Contents

1. Introduction

At the core of corrugator scheduling resides the following problem:

Given

• A set of orders, all of the same board type
• A set of available roll widths

Find

The optimum corrugator schedule that produces the board sheets of the given orders, using the available roll widths.

For a schedule to be considered optimum, there must be some optimization criterion on which different possible schedules can be compared. Various optimization criteria are traditionally used, including minimum trim ratio (min TR), maximum utilization (max U), and minimum cost (min C).
 

There are two optimization criteria which directly serve the main economic objective of a plant: maximum profit and maximum profit rate.

There are two optimization criteria which directly serve the main economic objective of a plant: maximum profit (max P) and maximum profit rate (max PR). The other commonly used optimization criteria (min TR, max U, min C) do not necessarily result in schedules that realize maximum profit or maximum profit rate. This simply means that by optimizing a corrugator schedule on a criterion other than max P or max PR, we may be missing an attainable higher profit or profit rate. This white paper focuses on showing why and how min C can deviate from max P.

2. The Profit of a Corrugator Schedule

Carrying out a corrugator schedule incurs certain costs and results in some revenues. The profit of a schedule is the difference between the revenues and the costs.

The cost elements are:

1. The cost of the raw material (paper, adhesives, etc.) required for carrying out the schedule
2. The cost of operating the corrugator

The revenue elements are:

1. The money value of the produced cardboard sheets
2. The resell value of the side trim (as waste)

We can, however, regard the corrugator as a separate production unit, and associate a money value with its products for the purpose of measuring and driving its economic performance.

The term “money value” of the produced cardboard sheets deserves careful attention. In sheet feeder plants, the produced sheets are the end product. Therefore, the money value of the produced sheets corresponds directly to their price. On the other hand, in box plants, the blank sheets produced by the corrugator are not the end product. We can, however, regard the corrugator as a separate production unit, and associate a money value with its products for the purpose of measuring and driving its economic performance.

The Cost/Revenue Parameters
To be able to compute the revenue and the cost elements of a schedule, certain parameters must be known:

1. The money value, per unit area, of the produced sheets
2. The resell value, per unit area, of side trim
3. The cost of raw material per unit area of the web
4. The corrugator operation cost per hour
5. The corrugator speed

Although the orders involved in a schedule are all of the same board type, the money value per unit area of produced sheets might still vary from an order to another. This is mainly because these orders might have been priced differently. Perhaps one client has obtained a discount, and another client was convinced to pay more than the standard price. Similarly, a quality-upgraded order usually has a lower money value, since it was priced according to its original lower quality.

The fact that the money value per unit area of sheets can vary from an order to another requires that such money value be specified for each individual order. An equivalent alternative is to use a single standard money value per unit area (for each board type), and just specify a relative value for each order. The relative value of an order expresses how the produced blank sheets of that order are priced or valued relative to the standard money value for the board type. For example, the relative value is 1.05 for an order whose sheets are priced 5% higher than the standard, and is 0.97 if the sheets are priced 3% lower than the standard. Expecting the majority of the orders to be priced at the standard, the relative value will be 1 for most of the orders.

 

3. The Profit Rate of a Corrugator Schedule

The profit rate of a corrugator schedule is the profit it realizes per unit time. For example, if a schedule has a profit of $7200 and carrying out the schedule takes 6 hours, the profit rate is $1200 per hour, or $20 per minute. There are certain circumstances that would call for maximizing the profit rate, rather than the profit, of a corrugator schedule.
 

At times of high demand for corrugated cardboard, maximizing the profit rate is preferred. On the other hand, when demand is slack, maximizing the profit can be more advantageous.

The difference between maximizing the profit and maximizing the profit rate is subtle. It is like when you choose between 2 jobs; in the first you earn $10,000 in one month, and in the second you earn $17,000 in two months. Being interested in earning as much as possible, you would choose the first job (higher profit rate, but less total profit) only if you have a good chance of getting a new job that provides you with more than $7,000 the next month. This is also the case in corrugator scheduling. At times of high demand for corrugated cardboard, maximizing the profit rate is preferred. On the other hand, when demand is slack, maximizing the profit can be more advantageous.

Even at times of high demand, there can be situations that would favor maximizing the profit over maximizing the profit rate. Consider, for example, a situation where the capacity of the available conversion machines is less than enough for keeping up with the corrugator outcome at high utilization. In this case, maximum-profit-rate schedules are likely to produce sheets out of the corrugator at higher rates than the capacity of the conversion machines. The corrugator will have to stop production from time to time to match the capacity of the conversion machine. Making the corrugator idle periodically defeats the purpose of maximizing the profit rate instead of maximizing the profit.

 

In fact, the overrun and underrun tolerances in cardboard manufacturing orders open the door wide for corrugator scheduling to affect the revenue.

4. How Scheduling Can Affect the Revenue

It is almost always that corrugator schedulers focus on reducing the production cost as means of increasing the profit. Although “profit = revenue – cost”, revenue is seldom attended to in attempting to increase the profit. Possibly, this is due to a supposition that the revenue is fixed anyway for a given set of orders. The basis of this supposition is that the price paid by the customer for each order has been already agreed on, independent of how the orders are scheduled. However, in most cases this reasoning is incorrect. In fact, the overrun and underrun tolerances in cardboard manufacturing orders open the door wide for corrugator scheduling to affect the revenue.

