Nicky D Van Foreest
email: n.d.van.foreest@rug.nl

Jacob Wijngaard

Taco Van Der Vaart

Van Foreest

Scheduling and Order Acceptance for the Customized Stochastic Lot Scheduling Problem

Keywords: Make-to-order production, order acceptance policies, customized stochastic lot scheduling problem, SLSP, simulation

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Introduction

Scheduling the production of multiple standardized items on a single machine with limited capacity and significant setup times under random demand is a classic operations research problem. The setups are required to switch production from one type of product to another, while at the same time any time spent on setups wastes scarce capacity.

One approach to reduce the fraction of time spent on setups is to combine individual orders with similar characteristics into production runs of a single product family such that intra-family setups are relatively short while inter-family setups are relatively long. Hence, production efficiency increases by spreading the interfamily setup time over the orders within one run. However, the problem becomes now to determine the size of these runs and the production sequence.

In situations that allow for make-to-stock production, for example when products are sufficiently standard, family runs can be reasonably large since the stock serves to decouple production from demand. Here the production control aims at using the inventory as efficiently as possible to meet demand.

However, many production situations, for instance when final products are highly customer specific, do not allow to use stock as a means to decouple demand from production. Rather than make-to-stock, the production policy is now make-toorder. The additional aspect of make-to-order considerably complicates the lot scheduling problem, as now not only the utilization expressed as the fraction of orders served should be acceptable, also the due dates of the customers should be met. We refer to this situation as the customized products stochastic lot scheduling problem (CSLSP) with strict due dates, c.f. [START_REF] Winands | The Stochastic Economic Lot Scheduling Problem: A Survey[END_REF].

One of the most interesting aspects of the CSLSP is to try to strike a balance between the conflicting internal and external objectives of the production process. On the one hand, since the loss of production time due to inter-family setups is significant, quoting long lead times becomes tempting: the longer the lead times, the more potential to combine orders into long family runs, hence the smaller the setup loss. Moreover, long lead times act as a buffer to smooth demand fluctuations. As such the lead times actually provide the advance demand information that enables to efficiently combine orders. On the other hand, customers require short lead times, immediate acceptance decision, and on-time delivery. Consequently, customer service puts an upper bound on the lead times. Therefore, the main problem for the CSLSP is to design a customer acceptance control policy that respects on the one hand buyer-supplier relationships and on the other realizes an acceptable utilization.

In this paper we develop and analyze several order acceptance control policies that appear suitable for the CSLSP. These control policies try to accept orders by positioning the orders into a schedule of finite length, which results from the (finite) lead time of newly arriving orders. If an order cannot be scheduled, it is rejected. Given the difficulty of the problem involved we use simple approximate models and simulation to analyze the system behavior.

We first study a straightforward, greedy, control rule that tries to adjoin each arriving order to a run of its family, when present. When such an insertion should occur somewhere in the 'middle' of the schedule rather than at the 'tail', the insertion pushes backward all jobs scheduled behind the 'selected' family run. This is only allowed when the pushed back orders can still be produced in time. In case this fails the greedy rule attempts to start a new family run at the end of the schedule. When the schedule has sufficient room for the required setup and the new order, the greedy rule accepts it. Although this greedy policy is appealing by its simplicity, we show that it is not robust in that the realized utilization is quite sensitive to the order arrival rate. In fact, for increasing arrival rates, the utilization first increases, but after reaching a maximum it decreases.

To realize more robust performance, in the sense of being less sensitive to variations in the arrival rate, it turns out necessary to anticipate on future orders to combine with already present family runs. We develop and analyze four different modifications of the greedy policy to incorporate the required anticipation effect in the acceptance policy.

The development and study of customer acceptance policies for the CSLSP gains further motivation by pointing out that it is actually not so clear to what extent the insights of the make-to-stock case to balance production objectives carry over to the make-to-order case. First, the make-to-order environment uses an order acceptance function whereas the make-to-stock case does not. Second, the make-to-order situation can use advance demand information, i.e., the positive lead-time between the moment of arrival and delivery of an order, to schedule orders. Third, the major performance indicators to evaluate the relevant scheduling policies are more or less 'orthogonal' for the two situations. The performance of policies for make-to-stock situations is mainly expressed in terms of inventory cost and delivery reliability, possibly combined with setup cost, whereas utilization is less important. Assuming that orders are backlogged and that the system is stable the utilization must equal the arrival rate of orders. As, typically, scheduling policies cannot change the arrival rate, they cannot influence this utilization, making this is a less suitable performance measure. On the other hand, for make-to-order situations the major performance metric is the utilization, i.e., the fraction of accepted orders, and the service level, i.e., the fraction of orders delivered in time. Each of these measures can be directly influenced by the order acceptance policy. Indeed, companies such as a make-to-order tube mill, see, e.g., [START_REF] Schmidt | Using cyclic planning to manage capacity at Alcoa[END_REF], focus on the utilization. Inventory cost, i.e., the cost of producing too early, is not the primary performance measure to evaluate customer acceptance control policies, as this cost is rather insensitive with respect to the choice of acceptance policy. When the utilization is high, the schedule is usually near to full and the most urgent order of each run is delivered just before its due-date. Only less urgent orders in one production run build up inventories. However, producing early is the consequence of forming runs of orders with different due dates. Inventories are as such inevitable for any policy.

The paper is organized as follows. Section 2 summarizes related literature. Section 3 explains the model and the greedy policy. In Section 4 we derive approximate lower and upper bounds on the resource utilization. In Section 5 we evaluate the greedy policy. As alluded to previously, this policy performs poorly when the arrival rate is high. Therefore, we investigate in Section 6 the performance of four other customer acceptance control policies. Section 7 summarizes the results and presents suggestions for further research.

Theoretical Background

The make-to-stock version of lot scheduling has received much attention in the literature. [START_REF] Elmaghraby | The economic lot scheduling problem (ELSP ): review and extensions[END_REF] provides an overview of the case with deterministic demand, i.e., the economic lot scheduling problem (ELSP), and [START_REF] Sox | A review of the stochastic lot scheduling problem[END_REF]) treats the case with stochastic demand, i.e., the SLSP. More recent overviews are provided by [START_REF] Brander | Inventory Control and Scheduling Problems in a Single-Machine Multi-Item System[END_REF] and [START_REF] Winands | The Stochastic Economic Lot Scheduling Problem: A Survey[END_REF] On the other hand, there is hardly any literature on the pure make-to-order case of the SLSP, i.e., the customized products SLSP. [START_REF] Schmidt | Using cyclic planning to manage capacity at Alcoa[END_REF] and [START_REF] Bertrand | Production Control, a Structural and Design Oriented Approach[END_REF] study tube mills in which the make-to-order property is the consequence of high variety in the last production stages and cyclic schedules control the mill. However, they do not discuss in detail the applied order acceptance policies. Closest related is the work of ten [START_REF] Kate | Order acceptance strategies and lead times[END_REF][START_REF] Ten Kate | Order acceptance and production control[END_REF] and [START_REF] Wester | Order acceptance strategies in a production-to-order environment with set-up times and duedates[END_REF] who compare order acceptance based on only aggregate information with order acceptance that is fully integrated with scheduling. [START_REF] Van Der Vaart | The Contribution of focus in collaborative planning for make-to-order production situations with large setup times[END_REF] consider the multi-machine situation and investigate the possibility to dedicate machine capacity to specific families and reduce the multi-machine case to the one-machine case. [START_REF] Kim | Finite-capacity Multi-class Production Scheduling with Setup Times[END_REF] and [START_REF] Baker | Heuristic Procedures for Scheduling Job Families with Setups and Due Dates[END_REF] concentrate on minimizing maximal lateness, but do not consider rejection as a strategy to meet order due dates.

Polling models may seem an attractive approach to tackle the CSLSP, see, e.g., [START_REF] Federgruen | Impact of adding a make-to-order item to make-to-stock production systems[END_REF] and [START_REF] Winands | The Stochastic Economic Lot Scheduling Problem: A Survey[END_REF]. Also interesting is the random polling model with setups analyzed by [START_REF] Kleinrock | The Analysis of Random Polling Systems[END_REF]. A basic assumption in these polling models is that the queue may have any length whereas in our case the queue length is finite due to the committed due dates. [START_REF] Takagi | Analysis of Finite-Capacity Polling Systems[END_REF] and [START_REF] Kim | Finite-capacity Multi-class Production Scheduling with Setup Times[END_REF] study polling models in which the queue length for each family is bounded. In our case the due date of each order may be interpreted as putting a bound on the total queue length, rather than on the queue length of each family. The former bound introduces interference between the queues of each family, thereby considerably complicating the analysis.

