£*Mfc4l — ■ PRODUOT AND INVENTORY PLANNING ~ R MANOFKTUR'^ ■^ * ' &: ";•:-■ v William McKillop . S. Hoyer-Nielsen /CALIFORNIA AGRICULTURAL EXPERIMENT STATION BULLETIN 837 Ihis bulletin reports on an exploratory investigation into the application of produc- tion and inventory planning techniques in the field of lumber manufacturing. Specifi- cally, it • Highlights the need for quantitative decision-making techniques for planning pro- duction and inventory levels in lumber manufacturing. • Provides representative cost data for stud manufacturing in the North Coast region of California. • Describes the important economic features of a representative stud manufacturing operation by means of a quantitative model. • Utilizes linear programming to specify optimal levels of production and inventories over a twelve month planning period. Although the study concentrated on the problems of a stud manufacturing operation the techniques used are applicable to all types of lumber manufacturing and to any type of wood processing where the producer is faced with a fluctuating demand for his product and has the possibility of adjusting output and inventory levels. Cover photo: James Gilligan. TABLE OF CONTENTS Quantitative decision making 3 Study procedure 4 The representative stud mill 6 The decision problem 6 The planning horizon 6 Data 6 Formulation of the model 10 Optimization technique 10 Definition of variables and parameters 11 The objective function 12 Constraints 12 Matrix representation of the problem 14 The computer runs 14 Optimization 14 Sensitivity analysis 17 Conclusion 20 Literature cited 21 Appendix A: Outcome of the interviews 22 Appendix B: Computer output 29 March, 1968 THE AUTHORS: William McKillop is Assistant Professor and Assistant Forest Economist in the School of Forestry and Conservation, Berkeley. S. Hoyer-Nielsen was formerly Research Assistant in the School of Forestry and Conservation, Berkeley. PRODUCTION AND INVENTORY PLANNING IN LUMBER MANUFACTURING 1 When demand and sales prices fluctuate significantly, as they do in the lumber industry, the individual company faces the so-called "production scheduling problem. This problem is concerned with balanc- ing of the cost of carrying inventories with that of changing the production level. Changes in production level require overtime, multishift operation, or enlarged crew size when production is high; and idle time or reduced work force when pro- duction is low. Production fluctuations can be reduced or completely avoided, how- ever, by holding large inventories or by allowing orders to be turned down. Pro- duction decisions for the sawmill thus become strongly related to inventory con- trol decisions. If production is controlled inventory is determined, and vice versa. The determination of an optimal pro- duction and inventory policy rests on two kinds of information: • A continually revised forecast of fu- ture demands and sales prices • The relative costs of overtime produc- tion, multishift operation, holding in- ventories, turning down orders, hav- ing idle time, and changing the work force. For even fairly small sawmill operations the production smoothing problem may be far from having an obvious solution. No matter how much experience a man- ager may have, his judgment and intuition may never uncover the best production and inventory decisions. However, mathe- matical programming methods are now available and have been applied with suc- cess in production planning in many in- dustries. The lumber industry has been slow to adopt these quantitative methods. This is partly due to the fragmentation of the industry into a multitude of firms and the existence of a low degree of con- centration in the industry. The typical smalless of firms makes the search for ap- propriate operations research techniques difficult for individual companies. The primary purpose of the present study was to investigate how mathematical program- ming may be used in the lumber industry to adjust production and inventories to varying market conditions in the most profitable manner. QUANTITATIVE DECISION MAKING The typical process by which decisions are arrived at in the lumber industry may be referred to as "qualitative decision mak- ing," with decisions made on an individual basis as need arises with judgmental weight given to the factors that seem important at the moment. Qualitative decision mak- ing may be more or less routinized by re- sorting to common sense rules-of-thumb. The rule of carrying three weeks' inven- tory is one example. Generally, there is no analysis available to indicate that three weeks are better than two or four, or that the same rule should be used under all 1 Submitted for publication May 3, 1967. conditions. The only claim that can be made is that the rule seems "reasonable." Such intuitive decision making can be very effective when done by skilled managers of small operations, but even if satisfactory earnings are achieved this is no guarantee that management has made the most of its opportunities. "Quantitative" decision making utilizes a systematic procedure whose central fea- ture is a mathematical model of the decis- ion problem. To construct such a model requires formation of a set of relationships that state rigorously and in mathematical terms what variables are important, how they are interrelated, and how they lead [3 to a solution of the problem. Unimportant details are usually omitted from the model to make it manageable. When constructed, the model is manipulated using mathemat- ical techniques to arrive at an optimal solution. As suggested by Holt et al. (1960), the following steps may be identified in a quantitative decision-making process. (1) An intensive study of the specific de- cision problem is the initial step in con- structing the model. The study must in- vestigate the effects of different courses of action upon costs, on the attainment of the firm's objectives, and on other impor- tant measures of performance. It must con- sider what details of the problem may be omitted from the model without hazardous over-simplification. Validity of assumptions about the problem must be explored. Any constraints that confine decisions (for ex- ample the maximum number of machine hours that can be made available) must be identified. Often there will be some im- portant variables that are beyond the con- trol of the decision maker (for example the amount of logs that can be hauled to the mill during the rainy season). Any possible disturbances due to such factors must be estimated. Because it is important that the study describes the decision problem factually without undue attention to the way in which it has been handled in the past, sources of information and their re- liabilities must be critically evaluated. (2) Relationships between the decision problem under consideration and deci- sions outside the problem must be ex- plored. Where strong interactions exist, the scope of the analysis may have to be extended to include these other decisions. For example, decisions on raw-material purchases are often closely connected with production decisions .However, substantial difficulties may be encountered in trying to solve a firm's entire chain of problems in a way that simultaneously takes into account every relevant factor. Lines must be drawn to separate one decision problem from others, and the choice must be made as to