[Home] [APPW 2004] [Journal papers] 

FAVORING PROMPT DELIVERY IN THE TRIM LOSS PROBLEM
Authors Adolf Diegel^{1}, Sias van Schalkwyk^{2 }and Olaf Diegel^{3} Companies ^{1}Cutline Research Keywords deckle, trim, loss, stock, deckling programme, prompt delivery
ABSTRACT Deckling, cutting stock or trim loss problems arise when small units are to be fitted into large ones. One aims to reduce stock usage and setups, then favors long runs, prompt delivery, surplus over waste and similar pattern loss. One may also want to enforce exact solutions, with zero tolerance for surplus, or those with a narrow limit on pattern loss. Prompt delivery, in particular, can be guaranteed by planning from day to day, for successive short periods. Otherwise, for long periods, one still seems to be on time by running first patterns with early orders. Either approach is costly. In the first case, short periods may not provide enough variety to make for efficient stock usage: the same data combined into one long period tend to use less stock. As for favoring patterns with early orders, such patterns tend to yield also reels for delivery on subsequent days. In turn, the more latedate reels come in early runs, the more early reels are pushed toward the end of the period. Latedate reels in early runs incur extra storage cost. Early reels in subsequent runs incur penalties for tardy delivery. We explore the interaction between stock usage, storage and delivery and suggest how to strike a balance: for each pattern, weight each order's volume by its age, then sort patterns accordingly. This approach may not be optimal, but it clarifies the forces at work. Prompt delivery vs stock usage Assuring prompt delivery by means of short planning periods tends to increase both stock usage and the number of setups. To illustrate this claim, first a minimal example. From 6500 mm stock one is to obtain reels of 800 and 700 mm. Conveniently, the 800s are to be delivered before the 700s, today rather than tomorrow or this week rather than next week. One may use any relevant time unit, but one must not avoid the issue by saying "a few days don't matter". To guarantee prompt delivery, one puts early periods before late ones, each period on its own, first week 1, then week 2, as in Table 1. Table 1. Prompt delivery assured by deckling each period on its own By the end of week 1, 800 mm reels are to be delivered, 160 units. For 700s, it is 560 units at the end of week 2. With a winder output of say 10 sets per day, each schedule is feasible . It is on time if one starts about 2 days before the end of week 1, then continues day by day. It is also costly. Below the weekbyweek plans compared to an optimal, least stock solution for the same demand planned in one period. Table 2. Planning for each week on its own vs for both weeks at once The ontime plans need 83 sets, with much loss, surplus and 2 setups. The other plan uses 80 vs 83 sets, with neither loss nor surplus and a single setup. This is achieved by combining 800s and 700s. In turn, the combined plan is optimal except perhaps for delivery: Instead of having a steady stream of 800*8 to be shipped as they come off the winder, 800s come 2 per run over 80 runs or 8 days. So, if production starts on the same day as in the weekbyweek plans, 800s will not be ready until the end of week 2, one week late. Simultaneously, 700s are produced as of day 1. They will be ready on time, but two day's production, in week 1, must be stored. One can eliminate late delivery by starting about 8 days before the end of week 1 and have both sizes ready at that point. This means even more storage for the 700s, those wanted only at the end of week 2. Penalties for late delivery and early storage may cost more or less than higher stock usage and one will act accordingly. This includes keeping in stock those sizes whose storage costs less than late delivery. In short, stock usage is almost certainly higher for short periods than for long ones. At least, this is so for the data as they stand. One may dismiss these as academic, but the problem remains. Comparative advantage Before adapting the data for further analysis, one may want to reflect on the principle of comparative advantage. It helps to understand why combining orders is so much more efficient than deckling them on their own, for short planning periods. Briefly, the principle is  sizes fitting poorly on their own may yet combine perfectly. For example, with 6500 mm stock, 800*8 leaves 100 mm loss, while it is 200 for 700*9. Each size on its own fits poorly, 700s being worse. Yet 800*2 + 700*7 combine perfectly. At 7/2, one has more units of the worse size: the ratio gives 700 not an absolute, but a comparative advantage. If 700 fits poorly into 6500, 600 is even worse, with 500 mm loss. Yet combined with the other sizes, it, too, can make good patterns. Indeed, 800, 700 and 600 in 6500 can form 48 unique patterns and, amazingly, 7 without any loss. The full number will be analyzed below, but, to begin with, one may prefer to work with perfect patterns. Having no loss, each uses the stock width fully, so net and gross weight are the same. This simplifies the analysis without affecting its