Other queueing models in which the arrival and service rate depend on the work load (or the queue length), see e.g. [START_REF] Bekker | Finite Buffer Queues with workload-dependent service and arrival rates[END_REF] at hand as now the acceptance depends on the contents of the queue rather than just the length.

There is also some research on hybrid situations (a combination between maketo-order and make-to-stock), e.g., [START_REF] Soman | Comparison of dynamic scheduling policies for hybrid make-to-order/make-to-stock production systems with stochastic demand[END_REF], but because of the possibility of building inventory in some products, these situations resemble more the maketo-stock situation than the make-to-order situation.

We conclude that the CSLSP with setup times and strict due dates is theoretically and practically interesting, but no substantial model-oriented research on the different aspects of this situation has been carried out as yet.

Model and Performance Measures

In Section 3.1 we introduce the simplest production system that features all the problems of the CSLSP with strict due dates. Then, in Section 3.2 we present the greedy control policy, according to which jobs will be accepted and released into the system. Section 3.3 defines the performance measures of relevance.

The Production System

A single server resource receives orders arriving at rate λ in accordance to an independent stationary Poisson process P (λ). Each arriving order belongs to one of N product families with equal probability. Let A i denote the arrival time of order i and d i = A i + h its due date; the lead time h is the same for all orders. Each order should be served before or at its due date.

The resource accepts or rejects orders according to some order acceptance control rule, to be described below. When the resource is busy at an arrival epoch, it queues any accepted order. The resource can serve one order per time unit, and delivers every accepted order before its due date. As such the resource behaves as a single server queue with maximal waiting time h. Between jobs of different families the resource inserts a setup of duration s.

For later reference we use the shorthand (N, h, s) to characterize the production system by its main parameter values. A (family) run is a set of consecutive jobs of the same family.

We assume that the arrival rate is such that the resource is heavily loaded if orders do not combine into family runs, that is, when a setup separates every two orders whether the orders are of the same family or not. In such cases the amount of work arriving per order is one unit of serve plus a setup, i.e., 1 + s, so that the assumption on λ can be formulated as

λ(1 + s) > 1.
(1)

The Greedy Acceptance Policy

Now that we have introduced the production system, we turn to describing the greedy acceptance policy.

The greedy policy contains two order acceptance steps: the combine step and the spawn step. In trying to accept a newly arriving order the resource first applies the combine step. If this step rejects the order, the resource tries the spawn step. If the order is also refused by this step, the resource rejects the order altogether.

To explain the acceptance steps, we refer to Table 1. The first step, i.e., combine, aims at reducing the fraction of setups by trying to combine each order with the last run of the already accepted orders of the same family. For example, suppose a 'red' order arrives. The schedule clearly contains two runs of the 'red' family, to wit, at positions 1, 2, and 10. Since the schedule has two free places, the resource accepts the new red order and schedules it at position 11, thereby enlarging the second run of the red family. Suppose that, instead of being red, the new order is blue. Then the resource tries to join the order with the run of its 'kin' by scheduling it at position 7, shifting the setup to 8, and the already present orders to positions 9 to 11. This is considered attractive (since it increases the effective processing time of the server) as long as the shifted orders can be produced in time. However, if one of the shifted order will be late, it is not allowed to insert this new order at position 7. Now the resource applies the spawn step by trying to adjoin the order to the end of the schedule so that it 'spawns' a new generation of its family. This acceptance will necessarily introduce a setup between the last family and the new order. Hence, the resource schedules the just arrived job at position 12 and insert a setup at position 11. Thus, to accept such an order the schedule should provide room for the setup and one order, as is the case in the instance of Table 1.

When there is only one place left, which would be the case if position 11 would be occupied by a red job, the setup would shift the new order past its due date. When this happens, the spawn step also rejects the order. It is of interest to observe that the greedy policy not only accepts orders, but also schedules accepted orders. In this sense, it would be perhaps more appropriate to denote such a policy as an acceptance and scheduling policy. Furthermore, within each run the combine step maintains the earliest due date first order, c.f. [START_REF] Baker | Heuristic Procedures for Scheduling Job Families with Setups and Due Dates[END_REF] for some insights into the consequences for optimality. 1 2 3 4 5 6 7 8 9 10 11 12 R R s B B B s G s R Table 1. An instance of a schedule of accepted orders. The symbols 'R', 'G', and 'B', refer to order colors, 's' to setup. The size of the schedule is 12 positions.

Performance Measures

As explained in the Introduction, the utilization is the primary performance measure of interest, as by assumption all accepted orders are served before their due dates. The utilization u is the fraction of time the server is working on jobs. Let N (t) and S(t) denote the number of orders and setups served during [0, t], then, since the service time per order is 1,

u = lim t→∞ N (t) t = lim t→∞ N (t) N (t) + S(t)s . ( 2
)
The second equality follows from assumption (1), as the idle time is assumed negligible.

We are also interested in the long term acceptance probability p. Since the resource serves every accepted order, the number of accepted orders equals the number of served orders N (t). Hence, It appears to be exceedingly difficult to provide an exact mathematical characterization of the dynamical behavior of the CSLSP, c.f., [START_REF] Winands | The Stochastic Economic Lot Scheduling Problem: A Survey[END_REF]. We therefore settle at deriving approximate lower and upper bounds of the utilization.

p = lim t→∞ N (t) λt . (3
We derive these estimates under the assumption (1), i.e., that the combined workload of orders and setups is in excess of the capacity of the resource. These bounds are of independent interest also, as the difference between these bounds provides an estimate for the potential increase in utilization offered by the combine step.

Lower Bound

To obtain an approximate lower bound of the utilization, we propose, as an extreme case, to make the combine step, explained in Section 3.2, so strict that it only accepts orders at the tail of the schedule, and never 'in the middle'. In this case the queueing policy reduces to FIFO and family runs only originate at the tail of the queue. This assumption minimizes the average run length, hence maximizes the average number of setups present in the schedule.

To compute the resulting run length, proceed as follows. Suppose that at moment t the resource starts a service, the last order is red, and the schedule is full. The next order to be accepted must necessarily arrive after t + 1. Moreover, when this next job belongs to another family it must arrive after t + s + 1. Thus, when no red order arrives in the interval [t + 1, t + s + 1] any order arriving after t + s + 1 will be accepted in the schedule. Now q = P (The next order belongs to another family) = P (no red arrival during[t + 1, t + s + 1])P (next order is not red) = e -λs/N N -1 N .

Since the run length of a family is geometrically distributed with probability 1 -q, the expected run length R = 1/q. Observe that a setup splits every two runs, so that the minimal production efficiency becomes in analogy to (2),

u min ≈ min λ, R R + s = min λ, N N + s(N -1)e -λs/N , (5)
as the minimal utilization can never exceed the total arrival rate λ of orders.

Upper Bound

At the other extreme, the combination gain is maximal if the combine step would accept every order and simply ignore whether orders will shifted past their due dates. As a consequence there will be at most one run of each family present in the schedule.

To begin, observe that since every family run starts with a setup, the expected number of setups present in the schedule should equal the expected number of present families. To compute this we proceed as follows. It is clear that the initial Poisson arrival process P (λ) is thinned by the acceptance policy. Now we approximate this thinned stream as a Poisson arrival process P (λp) as with probability p the order is accepted. Since on the long run each of the above policies is unbiased towards family acceptance, the arrival process of a single family must then be well approximated as P (λp/N ). The probability that some family i is present equals the probability q that the interarrival time T i between two accepted orders of family i is less that the length of the schedule, thus q = P (T i ≤ h) = 1 -exp(-hλp/N ).

As a second approximation we assume that the presence of some orders is of no consequence to the presence of orders of other families. From this, then, it follows that the number of families in the schedule is binomially distributed with parameters N and q, the expectation of which is N q. The operation is most efficient when the schedule is full, so that of the h positions in the schedule N qs are consumed by setups, and the rest by orders. Consequently, the maximal utilization efficiency u max = min {λ, (h -sN q)/h}, as the utilization can never exceed λ, that is

u max = min λ, 1 - sN h 1 -e -hλp/N .
On the other hand, by (4), we have that u = λp. Combining these two observations into a single equation leads to a fixed point equation for p:

λ p = min λ, 1 - sN h 1 -e -hλp/N . ( 6
)
That is, for given (reasonable) values of λ, N, h and s we solve (6) for p, and set u max = λp. We occasionally write u max (N, h, s) to express explicitly the dependence on the parameters N, h, and s.