which decisions are to be included in the analysis and which are not. ( (3) Once the decision problem and its boundaries are known, it is formulated in a mathematical model. The building of this model may appear to many people to be a translation of known and familiar phenomena into a foreign language where even obvious things become difficult to understand. However, this translation has several advantages. For example, it makes it necessary to evaluate critically all vari- ables and relationships and clarifies aspects that have previously remained hidden in a haze of verbal arguments. Furthermore, it makes it possible to obtain an optimal solution by the use of mathematical tech- niques through an electronic computer. (4) When the model is set up, it is em- ployed to obtain an optimal solution using, in most cases, an electronic com- puter. While access to such a computer is not absolutely necessary for the applica- tion of quantitative decision methods, mathematical computations can be per- formed much faster and with fewer errors than would be the case with other means. (5) Having the optimal solution in hand, some caution should be exercised in its evaluation. It should be borne in mind that an "optimal solution" is a deductive statement made within the context of the mathematical model. Thus, the only sense in which it is optimal is that it gives a "best" solution under the particular as- sumptions. Since the model never per- fectly captures reality, the optimal solu- tion, being a deductive consequence of the model, may not provide a completely valid basis for recommending management de- cisions. Therefore, the optimal solution is checked and whenever possible, simula- tion tests or post-mortem analyses are car- ried out. STUDY PROCEDURE The primary aim of this study was to investigate the feasibility of employing production scheduling techniques in the lumber industry. It was recognized from the outset that most lumber manufactur- ing enterprises, though perhaps simple in [4] comparison to many other types of manu- facturing, are nevertheless fairly complex operations. Because this was an explora- tory venture the decision was made to focus only on stud 2 manufacturing because this is by and large the simplest type of sawmill operation. In some ways it would have been pre- ferable to make an in-depth study of an individual stud mill but the approach taken was to portray a representative mill using data collected from a number of companies. One advantage of this ap- proach was that the feasibility of apply- ing operations research techniques could be assessed for the stud industry in general. The other major advantage was that de- mands on the time and patience of the management of any single mill were minimized. All lumber companies listed in the 1966 Directory of the Forest Products Industry as producing studs in the North Coast Region of California were contacted with regard to their willingness and ability to supply data. Thirteen were interviewed and information collected on their current production decisions and on how costs varied as levels of production and inven- tory changed. Each interview was systematized by means of a questionnaire investigating the following issues: (a) The type, size, and complexity of the mill operation. (b) Production costs per thousand board feet produced on different produc- tion levels (that is, on day shift, night shift, regular time, or overtime). (c) Inventory carrying costs per thou- sand board feet per unit of time for finished products and for logs. (d) Constraints that confine the produc- tion planning decisions. (e) Market conditions for studs. (f) Current decision-making practices in production planning. Questions falling in section (a) were de- signed to give a general picture of the type of companies that are involved in stud manufacturing. Sections (b) and (c) dealt with information necessary for building cost functions. Questions in section (d) focussed on restrictions usually constrain- ing stud mill operation. Questions in sec- tion (e) were designed to explore the possi- bilities for forecasting order quantities and sales prices and the effect of turning down orders. Section (f) was aimed at uncover- ing the most important difficulties in plan- ning and to indicate the potential for im- provement. Details on the outcome of the interviews are presented in appendix A. With regard to constraints on planning, not all constraints are absolute but more or less conditional depending on where companies draw the lines that separate the production smoothing problem from other decision problems. For example, pro- ductive capacity is restricted to the capac- ity of the present machinery when seen from a pure production smoothing point of view. However, if the production plan- ning decisions are extended to include decisions about purchase of new equip- ment, the problem may have capacity limits that are far beyond the limits of the present machinery. Like most studies in production smoothing, this one treated the planning problem separately, without con- sidering capital investment decisions. However, the solution of the production smoothing problem can be helpful in eval- uating potential expansions in capacity. With regard to current planning prac- tices, none of the mills in the study used any kind of a formal decision-making system to determine what sort of responses and market changes should be made. The interviews indicated that the moment's events dictate the mill's production deci- sions for two main reasons. First, day to day problems take up all time and energy of the staff and prevent management from concerning itself with plans for the future. Secondly, a strong and widespread disbe- lief in the possibility of anticipating com- ing market conditions makes many of the mills uninterested in looking ahead. As to the first point, one of the major pur- poses of formalizing decision-making pro- cesses is to make production planning easier and faster, so that management can have time to forecast and meet disturb- ances and problems that may arise later. A piece of construction lumber 2 inches by 4 inches by 8 feet. [5] As to the second point, it is certainly true that forecasting future demands and sales prices is the crux of almost any produc- tion planning problem, and that even the most approximate forecasting and plan- ning is better than no planning at all. THE REPRESENTATIVE STUD MILL The approach taken in this study was to construct a hypothetical stud mill broadly representative of the sample mills inter- viewed. THE DECISION PROBLEM The sales opportunities of the representa- tive mill were assumed to be such that a one-shift operation, even with full over- time, could not satisfy all orders. While orders and sales prices at the mill were assumed to follow general market fluctua- tions and hence to make sales and produc- tion for a one-shift operation unbalanced, especially in the spring and early summer, the mill's sales opportunities were not sufficient to justify a two-shift operation the whole year round. The planning prob- lem was therefore to determine how, when, and to what extent the mill should vary its production level and inventories so as to maximize profit. In this planning prob- lem, capital investment decisions, for ex- ample decisions about expansion of the capacity of the mill, were not considered. Demand and sales prices