principle. Conveniently, the 7 noloss combinations could be orders for as many days, say Monday to Sunday, as shown at left in the plan below. Table 3. Poor sizes can make perfect plans Monday's orders are for 800*50, 700*10 and 600*30, pattern (5 1 3) run 10 times. If that many sets per day are feasible, orders are shipped on time. This holds for Monday as well as for the rest of the week, with a new setup for each day. Yet the plan at right fills the same orders with only 3 setups and long runs, but not with all orders on time. Similarly, among the 7 noloss patterns on could retain only those which contain each size at least once, every day so to speak. There are 5 such patterns, so the example suits those who plan Monday to Friday. Table 4. Each size demanded daily One may be pleased with the daily plans. One may also be surprised that such different patterns add up to similar totals. One will be even more surprised to see that these totals can be had with longer runs and fewer setups by planning for the week as a whole: the comparative advantage of combining sizes helps to reduce setups as well as total loss, if any. In short, it may so happen that daily scheduling, with prompt delivery, can be done without extra stock usage. Yet even then, weekly planning is likely to obtain the same totals with fewer setups. This may hardly matter with automatic knife changes, but manual setups tend to cause waste and delay, disturbing the best laid plans. Planning for longer horizons The plans above were kept short and simple. What happens if one deals with larger sets of data? Such sets could involve genuine data. Yet a consistent and instructive case can come from the data at hand, simply by exploding them into all possible combinations. Data thus generated are not random. They proceed systematically from wide to narrow reels, with more wide reels in early patterns: if early patterns carry early dates, age is related to size. Such an age/sizerelation is not likely to exist for genuine data. Yet, precisely because we want to study it, and know that a relation exists, the generated data are more relevant than random ones. So, to continue with 800, 700 and 600 in 6500 mm, what if any size is on order, any number of times, on any day? Table 5. All possible combinations over 48 days The table was split crudely into halves. Still, it reads as those above: on day 1, only 800s are wanted, 10*8 or 80 reels. Then one proceeds to the next day and finally to day 48 with only 600s on order. As for size being related to delivery date, the trend is not simple. Yet we do start with many wide reels and end with many narrow ones. Filling all orders day by day has all on time, with 48 setups and the totals in the bottom line, for stock usage, reel numbers and loss. Similar totals for the period as a whole can be had from the following plans, with pattern numbers copied from the full plan. Table 6. Least stock usage plans for the same data in one period The upper plans waste 3000 mm, but fill orders exactly. They use 3 or 4 instead of 48 patterns. As amazing is that the lower plans use identical patterns, but with slightly different runlengths. They transform 3000 mm loss into as much surplus, 600*5 vs 800*3 + 600*1. Most amazing, however, is that the least stock plans do with 462 sets vs 480 for ontime delivery: 18 in 462 is 3.8961 or about 4%. This may seem low as a percentage, but the cost of wasting 18 sets is substantial in absolute terms. 45 extra setups, 48 vs 3, may add to this cost. Also, extra stock usage is as low as it is precisely because the data at hand allow so many different combinations. As that number decreases, as fewer reels fit across the winder, waste and stock usage increase. Summary on efficiency The comparative advantage due to combining different sizes is such that sizes which fit poorly on their own yet make good patterns, with little loss, minimal stock usage and surprisingly few setups. The disadvantage is that the sizes must differ, enough so to allow many combinations within a given stock width. Such sizes tend not be on order at the same time, reel width being unrelated to delivery date. In other words, it is unlikely that wide and narrow reels are wanted at the same time, nor one width consistently sooner or later than another. Therefore good cutting patterns combine/mix unrelated delivery dates. In turn, if one runs first patterns with earlydatereels, one tends to simultaneously get reels for subsequent delivery. In turn again, the more latedatereels appear in early runs, the more early reels are pushed into later runs. The conflict is not resolved simply by saying "first run patterns with early orders". Favoring those orders forces higher cost on others, for storage or late delivery. To reduce total cost, one must aim to balance storage against prompt vs tardy delivery. In doing so, one considers not only an order's date, but also its volume. A compromise: weighting plans by urgency Roughly speaking, one favors large orders for tomorrow over small ones for today. This idea is quantified by weighting an order's volume by its urgency: one computes the sum of weight*age for each pattern, then sorts patterns according to that sum, from