Combination Potential

A reasonable estimate for the potential increase in utilization due to the the combine step is the difference between ( 6) and (5), i.e., u max -u min . We refer to this difference as the combination potential. Figure 1 gives an impression of the combination potential for various values of h, N and s. Obviously, the difference is considerable, thereby making a detailed investigation of an optimal acceptance policy interesting. By intuitive reasoning we derive that the combination potential should be a decreasing function of N for N sufficiently large: there are simply too many product families to form runs of interesting length. Also, when s is small the gain in forming combinations is small. These observations find further support from the following limits of ( 5) and ( 6):

lim

N →∞ u max = lim N →∞ u min = 1 1 + s ,
where the left equality follows from an expansion of (6) up to first order in N , and lim s→0 u max = lim s→0 u min = 1.

Evaluation of the Greedy Policy

To evaluate the performance of the greedy policy we use simulations. The reference situation is N = 8 product families, setup time equal to s = 1, and lead time h = 30. The length of simulations is 100,000 time units, making the results reliable up to ±0.001.

Figure 2 shows some of the characteristic results for the utilization as a function of the arrival rate λ. The left panel compares the utilization for the case (8, 30, 1) to its approximate lower and upper bounds, c.f., ( 5) and ( 6). Clearly, we see that by using the greedy control the utilization is not an increasing function of the the arrival rate λ. In fact, for high arrival rates the greedy policy performs hardly better than the approximate worst case behavior of (5). To analyze this substantial drop in performance in a larger context we varied the lead time h from 10 to 100. The middle panel presents the results. Apparently, for all values of h we see worst case behavior for large values of λ.

It appears that when λ is large an unacceptably large fraction of time is waisted on setups. The reason is that the combine step cannot accept orders in the 'middle of the schedule' to form long runs, because, roughly speaking, the schedule is full with short runs. These short runs result from the fact that for high values of λ any room originating at the tail of the schedule is rapidly filled by either orders or setups. As a consequence most accepted orders have no or just small slack between the delivery date and the due date, thereby making it impossible for the combine step to squeeze in a new order. Actually, this type of acceptance behavior is at the base of the lower bound (5).

One direction to mitigate the problem of only forming small runs is to extend the lead time h. Indeed, the graphs in the middle panel show that this is effective, however, only up to some point. When λ is large, relative to u max (h), the utilization drops to worst case. To clarify this further, we show in the right panel the same results as in the middle panel but now as a function of λ/u max (h). We see that the larger h, the more sensitive is the utilization to λ in the neighborhood of u max . From this figure we infer that u max is a critical threshold for the arrival rate. Simulations with various combinations (N, h, s) show similar behavior. This leads us to conclude that the performance of greedy control is quite unsatisfactory and that long lead times do not solve the problem. Consequently, trying to maximize utilization for the short term is far from optimal. It may better to accept short term penalties in the form of rejecting to orders to achieve high long term performance. In other words, anticipating future combination possibilities with as yet not available orders might lead to better results than the greedy policy, which focuses on immediate customer satisfaction. This is the topic of the next section.

Four Alternative Acceptance Policies

The analysis in the previous section shows that the greedy policy fails because the spawn step accepts orders too easily when the arrival rate is high. To repair this flaw in the acceptance policy we propose to extend the spawn step with additional rules, which we present in section 6.1. The resulting heuristics for order acceptance aim to protect the combination potential that the combine step tries to exploit. We evaluate these heuristics in section 6.2 to see whether the adapted acceptance policies strike a good balance between the combine and the spawn step.

Improved Acceptance Policies

The problems with the greedy policy as outlined above leads us to investigate four alternative acceptance policies that enable to accept a larger fraction of orders in the combine step by making the rules at the spawn step stricter. Thus, all four policies keep the combine step but differ in the precise rules of the spawn step.

Here the description of these steps is rather informal. In Section A we present the details.

The spawn step of the setup policy is the simplest. An order is accepted at the tail of the schedule if the greedy policy would accept it and it the number of setups present in the schedule at arrival time is less than or equal to some threshold S.

The spawn step of the time distance policy only accepts an order at the tail if the order can be delivered ch time units before its due date. This creates a distance slack ch. We infer that there is some optimal c ∈ [0, 1], which we motivate as follows. The larger the distance slack ch, the more potential for future combinations. On the other hand, if c = 1 the spawn step will only start a new run when the schedule is empty, so that the fraction of idle time of the resource is no longer negligible. Observe that if c = 0 the slack policy reduces to the greedy policy.

The generation distance policy maintains a minimal distance between two subsequent generations of the same family. The spawn step in this case only accepts an order if a new family run does not get closer than the generation slack dh to the start of the last run of the same family. If d = 1, there are no two runs of the same family in the schedule; if d = 0 we have the greedy policy again.

The cyclic policy protects the distance in the same way as the generation distance policy, but also guarantees a fixed cyclic order. Here the spawn step tries to insert an order at some position p between two adjacent runs if the fixed cycle order requires this. When this is not possible because an accepted job will be late, the spawn step accepts the order to form a new run at the end of the schedule provided the time between the preferred position p and the end of the schedule is less than dh. Note that the case d = 0 does not reduce the cyclic policy to the greedy policy, as the cyclic spawn step is allowed to insert new family runs in the middle of the schedule, rather than only at the end. All four rules provide the possibility to anticipate on future combination possibilities. Time distance control anticipates directly by adding a new run only if the order starting that run has some slack, controlled by the parameter c, between scheduled delivery time and committed due time. The other policies anticipate indirectly by restricting the number of runs in total (setup) or per family through a proper choice of d (generation and cyclic).

Evaluation

In this section we investigate the performance of the above four policies by simulation. As in Section 5 the reference case is (8, 30, 1). Appendix B provides results for various other combinations of (N, h, s). The results shown there support the observations made here.

Figure 3 contains the graphs of the utilization u for the policies and various choices of the controls as a function of λ ∈ [0.5, 1.3]. We include as a reference the performance of the greedy policy as the cases c = 0 and d = 0 for time and generation distance, respectively.

The first observation is immediate: the striking improvement in utilization achieved by the four policies as compared to the greedy policy. For instance, around λ = 0.8 the utilization increases by approximately 10%. Secondly, the adapted policies are much more robust to changes in the arrival rate, provided the control parameters are chosen appropriately. Thirdly, there is not a single value of c, or S, for the time, or setup, policy that gives (near to) optimal performance for the entire range of λ. On the other hand, generation and cyclic control work (nearly) optimal with maximal slack, i.e., d = 1.0. A particularly relevant range for λ, from the practical point of view, is around u max , which is approximately 0.75 for the present case (8, 30, 1). By varying λ from 0.6 to 0.8 it appears that the (near to) optimal control values are c = 0.2, d = 1, S = 6, d = 1 for the time, generation, setup, and cyclic policy, respectively. parameters. The graphs show that time distance control leads to the highest utilization of all four policies, setup and generation are very close, except for relatively high values of λ, and cyclic performs worst. These results support the claim that the inflexibility resulting from producing according to a fixed cycle is somewhat costly. Thus, cyclic control appears to be only interesting if production regularity is important, which is typically the case if the fixed cycle is communicated to the customers so that they might be willing to adhere to this cycle and accept longer lead times to compensate for the performance loss. In case it is not possible to develop a close relationship with customers, the other three policies appear more interesting.

As a second step we investigate the dependence of the utilization u for the different policies for different values of h, N and s, just as in Figure 1. For all combinations we solve (6) for the largest λ such that the acceptance probability p ≤ 1. Clearly, this critical arrival rate equals u max (N, h, s). For time and generation distance, and cyclic, we take c = 0.2 and d = 1. The setup threshold S must of course depend on N and h. Some experimentation showed that taking S as the integer part of N exp (u max h/N ) -1, compare the derivation of (6), is (near) to optimal. In the plots we see that u max captures the trends of the simulations very well; the actual results are about 5% less than u max .

Generation distance and setup perform well over the entire parameter regions. Interestingly, for larger lead times, time distance with c = 0.2 has a lower utilization than the other policies. Other simulations (the explicit results are not included) show that c = 0.2 is too small for large lead times. By taking c larger for larger h time distance gives the same results as generation and setup. It would be of interest to find analytical methods to obtain good estimates for c as a function of h. Finally, the cyclic policy never achieves better utilization than the other policies with good control values, but when N is small, or h large, it is nearly as good.

Summary and Conclusions

In this paper we develop and analyze customer order acceptance policies to achieve high bottleneck utilization for the customized SLSP with setups and strict order due dates. To benchmark the performance of the policies we derive an approximate upper and lower bound for the utilization. The former bound is also an upper bound on the arrival rate if zero loss if required. To compare the policies we use simulation, as a detailed analysis appears very complicated. The order acceptance policies apply two subsequent steps to accept and schedule orders. The combine step, which is the same for all policies, tries to adjoin arriving orders to runs of the same family type. This is acceptable as long as orders that are pushed back due to the insertion will not be late. The details of the spawn steps differ among the policies.