were considered to be beyond the control of the mill and the optimal production plan for the whole planning period was accepted as the one which made total sales revenue exceed combined production and inventory cost by a maximum amount. Tackling the pro- duction smoothing problem was a matter of pinpointing the production level for each of a number of future time segments (months), and at the same time making sure, first, that the resulting decisions did not violate any constraints, and second, that the cost of changing from one pro- duction level to another was at least offset by additional revenues from the change. It was recognized that decisions made in this way would not be a once-and-for-all commitment, and would simply constitute a basis for more detailed scheduling as the planning period proceeded. THE PLANNING HORIZON Before a model can be formulated, the length of the planning horizon must be decided on. The annual cycle that prevails in the stud market made it reasonable to use a one-year horizon, beginning March 1 and having monthly time segments. By the end of February the old log deck is depleted and the busy period of the year is immediately ahead so that the top prices and demand can be forecasted with the greatest possible accuracy. It was also rec- ognized that, in reality, it would be con- venient to use the winter months for planning work. DATA The representative mill was taken to be of medium size with 22 men in the sawmill crew and an output of 400,000 board feet for a 40-hour week on a one-shift basis. This production rate is characteristic of a fairly modern mill. Workers strictly oc- cupied with inventory handling were not considered to be part of the crew. It was assumed the almost all the mill's production was in redwood sold green to California wholesalers. The grades con- sidered were the standard ones for red- wood studs, "one star and better," and "two star and better," In addition, it was assumed that minor quantities of 2 x 3 and 2 x 1 shorts were produced and that chips could be sold to pulp mills at a net value of $2.50 per thousand board feet lumber produced. The production costs used appeared to be typical for a mill of the given size. In practice, individual mills may extract their own cost figures from accounting records but care should be taken because such records are usually compiled for purposes other than cost analyses. In addition, because only future costs are of interest for decision making, historical cost data must be verified. For some costs, such as produc- tion costs for activity levels for which the [6] mill has no operating experience, records will not be available and estimates must be constructed. Although estimates of high accuracy are desirable, "best available" figures are preferable to none at all. Only "direct" costs, that is, those varying with the level of production, were considered. Costs that were "fixed," such as interest charges on borrowed capital, managerial salaries, and basic administrative expenses were not. considered. It was assumed that such costs would not be affected by the level of output and could be avoided only if the company went out of business. Production costs. Production costs per thousand board feet for the different "ac- tivities" used to produce studs are given in table 1. Only four activities were con- sidered to be feasible — day shift regular time, day shift overtime, night shift regu- lar time, and night shift overtime. It was assumed that the mill purchased only second-growth redwood logs at an average price delivered at the mill of $45 per thousand board feet log scale. Overrun was considered to be 30 per cent, regard- less of whether the mill operated on regu- lar time or overtime, or on one shift or two shifts per day. The log cost per thousand board feet of studs was therefore calcu- lated as $45/1.3, or about $34.60. Cost of saws and saw maintenance was taken to be $1.40 per thousand board feet of studs produced on regular time, and slightly higher for studs produced on over- time. The power cost was specified to be $1.25 per thousand board feet when the mill ran on one shift, regular time. At that produc- tion level, power consumption was as- sumed to be just below a figure that led to a lower rate so any production of more than 40 hours per week was assumed to have a new and lower power cost per unit of output. Cost of machinery maintenance and wear-and-tear was taken as $3.35 per thou- sand board feet when the mill was on a 40-hour week. Higher production levels were assumed to have a higher cost per thousand. Labor cost per thousand board feet was computed as (number of men in crew) x (average wage per hour)/(production rate [7 per hour). It was estimated that there would be something like a 10 per cent decrease in efficiency from regular time to overtime and from day shift to night shift, with the production rates per hour being 10,000 board feet for day shift regular time, 9,000 board feet, for day shift over- time, 9,000 board feet for night shift regu- lar time, and 8,100 board feet for night shift overtime. The weighted average labor wage was calculated as $2.90 per hour, so that the labor cost per thousand board feet turned out as listed in table 1 (figures are rounded off). Additional supervision costs when the mill's production level goes above 40 hours per week were also computed. First shift overtime was assumed to require $4.50 per hour for the present foreman, or $4.50/9 per thousand board feet. To run a night shift it was assumed to be necessary to hire two new supervisors at a total cost of $9 per hour, or $1 per thousand board feet. It was considered that expanding the night shift to include overtime would require 1.5 times as much. The additional office work and office supplies required when operating more than 40 hours per week was taken to be about $50 per month for full overtime pro- duction on either shift (office supplies only) and $285 per month for night shift production (half-time office girl $185, office supplies $100). These costs become, on a thousand board feet basis, approximately $.10 for studs produced on overtime and $.20 for studs produced on night shifts. Inventory carrying cost. Inventory carry- ing cost was specified to be $1 per thou- sand board feet per month for both studs and logs. The average value of studs in inventory was taken to be $50 per thou- sand and that for logs as $45 per thousand. The interest rate used in calculating carry- ing costs was 1 14 per cent per month which led to interest charges of 75 cents per thou- sand board feet per month for studs and 67.5 cents for logs. The remainder of the carrying cost was Assumed to consist of mainly inventory handling cost. The interest rate .was set purposely high to recognize the urgent need for operating capital experienced! by many stud mills. In practice, certain mills may be able to obtain bank loans to finance log and lum- ] Table 1 DIRECT PRODUCTION COSTS Cost component Day shift Night shift Regular time Overtime Regular time Overtime Dollars per thousand board feet 34.60 1.40 1.25 3.35 6.40 34.60 1.50 1.15 3.55 10.60 .50 .10 34.60 1.40 1.15 3.55 7.10 1.00 .20 34.60 1.50 1.15 Machinery maintenance and depreciation. . . . 