## INTRODUCTION

Product flow and production pull are key concepts of lean manufacturing (Womack and Jones, 1996). When product can flow through the factory at the same rate at which it is consumed by the customer then inventories can be kept low and capital that would have been tied up in work in process and finished goods inventory can be used more effectively. If customer demand is known exactly and production is set equal to this known demand then the factory can function like a production line only needing small amounts of inventory to accommodate possible yield variations in the production cells within the plant. However, even in the overly simple case, where demand is varying randomly around some known fixed mean, and the production rate is set to this mean rate, finished goods inventory can vary significantly. The reason for this can easily be proven by writing the inventory mass balance.

where *I*_{t} = inventory at time “*t*”; *P*_{t−1} = production decision for time period “*t*” made at “*t* − 1”; *D*_{t} = Demand from “*t* − 1” to “*t*”.

With demand varying randomly around a fixed mean, inventory can then be written as:

where *e*_{t} = the difference between our specified *P*_{t−1} and the mean of the sample demand that actually occurred during the “*t* − 1” to “*t*” time interval.

In this case the rate of change in inventory is random error, and inventory is made up of the sum of those random errors that have accrued over time. Mathematically this can be shown by defining a backshift operator, *B*, such that *BI*_{t} = *I*_{t−1}, Equation (1.1.2) could then be rewritten as

Dividing both sides of Equation (1.1.3) by (1 − *B*) permits us to write *I*_{t} in terms of only current and past errors.

This is significant because it mathematically demonstrates that inventory is a random walk (i.e., a sum of random errors) even when production and the true mean demand are known exactly.

Below are two plots. The 1st plot shows a generated time series of *N*(0,1) random deviates. The 2nd plot shows the sum of those random deviates. Even though the random deviates varied between ±3, the sum of those random deviates (i.e., its random walk) varied between about 5 and −25 (Fig. 1).

Because finished goods inventory is a random walk it can also wander randomly, as in the second plot shown below. Inventory is not a memoryless process, and it will continue to accumulate any set of small biases that might be present in the difference between production and demand. It cannot be assumed that these differences will rapidly cancel each other out. The finished goods inventory time series is a nonstationary process.

Since finished goods inventory is the highest value-added inventory in the supply chain it makes economic sense to keep it as small as possible. However, this cannot be guaranteed with constant production, as shown above, because finished goods inventory is a random walk when future demand is not known exactly. And production capacity cannot be used and planned most efficiently when it must be varied to maintain low finished goods inventory. Therefore, a trade off is needed between the level of finished goods inventory needed to service customers with an acceptable level of backorders and an appropriate level of production variability that can be efficiently managed. The manufacturing cost as a function of finished goods inventory, production variability, and customer backorders will not be derived in this paper. Instead, production design curves will be developed that can be used to specify the amount of time that it is possible to produce at a level rate with a user-specified inventory level, customer service level, and production flex capability. This will then allow each production facility to determine the mix of these variables that will work effectively with their own unique cost function and manufacturing constraints. These production design curves will be developed by deriving the inventory variance that will result when production is kept constant for an arbitrary period of length *N*. The distributional properties of inventory will then be used to set an inventory level that will meet a user-defined level of backorders (i.e., a desired service level). The inventory buffer will be kept as close as possible to this service-mandated protection level at the start of each level production period via a feedback control law. This control law will determine the constant production rate for the upcoming period of length *N*. It will be shown that this control law results in a pull production system that sets its new production rate to exactly compensate for any mismatch between actual production and true demand in the previous level production cycle and a forecast for demand for the next production period. However, this pull production rate might result in unacceptably large changes in this rate. Therefore, a control strategy that only corrects for a proportion of the deviation of actual inventory from its desired aim has also been developed. This strategy can maintain a desired customer service level with a large decrease in production variance and only a minor increase in the desired inventory aim relative to the pure pull strategy. The production design curves that have been developed give a clear graphical representation of the possible benefit of backing off slightly from the pure pull production strategy.

### Literature Review

The level mixed production JIT scheduling technique know as heijunka, Coleman and Vaghefi (1994), is aimed at trying to produce the right products at the right times with minimal inventory. Kovalyov et al. (2001) give a comprehensive review of the sequencing heuristics and algorithms that identify the production sequence that will result in a production rate that matches a predefined demand rate in some optimal manner (i.e., minimum sum of squares or minimum sum of absolute deviations). A simple method to determine a constant demand rate that can be used as a production aim is addressed by Kirwan (2002). Houghton and Portougal (2001) presents an optimum analytic production planning framework that develops a balance between JIT production that attempts to minimize inventory by varying production capacity versus batch production that attempts to maximize capacity by limiting the variability of the production process. Karmakar (1989) stresses the importance of hybrid systems that balance push and pull based on the complexity and dynamics of the products being produced. Groenevelt and Karmakar (1988) highlights the selection and maintenance of a target level of finished inventory and along with McCullen and Towill (2001) stresses the need to dynamically flex production to accommodate demand and yield variations. However, no details on how best to set the inventory target or flex production were presented. Bernegger (2002) highlights the problems caused by the inventory random walk and weaves an entertaining story of the exploits of an industrial engineer as he tries to apply Lean Manufacturing principles in the hypothetical KDK Corporation. The major thrust of his paper was to focus on the problems encountered when one attempts to use Ahead and Behind limits on inventory for JIT Make-To-Stock production applications. Similar to the work of Groenevelt, Bernegger suggests the importance of a target level of finished inventory and the need to control to this target on a regular basis. He gives no proof for his inventory random walk assertions and his results are entirely heuristic and apply only to the limited case where demand is constant with random error. Bernegger's unpublished work appears to be the only literature where the insidious nature of the inventory random walk has been addressed. Previous published work has assumed that inventory would remain under control if the production rate was aligned with the demand rate. This is not always true because of the stochastic nature of demand and its impact on inventory. This was demonstrated in the introduction to this paper. A method for determining reasonable production requirements and lead times that will result in acceptable production capacity variation and customer service levels under both stationary and nonstationary stochastic demand will be the topic of this paper. The control technique is analogous to the optimal averaging level control problem which arises in chemical process control when the manipulated flowrate exiting a surge tank feeds a sensitive downstream unit. Campo and Morari (1989) showed that a Proportional-Integral control law was the Linear Quadratic regulator that minimized the for this problem when demand could be modelled as a random walk disturbance and production could be adjusted at every sample instant. It was extended to the general disturbance case by Horton et al. (2003). In contrast to the work of Campo and Morari the control law developed here does not require that the production rate be adjusted at every sample instant. In addition, it develops design curves that will allow production management to choose an appropriate control interval, *N*, that will meet their production flex and inventory objectives.

### Overview of the Paper

The next section of this paper will develop the control law for production, *P*_{t}, that will allow manufacturing to produce at a fixed rate for a user defined period of length *N* and a user defined customer service level. This section will conclude with three examples using demand series with similar means and variances, but markedly different stochastic behaviour. The first demand series will represent an arbitrary stationary process and the second series will represent a nonstationary process. They were chosen to represent the robust nature of the derived control approach. There are many extensions to this JIT production control process that might be appropriate for different production environments, and the final section of this paper will serve to highlight these enhancements.