The spawn step of the greedy policy accepts orders whenever possible. It turns out that this policy performs poorly: when the order arrival rate is high the utilization decreases to worst case. The reason is that this greedy strategy does not introduce slack in the schedule, so that future orders cannot form runs of considerable size. This observation leads us to infer that a good spawn step should sometimes reject orders even when there is room at the tail of the schedule, with the aim to protect the combination potential for future arrivals.

We then compare four policies that introduce the required slack in different ways. The time distance policy rejects orders that cannot be served within (1 -c)h time units of their due date, thereby maintaining an amount ch of slack in the schedule. The generation distance policy introduces slack in the schedule by preventing runs of the same family to get closer than dh time units. The setup policy only accepts new family runs when the number of setups present is less then some threshold. Finally, the cycle policy is a variant of the generation distance policy that also tries to maintain a fixed production cycle.

Our main observation is that the time distance, generation distance and setup policy achieve high utilization, provided the control values are chosen properly, while the cyclic policy performs somewhat less. Moreover, the performance of these policies is not sensitive to variations in the arrival rate, in contrast to the greedy policy. Overall, we conclude that intelligently adding slack is good strategy to exploit the combination potential.

We envisage various ways to extend the present work. Firstly, we conjecture that introducing slack into the schedule is also necessary in more realistic, and hence more complicated, make-to-order situations with large setup times at a bottleneck resource. Simulations with non-identical product families provide some support for this conjecture already.

Our assumption that the resource is completely reliable enables us to use deterministic policies to schedule and accept orders. Whether such policies perform sufficiently well in the presence of random fluctuations in for instance processing and setup times remains for further research.

A general investigation of the influence of the policies on fairness when the requirements of the families may differ is also important. Suppose, for instance, that the lead time of one family is twice that of the other families, i.e., 2h rather than h. Clearly, this family does not have to compete for the schedule capacity in the interval [t+h,t+2h].

When the orders can be served by multiple resources, rather than one, many new problems come into play. Designing good customer acceptance control policies becomes now very challenging as the policy should also incorporate a production allocation decision, c.f. [START_REF] Van Der Vaart | The Contribution of focus in collaborative planning for make-to-order production situations with large setup times[END_REF]. the following set of subsequent steps.

F

(1) An order at an empty schedule is always accepted. A setup is inserted if the family of the order is different from the family of the last departure.

(2) Tail check 1: Check whether the order fits in the schedule at the tail. If successful, continue with the next step, otherwise reject the order.

(3) Apply the combine step (to be explained below). If successful, accept the job, otherwise proceed to the next step.

(4) Tail check 2: Check whether a setup and the order fit in the schedule at the tail. If successful, continue with the next step, otherwise reject the order.

(5) Apply the spawn step (to be explained below). If successful, accept the job, otherwise reject it.

The first tail check involves to see whether the order fits in the schedule at all. Suppose it would join the end of the schedule. If its due date t + h is smaller than its potential departure time t n + 1 the order cannot be served in time, nor can it be inserted at any other position in the schedule, as necessarily another job will then be late. Hence, the order must be rejected in this case.

The combine step accepts the arriving order of family r if inserting it at the end of the run of its family will not delay the present jobs too much. In formal terms, the order is accepted whenever R(r) = ∅ and t k + 1 ≤ d k for all k such that j r < k ≤ n.

The second tail check precedes the spawn step. As the combine step failed, the insertion of the order will necessarily insert a setup at some position in the schedule. Therefore we first check whether the latest possible departure time of the order is smaller than the due date, i.e., t n + s + 1 ≤ t + h.

The details of the spawn step depend on the actual policy. The setup policy is particularly simple. Whenever the number of setups present in the schedule, or equivalently, the amount of time spent on setups, does not exceed some threshold S the order is accepted. The time distance policy requires the presence of some slack at the end of the schedule in order to let the job spawn a new family run at the end of the schedule. More specifically, accept when the departure time t n + s + 1 ≤ t + (1 -c)h. When c = 0 this case reduces to the greedy policy, as t + h is the due date of the order. Observe that the order with the setup will fit into the schedule, for to arrive at Step 5 it is necessary to pass Step 4. The generation distance policy requires that subsequent family runs are not too near to each other. Formally, accept if R(r) = ∅ and t jr ≤ t + (1 -d)h, or accept as in the greedy policy if R(r) = ∅.

The details of the cyclic policy are slightly more complex than the other policies. In simple terms it tries to maintain a cyclic production scheme. An example serves perhaps best to clarify the rule. Suppose the families in the schedule are ordered as 6, 1, 4, and an order of family 7 arrives. To maintain a cyclic order we would like to insert this order between the families 6 and 1. In detail, we first look up the job k after which the arriving order of family r should be inserted according to the cyclic scheme, that is, we search for the first job k, starting from the last job in the schedule (which is of family f n ) such that

(f k + N -1 -f n ) mod N ≤ (r + N -1 -f n ) mod N. (A1)
Insertion behind job k is allowed if t j +s+1 ≤ d j for all j such that k < j ≤ n. If this insertion fails, the job is adjoined to the tail of the schedule when t k ≤ t + (1 -d)h.

Figure 1 .

 1 Figure1. The upper and lower bounds on u as a function of lead time h, the number N of product families, and setup time s, respectively. In all cases λ = 0.8. When N , s or h do not vary, they take the values 8, 1 and 30.

Figure 2 .

 2 Figure 2. The utilization as a function of the arrival rate λ for the greedy policy for N = 8 and s = 1.

Figure 3 .

 3 Figure3. The utilization as a function of the arrival rate λ for the four acceptance policies with various values of the protection factors for (N, h, s) = (8, 30, 1). The straight line y = x represents the zero loss situation; hence the difference between the graphs for the policies and this line shows the fraction of lost orders.

Figure 4 .

 4 Figure 4. The utilization as a function of the arrival rate λ for the time policy with c = 0.2, generation with d = 1, setup with S = 6, and cyclic with d = 1.

Figure 5 .

 5 Figure5. The utilization as a function of N , h and s while the arrival rate λ = umax varies according to the choice of (N, h, s). The top graph (with index up) corresponds to umax.

Figure B3 .Figure B4 .

 B3B4 Figure B3. The effect of the protection factor for the system (8, 15, 0.5).

http://mc.manuscriptcentral.com/tprs Email: ijpr@lboro.ac.ukInternational Journal of Production Research

Appendix A. Policy Details

Here we describe the customer acceptance control policies in detail. First we need some notation. A job of family r with lead time h arrives at time t at the resource, requires 1 time unit of service, and introduces s time units of setup if it spawns a new family run. The schedule contains n jobs with scheduled delivery times t < t 1 < t 2 < . . . < t n and due dates d 1 < d 2 < . . . < d n . Obviously, t i ≤ d i ≤ t i + h for all i. Let f j denote the family of job j. Define R(r) = {i | f i = r}, i.e., the set of jobs of family r present in the schedule. Let j r be the index of the last job of family r, i.e., for all i ∈ R(r), t i ≤ t jr .

The acceptance procedures we consider in the paper consist of the application of


Joakim Storck

Exploring improvement trajectories with dynamic process cost modelling: A case from the steel industry

Keywords: COST ANALYSIS, MANUFACTURING STRATEGY, SIMULATION, PROCESS Cost analysis, Manufacturing strategy, Simulation, Process industry, System Dynamics, Exploratory data analysis

Improvement trajectories are sequential managed chains of improvement initiatives required to handle changes in competition and market. This paper presents a five-step framework, based on dynamic process cost modelling, which was developed during a four-year research project at a major stainless steel producer, to support the selection of an improvement trajectory based on strategic requirements to combine high product diversity with cost reduction. The framework aims to develop insight into what manufacturing capabilities are required to reach the strategic goals by combining system dynamics simulation with process cost modelling and visual exploratory data analysis in an iterative modelling procedure. The applicability of the five-step framework is demonstrated through a case study from the steel industry, in which a goal driven analysis is used to assess process requirements based on performance and market considerations.

Introduction

Successful companies continuously develop new capabilities that allow them to handle threats and opportunities as they arise. [START_REF] Hayes | Manufacturing strategy: at the intersection of two paradigm shifts[END_REF] states that the new role of strategy is "to make possible dynamic tradeoffs through the selection, development and exploitation of superior capabilities". They argue that firms that manage to harness various improvement programs in the service of a broader manufacturing strategy can turn manufacturing into a competitive advantage. A central issue is to decide what capabilities to develop in order to transform manufacturing into a source of sustainable competitive advantage. [START_REF] Ohno | Toyota production system: beyond large-scale production[END_REF] considered most production initiatives to be cost-driven, stating that "all change should boil down to cost reduction". However, this does not in itself need to imply a narrow product range. Instead, it is necessary to properly assess the manufacturing flexibility requirements to support the contemplated range of product volumes [START_REF] Berry | Manufacturing flexibility: methods for measuring the impact of product variety on performance in process industries[END_REF]. Development of new capabilities along multiple dimensions can allow simultaneous improvement of cost and flexibility performance. Ohno's statement should therefore be understood as cost reduction compared to alternative configurations that satisfy the requirements of the chosen business strategy.