high to low. An order's volume speaks for itself: the greater its weight, the more important the order. Weight is computed for each order in each pattern, formally, because the same size may go to several clients, with the same or different dates: each matters on its own account. As an order's weight directly reflects its importance, so its date does the opposite if set to the corresponding day of the planning period. The most immediate or urgent order would be on day 1, a low age supposed to signify great urgency. Yet multiplying a large volume by a low date reduces an order's significance. This can be remedied by counting age backwards, from the end of a plan's horizon, so the order for delivery on the last day has the lowest age. More precisely, one sets the horizon 1 day beyond the number of days in a planning period, then subtracts the number of the delivery day/date. For example, for a period ending on day 48, the horizon is 49. Then age, lead time or timetohorizon for any given date is counted backwards as "horizon  date". 48 is 1 step below the horizon, 42 makes 49  42 or 7, and the most urgent order for day 1 has age 49  1 or 48: the earlier its date, the greater an order's urgency. So "importance = weight*age". This procedure consistently handles orders without any delivery date, to be shipped when expedient or produced for stock. Their age is zero, "not urgent", so weight*age is zero, too, regardless of volume. Patterns with such orders tend to move toward the bottom of the list. More precisely, with all orders undated, all have zero age and pattern sequence remains unchanged even if the datesorting routine is called. Another special case arises if all orders are dated for the end of the planning period: all have age 1 and weight matters as such. Patterns are now sorted by their weight but, weight depending on runlength, runlength tends to take over. In other words, sorting patterns with endofperiod dates may have the same effect as sorting them by runlength, explicitly. However, there is a subtle difference between sorting by runlength vs by weight. It arises from patterns with broke or cutreel trim for a second pass. Broke and trim reduce a pattern's net weight. The weight*age sums count precisely that net or useful weight, excluding broke and trim. So, other things being equal, highloss patterns end up with less importance and move toward the bottom of a cutting plan. In short, the weight*age idea promises not only to reflect common sense, but also to deal consistently with special cases. Unbiased data: a simple example To illustrate this idea, below the previous weekly plan with daybyday delivery. The patterns are first shown in the original sequence, Monday to Friday, then after reversing their order. Table 7. Each size demanded daily, forward and reversed In the forward sequence, at left, on Monday many 800s are wanted and few of the narrow sizes. Then the number of 800s decreases day by day. This relates size to delivery, with more wide reels on Monday than on Friday. The trend is irregular for the other sizes, but still inherent in the way the data were generated. Yet, instead of ignoring this fact, we can profit from it by reversing the plans, at right, then study both cases. Or rather, nothing needs to be studied if the plans are run as shown, 10 each on five consecutive days, with 5 setups: each size is ready when it is wanted. However, we know that the same reel numbers can be obtained with only 2 setups, one for 30 runs, the other one for 20. Table 8. Few setups for weekly plan: which pattern should run first? One will be tempted to align the optimal plans with the daily ones shown above. At left, one obtains first the many 800s for the beginning of the week and few of the narrow widths. At right, it is the opposite. Table 9. Raw data for 5 days with true dates, original/reverse sequence The intuitive approach seems plausible, but is it relevant? One cannot answer that question by inspection, without quantitative measurements such as the weight*age analysis in question. To illustrate it, Table 9 has the data from which the previous plans were derived. As they stand, both sections look deceptively similar. Indeed, demand is identical in either case and so are delivery dates. Yet the numbers to be delivered on each day differ: at right, it is 800*50 on Monday, but 800*10 at left. Reel numbers for given dates differ for each size, on all days other than the one in the middle, Wednesday. Computing weight*age How then do these data fare in terms of weight*age? Relevant sums come from the Master Production Schedule, first for the original data, with 800*50 demanded on Monday. The long run pattern 30*(5 1 3) yields 150 reels of 800 mm, 30 of 700 and 90 reels of 600 mm. Orders are sorted by date, so the 800s go to successive orders of 50, 40, 30, 20 and 10. Table 10. Weight*age analysis for forward data, pattern 1 As for the 700s, 30 are produced. As it so happens, 30 are demanded on Tuesday. So, should all 30 reels from pattern 1 go to Tuesday, leaving the Monday order for the next pattern? At the rate of 10 sets per day, it would be delayed by 3 days. Similarly, the 90 600s in pattern 1 fit orders for 20 + 70 