3.85 11.80 1.50 Administration and office supply .10 TOTAL 47.00 52.00 49.00 54.50 Net value of chips 2.50 44.50 2.50 49.50 2.50 46.50 2.50 52 00 ber inventories at considerably lower rates. In such cases, however, charges for the services of warehousing companies may be incurred. Furthermore the value of mort- gages on log decks or lumber inventories may be only a fraction of the market value. In this case, even a company with a good credit rating cannot avoid the problem of using operating capital to finance inven- tories. In addition to this normal carrying charge, a special carrying charge of $1 per thousand was specified for inventories on hand on March 1, to cover property tax. This figure was based on an assessed value of 22 percent of market value and a tax rate of 9.5 per cent of assessed value. Production constraints. The maximum number of machine hours available per month was computed on the basis of 22 workdays per month, except July and December which were considered to have only 17 workdays each because of vaca- tions during the week of July 4th and a week around Christmas. The upper limits on machine capacity were therefore speci- fied to be (figures rounded) for July and December, First shift regular time 40x17/5 = 136 hours, Second shift regular time 40x17/5 =136 hours, First shift overtime 14x17/5 = 48 hours, Second shift overtime 14x17/5 = 48 hours, and for all other months, First shift regular time 40x22/5 = 176 hours, Second shift regular time 40x22/5 =176 hours, First shift overtime 14x22/5 = 62 hours, Second shift overtime 14x22/5 = 62 hours. For regular time, the lower limit on day- shift operation was taken to be 32 hours per week on the assumption that if the mill's operation level goes below four full days per week, the employees would quit and look for other jobs or seek unemploy- ment compensation. For night shift opera- tion, the lower limit and the upper limit were taken to be the same on the assump- tion that management would not consider a second shift unless it could be run for the full 40 hours per week for some period of time. These specifications are in accord with the general view that when market conditions call for only a very low level of production at a particular mill, a simi- lar situation prevails at other mills in the area so that labor on the day shift will have difficulty in getting other employ- [8] ment and can be retained down to a pro- duction level of 32 hours per week. How- ever, when demand is normal or high, the requirements of other mills for labor tend to be substantial and it becomes difficult to retain the night shift at less than full employment levels. Limits on inventories. Upper limits were placed on the size of both log and lumber inventories. The maximum capacity of the log yard was set at 6 million board feet and that of the lumber yard at 1.8 million. Initial lumber inventories at the start of the planning period were specified to be 800 thousand board feet. The initial log inventory was assumed to be zero. In addi- tion to space limitations, the size of inven- tories in any one month was assumed to be restricted by the availability of credit or capital. This limit was set at $250,000. Restrictions on log supply. The maxi- mum amount of logs that the representa- tive mill could purchase during the year without cutting into the "quotas" of other mills was set at 27 million board feet log scale. A further restriction was imposed, namely on the amounts that the mill's logging contractor could supply in each individual month. The logging season was taken to be March through October, with the logging contractor able to supply at the most 2 million feet in March, 3 million feet in October, and 4 million feet per month from April to September. It was considered that logs could not be pur- chased in November, December, January, and February. Orders and prices. Forecasts of future prices and orders are critical in any pro- duction-scheduling and inventory-control study. Investigations are currently being undertaken in the School of Forestry and Conservation, University of California, with the objective of providing short-run forecasts for a range of types of lumber. When available these forecasts may be employed in conjunction with production scheduling. Alternatively the management of an individual company may prepare its own forecasts using elementary economet- ric or moving average techniques, or simply the average levels of prices and orders that have prevailed in the corresponding months over the past few years. Or man- agement may construct a market forecast in a judgment basis which represents its estimates or "hunches" as to how the mar- ket is going to develop in the next twelve months. Needless to say, the more accurate the set of forecasts the more effective the results of production scheduling will be. Some people may claim that the lumber market is so volatile that it is impossible to forecast changes. This is not so — and in any case, any course of action involves im- plicit assumptions about future markets in the sense that it is optimum only for a par- ticular pattern or set of patterns of prices and orders. So management might just as well be explicit in its estimates of future conditions. A market forecast constructed on a judg- ment basis appeared to be the type most likely to be favored by management in reality and was the type used in this analy- sis. The specific forecast used is presented in table 2. The price forecast followed the typical seasonal pattern suggested by price report- ing publications and discussions with per- sons interviewed. A high point occurs in April, with a slow decline through July and August and a slight pick-up in Sep- tember and October. A low is reached in December followed by improvement in the early months of the year. Price was con- sidered to be the key element in the fore- cast but forecasts of maximum levels of new orders were also prepared for each Table 2 FORECAST OF MAXIMUM NEW ORDERS AND PRICES Month Maximum new orders Monthly average sales price March April May June July August September . October November. . December. . January. . . . February. . . Thousand board feet 3,100 3,400 3,400 3,200 2,800 3,000 3,000 2,800 2,600 2,500 2,600 2,800 Dollars per thousand board feet 53 55 54 52 50 50 51 51 50 49 50 51 TOTAL . 35.200 [9] month. It was recognized that at certain times of the year the limiting factor on sales would not be the general market price for studs but the level of orders re- ceived by the individual mill. The pattern followed was similar to that for prices with a high in April and May, a slight decline in July, a pick-up in August and Septem- ber, and a steady fall-off to a low in December. FORMULATION OF THE MODEL are OPTIMIZATION TECHNIQUES Production planning over time is an in- tensively studied area of industrial opera- tions research and various techniques are available for obtaining optimal solutions. No approach is superior in all aspects. Each has potential strengths and weak- nesses that make it well suited to some situations and somewhat ineffective in others. The most common techniques available for solving production smooth- ing problems are: Dynamic programming Quadratic programming Linear programming Descriptions of these techniques given in Hansmann (1963). Dynamic programming is useful in for- mulating production smoothing problems and clarifying their nature conceptually, particularly in cases where the problem may be thought of as a sequential decision problem. However, the approach is not very efficient computationally