An improvement trajectory implies a managed sequence of initiatives to raise performance [START_REF] Da Silveira | Improvement trajectories in operations and technology management: concept, process and content issues[END_REF]. Central ideas within the theory of improvement trajectories are path dependence and dynamic capabilities. Dynamic capabilities contrast with ordinary capabilities in that they are concerned with change [START_REF] Winter | Understanding dynamic capabilities[END_REF]. Since dynamic capabilities are cumulative [START_REF] Ferdows | Lasting improvements in manufacturing performance: In search of a new theory[END_REF], the order in which strategic initiatives are applied matters. This is elaborated by [START_REF] Flynn | An exploratory study of the nature of cumulative capabilities[END_REF], who propose a sandcone model that links strategic initiatives to the sequential development of cumulative capabilities.

Performance measurement is an important instrument in a firm's internal process of strategy making [START_REF] Leong | Research in the process and content of manufacturing strategy[END_REF]. However, it is necessary to differentiate between manufacturing capabilities and manufacturing outcomes. Cost is a manufacturing outcome, flexibility is a manufacturing capability [START_REF] Swink | Core manufacturing capabilities and their links to product differentiation[END_REF]. Hence, assessment of investments in new manufacturing technology must be approached from both an engineering and financial point of view [START_REF] Busch | Cost modeling as a technical management tool[END_REF][START_REF] Field | Process cost modelling: Strategic evaluation of engineering and economic evaluation of materials technologies[END_REF][START_REF] Szekely | The integration of process and cost modeling -A powerful tool for business planning[END_REF].

The term capability awareness is proposed to denote the level of insight, based on the available knowledge, tacit or formal, that guides firm management's perception of the gap between current practice and the capabilities that are required to successfully realise the chosen business strategy. Development of routines that aim to raise the level of capability awareness addresses the issue of developing a "capability for capability building" [START_REF] Fujimoto | The evolution of a manufacturing system at Toyota[END_REF], to guide the selection of future improvement initiatives by providing better opportunistic learning capabilities. Process-based cost models can play an important role in this process.

The present paper discusses the potential to support the strategy process with dynamic process cost modelling. A five-step framework is proposed for assessment of current and future production capability requirements. A case study from the steel industry illustrates how the five-step framework can aid the development of capability awareness through an explorative search for improvement trajectories that support the goals of the chosen business strategy.

State of the art

Process cost modelling

The significance of cost estimations is emphasised by [START_REF] Field | Process cost modelling: Strategic evaluation of engineering and economic evaluation of materials technologies[END_REF], who argue that process based production cost models are vital to avoid that the engineers' technical measures are set aside by firm management "in favour of more poorly supported, but wholly familiar, cost metrics devised by those with only a passing familiarity with the technical issues at the real heart of the issue" [START_REF] Field | Process cost modelling: Strategic evaluation of engineering and economic evaluation of materials technologies[END_REF]. The use of cost models that fail to account for the effects of new process technology or operating practice from an engineering perspective, may in fact lead the firm towards an unintended change trajectory that makes it difficult or even impossible to reach the desired goals.

According to [START_REF] Szekely | The integration of process and cost modeling -A powerful tool for business planning[END_REF], neither physical engineering models, nor traditional financial cost models are particularly well suited for the task of guiding technology strategy and business development. Instead, building better, more robust, cost models requires the incorporation of physical modelling principles, while building better, more useful, physical models requires that they address pertinent business issues such as cost [START_REF] Szekely | The integration of process and cost modeling -A powerful tool for business planning[END_REF].

Technical cost modelling (TCM) is an extension of conventional process modelling with emphasis on capturing the cost implications of material selection, process variables and changing economic scenarios [START_REF] Arnold | Economic modelling of multi-sequential manufacturing processes: A case study analysis of the automotive door[END_REF][START_REF] Kendall | Discrete event simulation and cost analysis for manufacturing optimisation of an automotive LCM component[END_REF][START_REF] Park | Development of a production cost estimation framework to support product family design[END_REF]. TCM is designed to follow the logical progression of process flow. [START_REF] Busch | Cost modeling as a technical management tool[END_REF] identifies eight cost elements that typically suffice to describe most processes:

• Materials • Labour F o r P e e r R e v i e w O n l y • Energy • Capital equipment • Tooling • Building space • Maintenance • Time value of money
For each cost element, simple relations, or "identities", can be defined. Key variables within cost identities are then replaced with predictive equations derived from first principle theories and/or from the analysis of commercial practice. It is the linkage of cost identities to process parameters that differentiate TCM from standard cost estimation templates [START_REF] Busch | Cost modeling as a technical management tool[END_REF]. [START_REF] Wang | Process cost modelling using neural networks[END_REF] found that current cost model development processes did not possess the efficiency or responsiveness needed to cope with new materials and manufacturing processes, as well as with an increasing number of product variants. It was suggested that neural networks could help to identify cost drivers, simplify model development, and allow more variables to be included in cost models. [START_REF] Park | Development of a production cost estimation framework to support product family design[END_REF] suggested a three-stage production cost estimation framework which draws on TCM. Required production activities are identified in an initial resource allocation stage. The second stage is cost estimation, which involves modelling of resource consumption rates using the chosen product design variables. Finally, in the analysis stage, product design alternatives are assessed by varying design parameters.

According to [START_REF] Field | Process cost modelling: Strategic evaluation of engineering and economic evaluation of materials technologies[END_REF], cost can be identified as an emergent property, which depends on the context within which products are designed and manufactured. They propose a three-layered approach to cost modelling, based on 1. a process model, where fundamental engineering principles are used to assess how the necessary resources for production can be employed; 2. an operations model, in which the physical implementation of the manufacturing processes as well as the organisation of resources is considered; and 3. a financial model, in which resource requirements are converted to economic costs.

It is suggested that the operations model contains a dynamic aspect since it helps the manufacturer to "optimise its use of one resource that is most difficult to obtaintime" [START_REF] Field | Process cost modelling: Strategic evaluation of engineering and economic evaluation of materials technologies[END_REF]. This need is evident for cost modelling in the process industry. For example, a steel mill may produce a large number of product variants that differ in chemical composition, casting geometry and rolling temperature. However, since all variants undergo the same processing activities, it is primarily the dynamic behaviour of the manufacturing system that determines cost.

The need to include temporal dynamics is also noted by [START_REF] Kendall | Discrete event simulation and cost analysis for manufacturing optimisation of an automotive LCM component[END_REF], who found that TCM tends to underestimate manufacturing costs since "the traditional TCM approach assumes that each step in the manufacturing process operates independently from one another for a temporal standpoint of part flow" [START_REF] Kendall | Discrete event simulation and cost analysis for manufacturing optimisation of an automotive LCM component[END_REF]. This is due to the difficulty of modelling a dynamic course of events within a spreadsheet environment. Indeed, cost models have combined TCM with system dynamics [START_REF] Tucci | A simulator of the manufacturing of induction motors[END_REF], and with discrete event simulation [START_REF] Kendall | Discrete event simulation and cost analysis for manufacturing optimisation of an automotive LCM component[END_REF], in order to capture dynamic effects that the traditional TCM method failed to account for.

System dynamics

System dynamics is a generic modelling and simulation methodology with its roots in control theory [START_REF] Ortega | Control theory applications to the production-inventory problem: a review[END_REF]. The system dynamics methodology has developed from the [START_REF] Forrester | Industrial Dynamics[END_REF], who used computer based simulation to increase the understanding of dynamic behaviour in industrial supply chains [START_REF] Fisher | The prophet of unintended consequences[END_REF][START_REF] Forrester | System dynamics -a personal view of the first fifty years[END_REF][START_REF] Ortega | Control theory applications to the production-inventory problem: a review[END_REF].

System dynamics is based on continuous simulation. Hence, causation is not expressed in terms of events, but as effects, or influence, between variables. A model consists of a combination of differential equations and causal loop diagrams that show causal relations between model parameters using text labels and directed arrows. The model is solved numerically using e.g. Euler integration or Runge-Kutta methods [START_REF] Sterman | Business Dynamics[END_REF].