It is common practice in the corrugated cardboard industry that an order specifies a quantity with overrun and underrun tolerances. For example, an order may specify a quantity of 10,000 sheets/boxes with maximum overrun of 10% and maximum underrun of 5%. In this case, any number of sheets/boxes between 9,500 and 11,000 may be produced, and they will be accepted by the customer at the pro-rated price.

Using overrun and underrun tolerances to express a flexible quantity is customary in contracting and in placing orders. However, in the context of reasoning about corrugator scheduling, we prefer to express a flexible quantity as a pair: a must quantity and an optional quantity. The must quantity of an order is the minimum number of sheets to be produced for that order. The optional quantity of an order is the number of sheets that may optionally be produced, in whole or in part, in addition to the must quantity for that order. The optional quantity (or any part of it) will be accepted by the client at the regular price. Referring to our earlier example where the number of sheets should range from 9,500 to 11,000, the must quantity is 9,500 and the optional quantity is 1,500.

It should be clear that the more of an optional quantity is produced, the more is the revenue. Cost may also increase as more of an optional quantity is produced. However, it is not necessarily the case that cost and revenue both increase at the same rate. The rate of increase in cost depends heavily on the combination patterns of the schedule, and it can be highly nonlinear. Therefore, at certain combination of optional quantities, the difference between revenue and cost is at its maximum. This is the maximum-profit point. As will be demonstrated in the next section, the maximum-profit point is not necessarily the minimum-cost point. Moreover, maximizing the profit does not necessarily mean producing all the optional quantities. Similarly, minimizing the cost does not necessarily mean producing none of the optional quantities.

 

5. Illustrative Example

In this section we give a scheduling example that illustrates the ideas discussed so far.  The input for our example consists of a set of orders to be scheduled, the available roll size(s), and the optimization parameters.  The input data is shown below.

Next, we present 2 schedules for the given orders: the first schedule minimizes the cost, and the second maximizes the profit.

The production of each of the 2 schedules is shown in the following table:

The following table shows a comparison between the min-cost schedule and the max-profit schedule.

Those who minimize the cost – instead of maximizing the profit – would miss such higher profit.

A max-profit schedule does not necessarily produce all the optional quantities.

A min-cost schedule does not necessarily refrain from producing optional quantities.

Observations

1. The profit from the max-profit schedule is 10.6% higher than the profit from the min-cost schedule ($5,475 vs. $4,951). Those who minimize the cost – instead of maximizing the profit – would miss such higher profit.
2. A max-profit schedule does not necessarily produce all the optional quantities. In our example, the max-profit schedule produced only 72% of order #2. Attempting to produce more of the optional quantity of order #2 would decrease the profit.
3. A min-cost schedule does not necessarily refrain from producing optional quantities. In our example, the min-cost schedule produced only 53% of order #1.

A min-cost schedule can have a significantly lower profit rate than its max-profit-rate counterpart.

6. Minimizing Cost vs. Maximizing Profit Rate

We have already established that minimizing the cost is no substitute for maximizing the profit. But can it be a substitute for maximizing the profit rate? The answer is no. The following example demonstrates that a min-cost schedule can have a significantly lower profit rate than its max-profit-rate counterpart.

The input for our example consists of a set of orders to be scheduled, the available roll size(s), and the optimization parameters.  The input data is shown below.

Next, we present 2 schedules for the given orders: the first schedule minimizes the cost, and the second maximizes the profit rate.

The production of each of the 2 schedules is shown in the following 2 tables:

The following table shows a comparison between the min-cost schedule and the max-profit-rate schedule.

Observation
The profit rate of the min-cost schedule is $19.12/min. The maximum possible profit rate is $23.68/min, which is about 24% higher than the profit rate of the min-cost schedule.  Those who minimize the cost – instead of maximizing the profit rate – would miss such higher profit rate.
 

7. Minimum Cost per Unit Area

Instead of minimizing the total cost of the schedule, some corrugator schedulers minimize the cost per unit area of the produced sheets (min CPA) . Like min-cost, min-CPA can score poorly on both profit and profit rate. This can be illustrated by an example.

Consider the same orders, roll widths, and cost/revenue parameters of the example in the preceding section. In what follows we present the schedules for min-CPA, max-profit, and max-profit-rate.

The production of each of the 2 schedules is shown in the following 2 tables:

The following table shows a comparison between the min-cost schedule and the max-profit-rate schedule.

Observations

1. The profit of the min-CPA schedule is $6,986, while the maximum possible profit is $7,572. That is, the maximum profit is about 8% higher than that of the min-CPA schedule.
2. The profit rate of the min-CPA schedule is $20.19/min, while the maximum possible profit rate is $23.68/min. That is, the maximum profit rate is about 17% higher than that of the min-CPA schedule.
3. In min-CPA, the profit per 1000 m² of produced sheets ($89.7) is the highest among the three schedules. However, this does not do us any good regarding maximizing the profit or the profit rate.

Automated corrugator schedules should support maximizing the profit and the profit rate per se, not by attempting to minimize the cost.

8. Conclusion

In corrugator scheduling, maximizing the profit and maximizing the profit rate are prominent optimization criteria because they contribute directly to the economic objectives of the plant. In this paper we have shown that minimizing the cost can have adverse effects on both the profit and the profit rate. Therefore automated corrugator schedules should support maximizing the profit and the profit rate per se, not by attempting to minimize the cost.

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