when the problem contains as many stages, state parameters and decision variables as are involved here. Also, no computer code exists of sufficient generality to solve even a small subclass of dynamic programming problems, such as production smoothing problems. Quadratic programming refers to the problem of maximizing (or minimizing) a quadratic objective function subject to a set of linear constraints. Application of quadratic programming to the production smoothing problem is seen in the linear decision rules developed by Holt et al. (1960). Once the best decision function has been determined, decisions are made without further calculations by evaluating the model at each point in the planning horizon for which production is to be scheduled. The evaluation involves apply- ing to the function information on state parameters (for example, the inventory level) at each point and the latest available forecasts. This approach suffers from the same weakness as dynamic programming, namely that no computer code is general enough to handle even slightly differing situations. Linear programming optimizes a linear function subject to a set of linear con- straints. The approach assumes that pro- duction costs, inventory carrying costs, and sales gross income are linearly related to the decision variables. If they are not, the linear programming formulation may still be applicable because a nonlinear objec- tive function may be closely approximated by a series of straight line segments. The linear programming (LP) approach has several advantages. First, a general computational technique, the simplex algorithm, is readily available in computer codes. Second, it is normally very easy to revise the model or incorporate additional decision variables, cost components, and side restrictions. As pointed out earlier, this feature is important because an indi- vidual mill may easily improve its model as operating experience accumulates. Third, in addition to the optimal solution itself the LP model furnishes insight into the firm's production process through the cost-range table and shadow prices in the computer outputs. A description of the simplex algorithm is available in Dantzig (1963). One disadvantage of the LP ap- proach is that uncertainty cannot be con- veniently handled by incorporating prob- abilistic forecasts in the model. The ap- proach requires using expected values of future orders and prices as if they were known with certainty. However, uncer- tainty may be handled by determining the sensitivity of the solution to forecast errors, by scheduling in detail using weekly time segments, and by revising the twelve- [10] month production plan as time goes by. Because of these advantages linear pro- gramming was selected as the optimization technique for this study. It was recognized that it was not completely satisfactory, especially because of its inability to handle set-up costs but it was felt that no other technique could yield as much information for the same expenditure of time and money. The problem of set-up costs is dis- cussed later. Selection of linear programming as the optimization technique required that the model be structured in two parts (a) a linear objective function and (b) a set of linear constraints. Details of these com- ponents are given below, following the definition of variables and parameters of the model. DEFINITION OF VARIABLES AND PARAMETERS Variables and parameters used in the model are defined below with the index "t" referring to the month and "i" refer- ring to the "activity" used to produce studs. The index "t" varies from 1 to 12 with 1 referring to March, 2 to April, 3 to May and so on. The activity index "i," varies from 1 to 4 with 1 referring to day shift regular time, 2 to day shift overtime, 3 to night shift regular time, and 4 to night shift overtime. The term "variable" refers to a control variable such as inventory level, or output that the mill can adjust. The term "param- eter" refers to such things as prices and costs which are "given" to the firm within the context of the production scheduling problem. F(t,i) - the volume of studs (thou- sands of board feet) produced during month t by activity i. FS(t,i) - the number of slack hours in month t for activity i. SA (t) = the volume of studs (thou- sands of board feet) sold in month t. DS(t) = the amount by which maxi- mum level of new orders ex- ceeds actual sales in month t. SI(t) - the stud inventory (thousands of board feet) at the end of month t. For t-0, SI(t) repre- SIS(t) LI(t) LIS(t) CRS(t) FLPO(t) FLPOS $DC(t,i) $BP $PR(t) $SH $CARR.S %CARR.L %TS sents the assumed initial in- ventory level of 800 thousand board feet. the slack (thousands of board feet) on storage space in the lumber yard at the end of month t. the log inventory (thousands of board feet, log scale) at the end of month t. For t-0, LI(t) is the log inventory at the start of the planning period. This was assumed to be zero. the slack (thousands of board feet) on storage space in the log yard at the end of month t. the amount of credit (dollars) available to the mill but not utilized at the end of month t. the logging contractor's haul- ing capacity in month t less the log quantity bought by the mill in month t. FLPO(t) may be interpreted as the fore- gone log purchase opportu- nity in month t. the mill's total log purchase opportunity for the year less the amount of logs that it actually buys. direct production cost per thousand board feet in month t for activity i. the net value of chips per thousand board feet of lum- ber produced. the average sales price per thousand board feet of studs in month t. the cost of turning down orders per thousand board feet. the monthly cost of carrying lumber inventory per thou- sand board feet, the monthly cost of carrying log inventory per thousand board feet. the property tax per thou- sand board feet of studs in stock on March 1. the property tax per thou- sand board feet of logs on hand on March 1. [ii] THE OBJECTIVE FUNCTION cost , In addition , the cost of refusing The objective function was basically for- orders and the value of closing inventories mulated as gross receipts from sales less of studs were also considered. The objec- total direct production and inventory tive function may be represented as 12 12 4 Z = X $PR(t) * SA(t) - EE [%DC(t,i) - $BP] * F(t,j) 11 11 12 - £ $CARR.S * SI(t) - 2 &CARR.L * LI(t) - £ $SH * DS(t) - L/(12) * [%CARR.L + %TL] + £7(12) * [$P#(12 + 1) - $CARR.S - $TS]. The symbol * signifies a multiplication sign. The first term on the right-hand side represents gross revenue from sales over the twelve-month period. The second term consists of production cost less revenue from chips per thousand board feet multi- plied by level of output and represents total net production cost over the plan- ning period. The third and fourth terms represent inventory carrying costs for studs and lumber, respectively. The fifth term represents the cost of turning down orders. The final terms represent, in turn, carry- ing costs for closing inventories of logs and the value of closing inventories of studs less carrying costs. This type of objective function is likely to be suitable for a wide range of condi- tions where profit maximization subject to various constraints is set up as a goal. For instance, it would be applicable to a stud mill that was designed to utilize peeler cores from a