Discrete event models differ from system dynamics models in that they concern instantaneous change of state [START_REF] Law | Simulation modelling and analysis[END_REF]. Event based modelling can be introduced in system dynamics models through the use of step functions and pulse trains. It is sometimes useful to use mixed continuous and discrete event modelling.

System dynamics uses an analogy where generic systems are modelled as stocks and flows [START_REF] Sterman | Business Dynamics[END_REF], which allows a system to be modelled in terms of high-level variables such as throughput rates and inventory levels. Reusability and rapid model development are characteristics of system dynamics modelling [START_REF] Tesfamariam | Aggregate analysis of manufacturing systems using system dynamics and ANP[END_REF].

Applications of system dynamics are found in diverse fields of industrial research. Examples include inventory management, demand amplification, supply chain reengineering and management of international supply chains [START_REF] Angerhofer | System dynamics modelling in supply chain management: research review[END_REF], mining and exploration [START_REF] Moles | Using system dynamics to model the interaction between environmental and economic factors in the mining industry[END_REF], production systems engineering [START_REF] Stamboulis | A system-dynamics study of a resource-based approach to process development strategy[END_REF] and manufacturing strategy and technology evaluation [START_REF] Lyneis | System dynamics for business strategy: a phased approach[END_REF][START_REF] Wolstenholme | The use of system dynamics as a tool for intermediate level technology evaluation: three case studies[END_REF].

System dynamics simulation can be combined with Monte Carlo sensitivity simulation, a process that can be automated in order to conduct a large number of replications. A difficulty that is often encountered is the large amounts of data that results from sensitivity simulations. However, evaluation of sensitivity simulation results can be simplified through the use of visual exploratory data analysis as discussed in the following section.

Visual explorative data analysis

Exploratory data analysis (EDA) was introduced by [START_REF] Tukey | Exploratory Data Analysis[END_REF] as a method to support hypothesis formulation. It differs from confirmatory data analysis, which is mainly used to verify hypotheses after they have been formulated, in that the primary aim is to discover interesting features in the data [START_REF] Cook | Interactive and dynamic graphics for data analysis[END_REF]. During the last twenty years, development of specialised software changed the nature of graphical plots from formerly being a final product to now being an analysis tool where data views can be modified and adapted interactively on the computer screen [START_REF] Wilhelm | Interactive Statistical Graphics: the Paradigm of Linked Views[END_REF].

Linked views give graphical methods the capability of multivariate data analysis. A database is visualized simultaneously in different views, and changes and selections made in one plot are immediately reflected to all other connected plots [START_REF] Wilhelm | Interactive Statistical Graphics: the Paradigm of Linked Views[END_REF]. This technique is known as brushing. The brushing operation immediately reveals how a region in one graph maps to different views of the same data. Nodes and edges in the data can be highlighted, labelled or painted in order to explore the distribution of values in the graph [START_REF] Swayne | Exploratory visual analysis of graphs in GGobi[END_REF].

Plots produced during this process are usually of a temporary nature, since new views are produced rapidly in search of clusters or interesting patterns. Data visualisation software typically supports linked views with histograms, scatterplots and parallel coordinate plots [START_REF] Inselberg | Parallel coordinates: a tool for visualizing multidimensional geometry[END_REF]. Some have features to display random sequences of projections in order to simplify the search for structures in high-dimensional space. The search for interesting views can be guided by an optimisation algorithm [START_REF] Cook | Interactive and dynamic graphics for data analysis[END_REF]. Most literature on system dynamics is concerned with horizontal integration across supply chains. However, [START_REF] Ortega | Control theory applications to the production-inventory problem: a review[END_REF] identified the need for an uniform generic modelling framework that is capable of integrating control theory based models vertically (within plants) and horizontally (across factories). A complication is the lack of analytical support, which is a problem of simulation based research in general [START_REF] Ortega | Control theory applications to the production-inventory problem: a review[END_REF]. The five-step framework proposed in this paper, with its emphasis on sensitivity simulation and explorative data analysis, targets these issues.

Proposed framework

The following section presents a five-step framework, based on dynamic process cost modelling, that aims to aid the development of capability awareness. The proposed framework consists of five steps (Figure 1), which are carried out iteratively until the model reaches an acceptable level of maturity:

1. Problem identification. The problem structure is established and relevant processes, operations and cost elements identified. 2. Process modelling. Manufacturing processes are modelled with an emphasis on technology. Models are validated, e.g. through discussions with domain experts, comparison with experimental or in-process data, or comparison with reference models. 3. Operations modelling. Materials and information flows are introduced to link individual processes. Buffers are modelled as levels. Some decision processes may involve discrete event modelling. 4. Sensitivity simulation. The model is exposed to Monte-Carlo simulation with different ranges and combinations of input, which captures the characteristics of the model and often reveals inconsistencies that are otherwise hard to detect. 5. Exploratory data analysis. Linked data plots are used to find patterns and anomalies in the sensitivity results. This typically leads to the formulation of new hypotheses that serve as input to the problem identification stage of the next iteration.

(Figure 1)

The system dynamics paradigm emphasises continuous testing of models, which is particularly important in situations where historical data is unavailable or difficult to obtain [START_REF] Sterman | All models are wrong: Reflections on becoming a systems scientist[END_REF]. The framework does not include validation as a separate activity. Rather, as seen in Figure 1, it is integrated in all steps of the framework.

Figure 1 emphasises the iterative nature of the framework. The cost model is gradually refined as process models are developed and linked with operations models. As the model matures, new insights can lead to a revised problem statement, e.g. when a cost element that initially was thought to be important turns out to be small in relation to another cost element that was excluded at first.

During later stages, as the model reaches a state where it can be assumed to properly reflect the characteristics of the production system, the focus of exploratory data analysis shifts towards a search for improvement trajectories.

Case study from the steel industry

This section presents a case study where the five-step framework is used to assess improvement potential in an integrated mill for production of stainless steel strip.

The case study is based on a four year study, from 2004 to 2007, at a major stainless steel producer. During this time, the author had the role of embedded researcher within the process development group at the hot rolling mill.

The production environment

The plant, shown schematically in Figure 2, is representative for many integrated steel mills around the world. Liquid steel is produced from melting and refining of scrap. The metal is cast into workpieces (called slabs) of approximately 25 tons each through continuous casting (CC). Slabs are then hot rolled to strip in a hot rolling mill (HRM).

(Figure 2) Hot rolling is strictly in response to customer orders, but since customers often order less than 100 tons, which is the minimum batch size of the meltshop, an inventory of excess slabs is kept in anticipation of future sales. Transfer times for slabs from casting to rolling are in the order of a few days for ordered slabs, while excess slabs may remain in the slab yard for weeks or months.

Before hot rolling, slabs are reheated to about 1250°C. This is where the potential for energy reduction is most evident since shorter transfer times yields a higher charging temperature. So-called hot-charging, defined as continuous operation at more than 400°C charging temperature [START_REF] Tang | A review of planning and scheduling systems and methods for integrated steel production[END_REF], is an attractive option to reduce energy consumption. However, like many others [START_REF] Beck | Hot charging: Easier said than done[END_REF], the plant has had difficulties to achieve the required reductions of work in process (WIP) and transfer times.

The product range is highly diversified. Production data from a period of six months showed that 7% of the product variants represented 50% of the produced volume (Figure 3). Products can be classified as either high-volume standard grades or low-volume special grades. Additionally, it was possible to differentiate products as either narrow or wide. Four product groups were thus identified, classified by turnover and width.

(Figure 3) Management recognises that the difficulties to implement hot charging are caused by high variability due to the diversified product range, and that existing production equipment may not be optimally designed to handle this variation. Product range was therefore a key design variable in the case study. The question is what capabilities to develop in order to follow an improvement trajectory towards low-cost high diversity production.

Step 1: Problem identification

It was clear that energy consumption in the slab reheating furnaces was a key contributing factor to production cost. Energy consumption in the furnaces depend on many factors, e.g. furnace throughput rate [START_REF] Chen | Analysis on energy consumption and performance of reheating furnaces in a hot strip mill[END_REF], which is in turn influenced by other factors that are normally not regarded to belong to the furnace subsystem. Hence, the performance of the slab reheating process depends on work practice (operations) as well as choice of equipment (processes). Initially, key factors were identified and a conceptual system dynamics model was produced during a project that involved members from the project engineering, process development and logistics functions in the rolling mill. The model was based on high level assumptions on how operations and manufacturing processes influence each other. Despite the crudeness of this model, it provided valuable insights that guided the further analysis.

The development of the first model was followed by the development of improved cooling and reheating models. This involved numerical simulation of the cooling and reheating processes. The cooling model was validated against temperature measurements on slabs in the slab yard, while the reheating model was validated against data from the reheating furnace control system. Equations were adapted to the validated models and integrated in the cost model.