units on Tuesday and Thursday, with the 20 units on Monday and 10 on Wednesday waiting for the next run. Such an analysis reduces orders being split over several patterns, but it is hardly suitable for routine sorting. It entails the evaluation of orders not only for the pattern at hand, but also for those to follow: assessing one pattern simultaneously involves all other patterns. For the moment we aim for a more robust approach, allocating production to orders strictly by date, to fill early orders first, as shown above. Of the 30 700s, 10 are for Monday and 20 for Tuesday. The Tuesday order is for 30 units, so 10 remain for pattern 2. Similarly for the 600s. At least, this allocation and splitting of orders occurs if pattern 1 runs first. To decide whether it should, two columns are added to the schedule, for Age and Weight*age. Specifically, Monday is 5 days below the horizon and 40 tons translate to 200,000 units. The total of all orders potentially filled by pattern 1 comes to 723,000. As for pattern 2, 20*(0 5 5) yields 100 reels each of 700 and 600 mm. We allocate these as before, to the original orders (not the remnants left after pattern 1), to obtain a comparable sum of 442,000. Table 11. Weight*age analysis for forward data, pattern 2 20 runs of pattern 2 fill orders for 700 and 600 Monday to Thursday, but 600*30 on Thursday remain for pattern 1, or some other pattern. In short, for either pattern running first, the sums for weight*age are 723000 vs 442000: clearly pattern 1 carries more weight, literally. It carries more weight because it has a longer run, at 30 vs 20. Is runlength therefore a dominant factor? Not quite. A closer analysis should involve not only total weight, but also the weight*age sums. In the case at hand, with zero loss patterns, tonnage is proportional to runlength, with a ratio of 30/20 or 1.5000. Yet the weight*age ratio is 723/442 or 1.6358. It is higher because the long run is not only longer, it also has comparatively more early orders. Counterproof: the reversed data Clearly, for the original data with 800*50 on Monday, pattern 1 should run first. What about the other arrangement, with this order on Friday? Table 12. Weight*age analysis for reverse data, both patterns Table 12 shows all computations so one can study the effect of the age factor day by day, for 30*(5 1 3) then 20*(0 5 5). The first pattern yields 150, 30 and 90 reels of successive widths, the other one has no 800s, but 100 each of the other sizes. Which should run first? Its sum being higher, the long run should still precede the short one! This result disappoints those who bet intuitively on the other outcome, but it is the more significant as it is unexpected. It also agrees with common sense once one appreciates the relation, or lack of it, between order size and delivery date. As long as order size and delivery date are unrelated, large and small orders occurring anywhere in the planning period, any one pattern has a fair mix of early and latedate orders, so a long run will carry more weight than a short one, literally. As explained previously, the data at hand are not random, but there is some relation between size and date, forward and reverse. This shift in age patterns is captured by weight*age: the formula is sensitive enough to be relevant. Its sensitivity is evidenced by the ratios shown in a summary for both data sets. Table 13. Sums for weight*age, for forward and reverse data In the first case, with the large order for 800*50 on Monday, the long run beats the short one by a multiple of 1.6358. As that order shifts to Friday and others move correspondingly, the ratio falls to 1.2064, well below the 1.50 tonnage ratio. A further shift in dates would eventually favor the short run, as will be seen from the case based on Table 14. Counterproof: equal runlengths Table 14. Raw data for 6 days, Tuesday to Sunday, forward and reverse To test the weight*age approach on neutral ground, so to speak, one may want to examine a case with runs of equal length. Such a case happens to evolve from the original data for 7 days if one omits Monday. Thus, at left, wide reels appear on Tuesday, but there are no 800s on Sunday. The opposite applies at right. Does our approach measure the bias? Table 15. Optimal runs of equal length: which should come first? Instead of 10 runs of 6 patterns on as many days, one can do with only two runs of length 30. Which of these should come first? Table 16. Sums for weight*age for equal runlengths Of course, the data being symmetrical, so are the sums. This confirms that the weight*age formula is consistent. Also, the tonnage being the same in both runs, with a ratio of 1, the higher and lower weight*age ratios show that age does matter. Orders with open delivery What if some orders can be delivered anytime within a planning period, or are produced for stock? They have zero age and do not affect pattern sequence. Thus the plans for Monday to Friday yield the sums below. Table 17. Weighted sums with 800s dated and undated Demand remains as before for the 5day data, but we removed the date from the 800 mm reels, the new sums appearing in the row "No date". Previously volumes were such that the 30run