plywood plant owned by the same company. The major difference in the model would be in the constraints. For example, the supply of cores from the ply- wood plant to the stud mill would not be limited to the months March through October as was the case for log supply. Furthermore it might be necessary to insert restrictions to ensure that the core inventory at the plywood plant would never exceed a certain limit. Parameters of the model need not repre- sent phenomena that are directly observ- [12 able. The cost per thousand board feet of turning down orders is an example of this because it represents the value that mill management attaches to good customer relations. In computer runs of the model this value was set at zero but was con- sidered in the course of a sensitivity analysis. Net revenues obtained early in the plan- ning horizon were given equal weight with later revenues. No attempt was made to maximize discounted net revenue though this would have been a relatively easy modification of the present model and would simply have required substituting discounted prices and costs for actual ones. CONSTRAINTS Constraints were classified in three ways: • Production capacity equations • Sales capacity equations • Inventory and log supply equations. Production capacity equations. Denot- ing the time requirement for producing 1,000 board feet of studs in month t by activity i, by TMRQ (t, i), and the number of hours available in month t for activity i by H(t, i), production capacity equations may be written as TMRQ(t,i) * F(t,i) + FS(t,i) = H(t,i). It was assumed that the time require- ments for a given i were the same for all t. ) Since each of the 12 months in the plan- ning horizon has four different activities, there were 48 production capacity equa- tions. Sales capacity equations. Denoting the maximum volume that can be sold in month t by D(t), for any given t, the sales capacity equation is SA(t).+ DS(t) = D(t). Inventory and log supply equations. The stud inventory level at the end of any given month t. (t not equal to 1) is deter- mined as production in month t less sales during month t plus inventory at the end of the previous month. Thus, the equa- tion giving the inventory level SI(t) at the end of month t reads as ]C F(t,i) - SA(t) + SI(t - 1) - SI(t) = 0. For t = 1 the equation is - E F(1A + SA(1) + 5/(1) = 5/(0). The constraint equation to ensure that the model did not come up with a solution which called for more logs than the mar- ket could supply, reads as follows: 8 2 FLP0(t) - FLPOS = 2,000 where the right-hand side is the total hauling capacity of the logging contractor less the mill's annual log quota, in thou- sands of board feet. The inventory of logs by the end of any given month t was determined as the in- ventory at the end of the previous month plus the logging contractor's hauling ca- pacity during month t less the unused part of that hauling capacity less the log con- sumption during month t. Since the over- run was 30 per cent, the log consumption was .77 thousand board feet for each thou- sand board feet of studs produced. Thus, the equation giving the log inventory LI(t) at end of month t reads as follows: 4 £ F(t,i) * .77 - LI(t - 1) + LI(t) + FLPO(t) = hauling capacity in month t. For months when logging contractor could not deliver any logs, the equation simply reads £ F(t,i) * .77 - LI(t - 1) i=l + LI(t) = 0. For t - 1 the equation is E F(l,i) * .77 + LI{\) = LI(0). The lumber and log inventories at the end of each month of the planning horizon were thus determined in 24 equations. The constraint on storage space in the lumber yard is given in the equation SI(t) + SIS(t) = 1,800 t= 1, 12. The constraint on storage space (in thou- sand board feet) in the log yard is given in the equation LI(t) + LIS{t) = 6,000 t= 1, 12. At some mills, the lumber and log yard may be joint in the sense that lumber may conveniently be stored in the log yard and logs may be placed in the lumber yard. In such cases the 24 constraint equations on storage space would reduce to 12 equa- tions. If the joint space limit is expressed in square feet, SQFT(MAX), and the areas required to store 1,000 board feet of studs and 1,000 board feet of logs are P and Q [13 square feet, respectively, these 12 equa- tions would be of the following type: P * SI(t) + Q * LI(t) + SQFT.S(t) = SQFT(MAX), t = 1, • • -, 12 where SQFT.S(t) is the inventory space (expressed in square feet) not utilized in month t. The upper bounds on inventories set by the limit on the mill's credit availability are written as follows: 50 * SI(t) + 45 * LI if) + CRS(t) = 250,000, t = 1, • • ;, 12 where 50 and 45 are the average dollar values per thousand board feet of studs and logs, respectively. MATRIX REPRESENTATION OF THE PROBLEM The matrix representing the programming problem had 190 columns and 122 rows. Columns, listed from left to right consisted of 1 Objective column 48 F columns (relating to the volume of studs produced) 48 FS columns (relating to amount of unused production capacity) 12 SA columns (relating to the volume of sales) 12 DS columns (relating to the amount of orders refused) 12 SI columns (relating to the volume of the stud inventory) 12 SIS columns (relating to amount of unused storage space for studs) 1 2 LI columns (relating to the size of the log inventory) 12 LIS columns (relating to amount of unused storage space for logs) 8 FLPO columns (relating to the level of foregone log purchase opportunity for each month) 1 FLPOS column (relating to the level of foregone log purchase opportunity for the year) 12 CRS columns (relating to the amount of available credit not utilized) In addition there was the RHS (right- hand side) column specifying the upper limits of production levels, sales oppor- tunities, storage space, log supply, and credit availability. The rows, listed from the top of the matrix, were: 1 Objective row 48 Production capacity rows (setting the maximum number of hours available for each activity) 12 Sales capacity rows (setting limits on the volume of sales) 12 Stud inventory rows (relating produc- tion and past and present inventory levels) 12 Stud inventory constraints (setting upper limits on stud inventories) 1 Log availability constraint (specify- ing the quantity of logs available for the year) 12 Log inventory rows (relating log use, supply and past and present inven- tory levels) 12 Log inventory constraints (setting upper limits on log inventories) 12 Inventory financing constraints (set- ting upper limits on credit avail- ability The computer program utilized was the M3 system prepared by the Standard Oil Company of California (1962). An ex- ample of the computer output for this program is given in appendix B. Certain features of the output are described in the following section. THE COMPUTER RUNS OPTIMIZATION The main problem in obtaining an opti- mal solution was ensuring that no shift was prescribed to be operated at less than the minimum acceptable level indicated earlier. First run. The solution turned out by the first computer run was not feasible because [14 it violated the requirement that there should be either full capacity utilization on the night shift regular time over a sequence of weeks, or no night shift at all. The solution utilized all regular night shift time from approximately the last two weeks of March through the first three weeks of August. However, after that