rate (units/time)

Initial value of order rate

Price of product = $10000

Printed circuit board

Pressure to expand capacity

Proportional, Integral, and Derivative

Production, Planning, and Control system

Sales rate

Laplace transform

Sales budget

Sales force adjustment time

Shipment rate (units/time)

Time to perceive capacity deficit = 3 months

Time for capacity acquisition delay = 18 months

Time for company to perceive delivery delay = 3 months

Time for market to perceive delivery delay = 12 months

Revenue reporting delay = 3 months

Sales force adjustment time = 18 months.

The authors declare that they have no competing interests.