pattern came first for both data sets. With the 800s undated, that pattern's importance falls from 723 to 283 and from 637 to 357 for forward and reverse, respectively. Conversely, the weight*age sums remain the same for the 20run pattern because it does not include 800s, dated or undated. Its sums now being higher, the short run comes first for both forward and reverse data. Results are equally consistent if one adds an undated stock order to the data. For example, 1300 mm go perfectly into 6500, so, if a given number of that size is wanted, one might simply add a pattern to the previous solutions, as shown at left below. Table 18. Naive and optimal plans after adding a stock order The naive solution fills all dated orders first, in roughly 5 days. Then comes an independent run for stock only. That third setup can be avoided by the plan at right. With delivery dates as before, the weight*age sums move the 30run pattern to the top as in the original plan. However, the 700s and 600s combined with 1300 come at the rate of 2 vs 5 per run and finish only on day 8. Those who can quantify the cost of that delay will balance it against the cost of a setup and decide accordingly. Poor patterns The previous examples were easily checked because they used only perfect patterns, those without loss. What happens if patterns do involve loss? Two observations are relevant. Firstly, loss leaves less useful weight in a pattern. Weight*age sums count precisely that useful or net weight, excluding broke and trim. So, all other things being equal, poor patterns end up with less importance. They move to the bottom of a cutting plan  as they should. Thus our approach deals as consistently with poor patterns as with good ones. Secondly and on the other hand, there should be few poor patterns, if any. Their presence means that the planning period is too short and we return to the beginning of this argument. To summarize it again, the comparative advantage inherent in combining different sizes promises patterns with negligible loss, provided – the sizes are different enough, variety, and there are enough of them, quantity. Either condition tends to presuppose long planning periods. We demonstrated this with the plans for 48 days, made for each day, then for the period as a whole. That example was academic, to be sure, with generated data, but genuine data confirm its evidence. The only exception to this argument arises from a standard size such as A4: it is the only size of its kind. Yet such sizes tend to be made on dedicated machines, designed for the task and selfdeckling. In turn, if a standard size does entail waste, equipment should be redesigned. Either suggestion, long planning periods or good equipment, is easy to make, difficult to execute. Yet being unaware of alternatives is worse. One may lack the time for an exhaustive analysis, but a rough assessment will be better than no assessment of the factors in question. Exhaustive analysis To explore alternatives objectively, one should know each cost element: stock, setup, storage and penalty for late delivery, both per unit per time unit. One also needs to know the rate of production per time unit to count how many units are early or late for how many days. Given these data, one can establish the totals for each cost element and ultimately find the optimal planning period. It minimizes total cost by balancing savings in stock usage and setups against the costs of storage and of penalties for late delivery. Late penalty may in fact be prohibitive or so high that one must deliver on time. This is assured by starting each run/setup soon enough, working backwards: if a set's earliest order is for day 8, comprises 100 reels produced at the rate of 20 par day, then it takes 5 days and must start on day 3. Being ready on time, it involves only storage cost. Either approach must be judged on its merits. Yet costs should be lower if patterns are sorted by weight*age rather than by run length. Summary and conclusion We suggested a practical approach to favoring prompt delivery in plans with minimal stock usage. It begins with sorting reels by delivery date before allocating them to patterns. Then one repeats these steps: Compute each pattern's importance, the sum of weight*age per order. Run the pattern with the highest importance and delete it from plan. Update orders, subtracting reels done, then repeat step 1 until all patterns are done. In computing weight*age, one allocates remaining reels to orders as they come, sorted by date. One ignores splitting of orders between patterns, but possibly resumes that aspect once all patterns are sorted. This approach obtains the benefits of least stock usage and few setups while yet arranging patterns so as to favor prompt delivery. One also has a framework for computing total cost if one can assess the cost of each element, of stock, setup, storage and penalty for late delivery as they arise from the actual rate of production per time unit. References There is an abundance of references on cutting stock problems, yet the authors did not find any on prompt delivery. They will appreciate help at cutworks7@iafrica.com , www.cutworks7.com or www.cds.co.nz/cutline . 