the solution called for only part capacity uti- lization in September, October, and November. The optimal value of the solu- tion for the first run was $219,072. Second run. In order to get a solution which did not violate the restriction about unbroken night shift operation, a second computer run was made in which an attempt was made to utilize the slack in August and September by bringing for- ward an adequate amount of the produc- tion of the December and October night shifts. This was done by deleting com- pletely the night shift capacity in Novem- ber, and by limiting it to the first week in October, by setting the limits on produc- tion capacity in these months at zero and 40 hours respectively. Part of the computer output for the second run is given in appendix B. The new solution utilized the regular night shift capacity fully from the middle of March through the first week of October except for about an hour in August (see appendix B page 30 where FS63 is in the basis with a value of ap- proximately 0.9). The one hour slack, however, is not of any practical impor- tance but merely an indication of the fact that if the night shift capacity in October is extended to more than the first week, additional slack in August would build up. The value of the solution for the sec- ond run was $218,597, only $475 less than the previous solution. Third run. The cost establishing and inter- rupting the night shift could not be in- corporated conveniently in the linear programming model, so in order to get a basis for evaluating if it pays to set up a night shift for a 614 month period, a third computer run was carried out in which production was limited to the day shift with both regular time and overtime per- mitted. The night shift capacity was de- leted by removing all appropriate RHS cards from the punched card deck. The solution which was obtained called for full capacity utilization from the begin- ning of the planning horizon through September and only regular time produc- tion for the rest of the year. The value of the solution was $176,400 which was $42,200 less than when night shifts were operated. Thus, $42,200 appeared to be the maximum amount which could be spent on a hiring, training, and layoff cycle for the year under consideration. It was noted that training costs would depend pri- marily on the state of the labor market since a labor market with an excess supply of workers would be able to provide ex- perienced sawmill labor. It might even be possible to rehire workers previously employed in the company. On the other hand, in periods of labor shortage experi- enced workers might be difficult to find. In this case a mill would have to resort to whatever is available in the form of un- skilled labor and since training would take place with the worker on the job, the training cost might take the form of hav- ing less than full efficiency initially during the night shift. This aspect was examined in a fourth computer run and is discussed later. Only hiring and lay-off costs were considered in comparing the second and third runs. The hiring cost was considered to be made up of the cost of advertising for workers, interviewing for selection, general accounting, setup of payroll, and medical examination. It amounted to approxi- mately $500. The layoff cost included the cost of office work and a component re- lating to the unemployment compensation tax. The expense in the office was esti- mated to be about $75. That part of the unemployment compensation tax which could be avoided by not allowing work- force fluctuations represents a direct cost to a company for operating night shifts part of the year. The magnitude of the unemployment compensation tax levied on a company depends on two factors — the amount of money in the company's unemployment fund (kept by the state in the company's name) and the average annual base payroll of the company. If the amount in the reserve fund is less than 7 per cent of the average annual payroll of the company, the tax rate is at its [15] maximum of 3.7 per cent of $4,100 per year per employee. If the amount in the reserve fund exceeds 7 per cent of the payroll, the rate decreases in regular in- crements down to 1.8 per cent of $4,100 per employee. 3 In the case of the sharp deduction in the labor force which would be the result of interrupting the night shift in October, the reserve fund will be drained by the amount paid out in un- employment claims. Therefore, the fund will be below the 7 per cent level, and hence the 3.7 per cent rate will ^apply. Alternatively, if the annual work force is stabilized at one shift, the minimum rate will apply. The difference is 1.9 per cent of $4,100 per employee, or approximately $4,125. Total hiring and layoff cost was thus computed as approximately $4,700. This amount should be subtracted from the return of $42,200 from the night shift operation. Assuming no efficiency loss in the first time of operating the night shift, the solution to the second computer run (night shift allowed) is then $37,500 higher than the solution to the third run. (no night shift). Besides giving the dollar value of the optimal solution, the computer outputs furnish a wealth of detailed information about the production plans and their consequences. With regard to the second run (pages 29-31), the column X (I) gives the optimal values of all the variables in the model. All variables with non-zero values are in the basis. First are listed the production quantities in thousands of board feet by month and type of process (page 29) and then the amount of slack hours. It can be seen that, due to the in- ventory carrying cost for logs, it does not pay to produce on night shift or overtime when the logging season is over, that is, after the end of October (month 8). The order quantities accepted, SA(t), and those turned down, DS(t), are given in thou- sands of board feet month by month. The optimal stud inventories to carry over at the end of each month are listed next. For most months the inventory is shown to be zero. This is partly a consequence of treating the demands as known with certainty and must be interpreted with reservation. The zero inventories indi- cate that in certain months no stocks should be held for sale in later periods. Nevertheless it might be advisable to hold a certain amount of buffer-stocks to take care of unforeseen production de- lays and minor random demand fluctua- tions. Next, the unused storage space in the lumber yard, SIS(t), is given and it can be seen that the constraint on storage space is far from being binding. Optimal log inventories, LI(t), show that no logs should be put into inventory in the first four months. In month 5 (July) the building up of the cold deck begins. It reaches its maximum by the end of October (month 8) at 5.1 million feet. From then on it drops month by month until it is depleted at the end of February. The FLPO(t) values show full utilization of the logging capacity in August, September, and October (months 6, 7 and 8). Unused storage capacity in the log yard is indi- cated by the values of the LIS(t) variables. Except for the month of October the log yard is far from being full. The variables for the credit slacks, CRS(t), show that particularly in October the credit account is heavily depleted. The Delta (J) column (page 29) is the printout of the objective function row as it found in the