In retrospect, the problem identification phase can in itself be seen as an iterative procedure that contains all stages of Figure 1. Development of conceptual models involved aspects of both process and operations modelling. New insights resulted in the formulation of new hypotheses, and completed iterations yield input to the problem identification stage of later iterations.

Steps 2 and 3: Process and operations modelling

As mentioned initially in Section 4, the influence of product diversification was a key issue. The full range of products was aggregated into four groups, and implemented in the model with subscripted variables to create four parallel flows.

The structure of the model in its later stages of development is shown in Figure 4. It includes production scheduling, product range and volumes, energy consumption and throughput. It also includes setups due to work roll changes and caster restarts. Newly cast slabs are transferred to the slab yard before they are charged into the reheating furnace. They are then reheated to rolling temperature, discharged from the furnace, and processed in the rolling mill.

(Figure 4)

Figure 5 shows how slabs are first given the status of unordered slabs. The lead time calculation for each group is conceived to preserve the mean customer order frequencies for product groups as discussed in Section 4.1.

Ordered slabs that have been scheduled on the rolling mill are transferred to a reheating buffer, linking Figure 5 to Figure 6, from where they are charged into the reheating furnace.

(Figure 5) Discrete event modelling techniques was used to model production scheduling, since:

• scheduling operations occur infrequently in comparison to the time resolution of other processes in the model, and • the influence of scheduling on aggregated production flow was unclear.

It is assumed that material is perfectly blended in the slab yard. The average age of material in stock, buffer age in Figure 5, is then calculated for each product group.

Figure 6 shows the furnace model with reheating modelled as a two stage process. Cold slabs enter the reheating stage and undergo a gradual change of state to become reheated (Figure 6)

The final model is the result of an evolutionary development where many, often more complex, intermediate solutions have been rejected. It was sometimes necessary to conduct detailed analyses in order to establish how variables influence each other. For example, a 2D finite difference (FD) model was used to estimate the charging temperature. The FD model was tested through pyrometer measurements on real slabs in the slab yard in front the hot strip mill.

Another 2D FD model was also used to predict the reheating time as function of charging temperature. Simple mathematical expressions were then fit to the simulated cooling and heating curves and inserted into the system dynamics model. Figure 7 shows an example of a curve produced during validation of the reheating model. Charging and discharging rates from 48 h production was taken from the furnace control system in the rolling mill and supplied to the model. A time series of actual reheating power was then plotted together with the simulation output as shown in Figure 7. Considering the simplicity of the model, the comparison shows a reasonable agreement between simulation results and measured data.

(Figure 7) Capital, labour and reheating cost rates were integrated over time to estimate the margin cost of each resource. Each cost component is a product of a resource price and its cost driver. Hence, the reheating cost was calculated from the energy price, multiplied by the furnace power (cf. Figure 7). The energy price was in turn based on the price of propane gas and the heat content of the gas.

Step 4: Sensitivity simulation

Once the model has reached a state where it executes properly, it is subjected to Monte Carlo sensitivity simulation. Several parameters are varied simultaneously according to random uniform distributions within chosen ranges. The values of key parameters, such as the different cost components, WIP, furnace utilisation and mean charging temperature, are then exported to a data table.

As discussed below, specialised EDA software is then used to view and evaluate the results.

Step 5: Exploring sensitivity data

The sensitivity simulation results were screened visually with the aid of linked plots using visual exploratory data analysis software. The sensitivity simulation results were pruned to remove branches that represent unfavourable configurations with low production capacity and high cost (Figure 8). A plot of mean reheating cost against mean capital cost (Figure 8a) shows two branches that extrude from the main cluster of points. One is characterised by extreme reheating costs and the other by extreme inventory costs. These branches were brushed to distinguish them from the main cluster.

(Figure 8) A plot of total mean cost against total production volume (Figure 8b), shows how the brushed regions stand out with low to intermediate production volumes and high cost.

Plotting production capacity against furnace utilisation (Figure 8c) shows that low furnace utilisation is the main cause of high reheating costs. Contrary, high inventory costs are caused by over-utilisation of the furnace, turning reheating into a bottleneck.

Figure 8d shows that, in general, high mean reheat costs (Figure 8a) correspond to low charge weights and long setups in the caster. The result is low production capacity in the caster, which may cause poor utilisation in the reheating furnace if there is a capacity imbalance. At the same time, high inventory costs are associated with under-capacity in the reheating furnace. The rolling mill thus becomes a bottleneck, causing in-process slabs to stock up in the slab yard.

Once unfavourable configurations representing apparent capacity imbalance have been removed, the opportunities to reach the strategic goals by following a planned improvement trajectory can be investigated. This is done by brushing a desired future cost and product range region, and exploring which mill configurations yield the desired results.

Figure 9 illustrates how design parameters can be identified based on prior brushing of target regions in linked data plots. The figure shows a two-step trajectory towards increased product range with maintained or reduced cost.

(Figure 9) Starting from Figure 9a, which shows the remaining data points when the high-cost branches seen in Figure 8 have been removed:

• A region corresponding to the current configuration and product range is brushed.

• A second region, representing the first step towards lower cost and increased diversification, is brushed with differing colour and symbol. • A third region, representing further cost reduction and increased range, is brushed.

All remaining points except those that belong to the desired future configurations are then hidden (Figure 9b).

Figure 9c shows that, despite a significant spread, there is a trend that reduced mean cost is linked to increased production volumes. This is sometimes satisfactory, but if the firm's strategic goal is to maintain its current market share, cost may have to be reduced in absolute numbers instead.

A plot of setup times in the caster against charge weight (Figure 9d), shows that reduced setup times in the caster are required. In order to evaluate the effects of the charge weight, configurations with charge weights below the current 100 tons are brushed to distinguish them from higher charge weights as shown in Figure 10.

(Figure 10) Figure 10a confirms that increased production volumes on average yield lower mean costs. However, this figure also shows that reduced costs can be accomplished without, or with little, volume increase if the charge weight is reduced. Increased charge weight typically increases the production capacity for a given product range (Figure 10b), but it must be accomplished by setup time reduction (Figure 10c).

Setup time reductions increase the production capacity and, as expected, there is also a correlation with cost (Figure 10d). However, in order to accomplish real cost reductions, it is necessary to also reduce the charge weight. The elimination of excess slabs results in reduced WIP and inventory costs in the first step. Charging temperatures can then be increased in the second step, with lower reheating costs as a result (Figure 10e). Production capacity can be maintained, or even increased, since setup times are reduced.

Identifying an improvement trajectory

Based on the results of the sensitivity analysis, a two-step program to reduce cost and increase diversification was outlined. Each step involves a number of initiatives to develop new manufacturing technologies and operational practice. The suggested changes, based on the analysis in Section 4.5, are summarised in Table 1.

(Table 1) New sensitivity simulations were conducted in 250 replications, with the design variables varied according to Table 1. The output was compiled in a parallel coordinate plot (Figure 11), which shows how configurations B and C permit increasing production volumes and reduced cost despite increasing product range.

(Figure 11) Table 1 specifies long-term targets for improvement in terms of generic capabilities of the manufacturing system. These new capabilities can only be the result of long-term organisational commitment. Thus, the organisational capabilities needed to implement the suggested change must be developed first.

Table 2 lists development of setup time reduction, innovative scheduling policies and new process technology as key organisational capabilities that must be developed before the desired improvement trajectory can be realised.

(Table 2)

Reduction of setup times in the hot rolling mill facilitates more frequent roll changes, thereby shortening the scheduling horizon. Transfer times can then be reduced, which increases the mean charging temperature. Dynamic updating of schedules ensures that slabs reach the rolling mill with as little delay as possible. The responsibility for production scheduling can be decentralised and new scheduling policies introduced.

Reduced charge weights may involve making larger batches of a few base grades, while alloying to final composition is delayed. This can be accomplished through ladle metallurgy, or even though just-in-time alloying using techniques such as tundish metallurgy or mould metallurgy [START_REF] Bentell | Just-in-timelegering av stål[END_REF].

Discussion

The choices of market segment and product range are key decisions for any firm competing on an international or global market. According to the capability based view of the firm, manufacturing cost is limited by the existence of competitive frontiers, which are given by current technological and operational practice. This current 'best practice' defines the competitive frontier that limits the possibilities to obtain competitive advantages in relation to other firms [START_REF] Hayes | Manufacturing strategy: at the intersection of two paradigm shifts[END_REF]. Since most firms in a particular industry use similar equipment and organisation, competitive advantage is based on optimal utilisation of resources such that unit cost is minimised in relation to firms competing on the same frontier. This is sometimes interpreted as if a wider product range must come at the expense of increased unit cost. However, development of new production capabilities can increase the level of process flexibility such that production cost decrease despite increased product range.

The presented framework emphasise the role of model testing. It is argued that learning is primarily taking place within the testing loop. The existence of transitory stages in modelling is a natural part of scientific inquiry [START_REF] Popper | The Logic of Scientific Discovery[END_REF]. The model matures for each testing loop, which means rejection of some model features while others are strengthened or introduced. In this way, an awareness of what manufacturing features and capabilities that are needed to support the strategic goals can develop gradually. The use of system dynamics allows production to be interpreted in terms of flows instead of individual objects that move through the plant. Hence, it was easy to adopt a high level perspective, where aspects of production could be discussed in terms of generic features instead of the specific choice of technology to be used. Still, this paper has been concerned with low-level analysis of manufacturing in the sense that it concerns choices of manufacturing technology and operational practice on the plant level as opposed to the supply chain perspective.

Product groups were implemented as parallel flows within the model. This technique allows high and low volume products, as well as other groups distinguished by some characteristic property such as slab width, to be easily introduced in the model. The effect of varying degrees of product differentiation can be assessed without including detailed data for individual products in the model.

In the case study, improvement goals were given by the firm's business strategy, and involved requirements for increased diversification with simultaneous cost reductions. A goal driven explorative procedure was used to reduce the data set such that only points representing the current state and target regions remained. This process is illustrated by Figure 12.

(Figure 12)

Figure 12a shows how a current state region, A, has been brushed, along with two desired future state regions (B,C). The frontiers indicate the limitations of cost versus range at each stage. The plant can operate anywhere above its current competitive frontier, but in order to open up a new frontier, innovations in manufacturing technology or routines are required [START_REF] De | Definitions and linkages between operational and strategic flexibilities[END_REF][START_REF] Hayes | Manufacturing strategy: at the intersection of two paradigm shifts[END_REF].

Figure 12b shows how the required change in design parameters can be identified. At the same time, the cost profile changes as seen in Figure 12c, with a reduction of tied capital in the first step, followed by a reduction of reheating costs in the second.

The combination of Monte Carlo simulation followed by goal-driven data analysis reveals the potential in flexible small lot steel refining and casting technology. This technology may not be available off the shelf, and the strategic implication must be the need to invest in the organisational skills needed to lead the development of new technology. Hence, change is not primarily about choosing what equipment to invest in; rather it is about developing the required organisational capabilities needed to identify how to best tailor manufacturing to handle the variation that follows from the chosen business strategy.

Conclusions

This paper presents an attempt to bring together system dynamics, cost modelling and explorative data analysis into a coherent framework that can aid the selection of improvement initiatives within the chosen manufacturing strategy.

The proposed five-step framework provides a structured method for evaluation improvement initiatives concerning new manufacturing technology and routines through dynamic cost modelling.

• A case study from the steel industry demonstrates the applicability of the five-step framework to support the choice of an improvement trajectory (Section 4). • The use of visual explorative data analysis speeds up model development and improves model quality. • Interactive linked plots facilitate a goal driven analysis. Regions of desired performance and product range are identified, and required changes in design parameters can be identified directly in corresponding linked views.

In summary, the use of dynamic cost models, within the presented five-step framework, provides a means to interpret the strategic goals, expressed e.g. as production capacity and cost, in terms of required capabilities. Strategic manufacturing initiatives can thereby be directed to support the chosen business strategy.

  International Journal of Production Research slabs. The transformation is modelled as a delay, where the transfer rate is influenced by a number of environmental factors as seen in the figure.

Figure 1 .

 1 Figure 1. The five-step dynamic cost modelling framework.

Figure 2 .

 2 Figure 2. Steel production with continuous casting, reheating and hot rolling. The slab yard serves as buffer and inventory.

Figure 3 .

 3 Figure 3. Order frequency for products entering the hot rolling mill in the case study: 7% of product variants represented 50% of the volume.

Figure 4 .

 4 Figure 4. The three-layered cost model with processes, operations and cost components. The del contains four parallel product flows.

Figure 5 .

 5 Figure5. The interface between continuous casting and hot rolling, with order matching and scheduling.

Figure 6 .

 6 Figure 6. Reheating modelled as a two step process.

Figure 7 .

 7 Figure 7. Validation of simulated furnace power (solid) against plant data (-○-). Charging and discharging rates in the model were based on smoothed data from the production database.

Figure 8 .

 8 Figure8. Pruning of sensitivity results to remove configurations with high capital cost (+) and high reheating costs (×) from the main cluster (○). High-cost branches (a) are generally associated with reduced production capacity (b). Capacity imbalance may result in low furnace utilisation, or high buffering costs if the furnace is a bottleneck (c). Corresponding caster configurations fall in two distinct regions (d).

Figure 9 .

 9 Figure 9. Brushing a desired two-step improvement trajectory towards increased product range and lower cost. Current configuration indicated by (×), first step by (•), and second step by (■).

Figure 10 .

 10 Figure10. Exploring simulation results for configurations that combine increased product range with low cost. Current configuration indicated by (×). Configurations satisfying first goal with increased charge weights indicated by (•), and with reduced charge weights (○). Second goal with increased charge weights (■), and with reduced charge weights (□).

Figure 11 .

 11 Figure11. Parallel coordinate plot showing design parameters and simulated performance in terms of total mean cost and production volume for current (A) and future (B, C) plant configurations according to Table1.

Figure 12 .

 12 Figure 12. Goal driven explorative dynamic cost modelling: a) Current state (A) and improvement goals (B,C) are brushed. b) Process requirements are identified. c) Change in margin costs for tied capital, c WIP , and reheating costs, c E .

  Order frequency for products entering the hot rolling mill in the case study: 7% of product variants represented 50% of the volume. 197x93mm (600 x 600 DPI)The three-layered cost model with processes, operations and cost components. The del contains four parallel product flows. 277x133mm (600 x 600 DPI)The interface between continuous casting and hot rolling, with order matching and scheduling. 84x64mm (600 x 600 DPI) Validation of simulated furnace power (solid) against plant data (-○-). Charging and discharging rates in the model were based on smoothed data from the production database. 265x147mm (600 x 600 DPI) Pruning of sensitivity results to remove configurations with high capital cost (+) and high reheating costs (×) from the main cluster (○). High-cost branches (a) are generally associated with reduced production capacity (b). Capacity imbalance may result in low furnace utilisation, or high buffering costs if the furnace is a bottleneck (c). Corresponding caster configurations fall in two distinct regions (Brushing a desired two-step improvement trajectory towards increased product range and lower cost. Current configuration indicated by (×), first step by (•), and second step by (Exploring simulation results for configurations that combine increased product range with low cost. Current configuration indicated by (×). Configurations satisfying first goal with increased charge weights indicated by (•), and with reduced charge weights (○). Second goal with increased charge weights (■), and with reduced charge weights (Parallel coordinate plot showing design parameters and simulated performance in terms of total mean cost and production volume for current (A) and future (B, C) plant configurations according to Table 1. 262x68mm (600 x 600 DPI) Goal driven explorative dynamic cost modelling: a) Current state (A) and improvement goals (B,C) are brushed. b) Process requirements are identified. c) Change in margin costs for tied capital, cWIP, and reheating costs, cE. 172x57mm (600 x 600 DPI) Page 29 of 29 http://mc.manuscriptcentral.com/tprs Email: ijpr@lboro.ac.uk International Journal of Production Research

Table 1 .

 1 Values of design variables for current configuration (A), and for future configurations (B, C) on suggested improvement trajectory.

	Configuration	A	B	C
	Number of variants	150-200	200-300	300-400
	CC setup time	1-1.5	0.6-0.9	0.33-0.5
	Charge weight	90-100	70-90	50-70
	HRM setup time	1-1.5	0.6-0.9	0.33-0.5
	HRM schedule updates	0-1	2-3	4-6

Table 2 .

 2 Manufacturing competence and capability requirements.Steel production with continuous casting, reheating and hot rolling. The slab yard serves as buffer and inventory. 251x80mm (600 x 600 DPI)

	Manufacturing competences Capability requirements
	a) CC setup time reduction	Setup time reduction, development of flexible casting
		technology
	b) HRM setup time reduction Setup time reduction, development of flexible rolling
		technology
	c) Feasible charge weights	Setup time reduction, development of flexible
		refinement and alloying technology
	d) Dynamic scheduling	Decentralised scheduling systems, review of
		scheduling policies

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Acknowledgement

This research was funded by the Swedish Knowledge Foundation (KK-stiftelsen) and the Swedish Steel Producers Association (Jernkontoret).