final simplex tableau. The figures in it show the increases (or de- creases) in the cost coefficients that are necessary to bring the corresponding vari- ables into the basis. For all variables al- ready in the basis the Delta (J) figures are zero. The information in this column is valuable in sensitivity analyses. A cost range table is given on page 32. When reading it, the positive and neg- ative signs must be exchanged because the M3LP-computer code is based on min- imization problems and maximization is carried out in the present model. This table also is valuable in sensitivity an- alyses. For example the first two rows on page 32 may be used to make the follow- ing observations: If the cost on night shift production, regular time, in March goes down more than $1 per thousand board "Information obtained from the State of California Employment-Unemployment Insurance Area Office, San Francisco. [16] feet the nonbasic activity, FS23, will enter the basis. That is, there will be slack on the night shift in April. Alternatively, if the cost on F13 increases by more than $1 per thousand board feet, F32 will enter basis, that is, there will be overtime pro- duction on the day shift in May. If the cost on day shift operation, regular time, in March decreases by an infinitely small amount the basis will not change because Fll is already at full capacity. Alter- natively, if the production cost for Fll goes up by more than $2 per thousand board feet, FS11 will enter, that is, there will be slack on the day shift in March. SENSITIVITY ANALYSIS A linear programming problem such as the present one is usually not completely solved as soon as the simplex method iden- tifies the optimal solution for the model. The reason is that the data for the cost parameters in the objective function, the input-output coefficients in the matrix, and the resource limitations are generally estimates rather than exact values. This makes it desirable to perform sensitivity analyses to determine what happens to the optimal solution if particular parameters take on other possible values, or if some of the assumptions fail to hold. Several aspects are explored in the following sec- tions, including training costs when start- ing up a night shift, costs of turning down orders, and deviation of market conditions from those forecast. Training cost. As mentioned previously, it was considered that the cost of training inexperienced workers when starting up a night shift would take the form of decreased efficiency. It was assumed that the starting production efficiency would be 30 per cent below normal efficiency in March and an average of 15 per cent below standard in April. It was assumed that by May full efficiency would be reached with a normal night shift produc- tivity of 9,000 board feet per hour. This modification was made by changing the objective function and the appropriate production capacity constraints. The alter- ations that were performed in the matrix were to change the TMRQ(t,i) cards to [17 allow for the increased time requirement per thousand board feet. The alterations in the objective function were changing the $DC(t,i)-$BP cards to reflect, princi- pally, increased labor costs per thousand feet. It was not necessary to change the cost coefficients corresponding to overtime production on the night shift, but because the changes resulted in the cost for F13 (night shift regular time) becoming the same as that for F12 (day shift overtime) the day shift overtime cost was increased by 1 cent per thousand feet as a convenient way of making sure that the new solution would use up all night shift capacity be- fore it started using overtime on the day shift in March. The model with the re- vised data was utilized in a fourth com- puter run. The value of the optimal solu- tion was $214,400, representing a decrease of $4,200 compared to the second run. It is worth noting that while the assumption about full night shift efficiency all the time led to a solution with inventory at the end of March and April and overtime on the day shift in July only, the assumption about subnormal productivity in March and April led to overtime on the day shift in April and May also, and no inventory until the end of June. The low efficiency start called for the night shift operation to begin only two days earlier. In relation to a one-shift plan, the solution for the fourth run showed an increase in value of $33,300. Cost of turning down orders. The com- puter runs previously mentioned assumed that turning down customer orders in- volved no cost. For some companies this assumption may be quite unrealistic, how- ever, since it is possible that customers may favor other sawmills if they find that a particular mill is frequently unable to handle their orders. It is somewhat diffi- cult to specify what penalty should be assigned to refused orders, but one way is for management to consider the size of an expected price rise that would lead it to refuse to book orders, say, for delivery next month. This can be taken to be the value that management attaches to refusing orders. Sensitivity analysis may at times be car- ried out without making further computer runs. Existing computer outputs for the second and third run can be used to make approximate estimates of optimal values that would have occurred if a cost of $1 per thousand board feet had been specified as the cost of turning down orders and included in the objective function in asso- ciation with the DS(t) variables. Looking at the second computer run (page 31) it can be seen that DS(t), for t = 9, 10, 11, 12, are in the basis with approximate values of 840, 1 140, 840, and 1040 thousand board feet, respectively. This means that the value of the second run would decrease by $840 + $1140 + $840 + $1040 = $3860 if $1 was used and if the same basis were main- tained. The cost-range table (page 32), however, shows that a cost of $0.27 or more on DS(9) will change the basis and that 57(8) (stud inventory at the end of Octo- ber) will enter. That means that if the pro- duction capacity for F83 (October, night shift, regular time) is expanded above the present first week limit, additional F83 time will be used. The extra production will be put in inventory for sale in month 9 (November). However, as the amount remaining in the mill's credit account at the end of October is only $19,924 (page 31, variable CRS8) it is not possible to stock more than a bare 400 thousand board feet of studs for the November orders. Consequently, DS(9) will remain in basis with approximately 440 thousand board feet. From page 32 it can also be seen that DS10, DS11, and DS12 will re- main in the basis with their previous quantities. DS10, for example, requires a cost of at least $1.35 per thousand board feet to create any changes in the basis. The previous value of the second run would thus decrease by approximately $3,860 minus $400, that is $3,460. Errors in forecasting. The effect of the sales price being other than expected in just a single month can be determined from cost range information. In the case of the third run it was found that if the sales price in September increases by an (infinitely small) amount, stocks should be built up in August for sale in September. Alternatively, if the sales price becomes lower than forecast by more than $.73 per thousand board feet, some of the overtime available in September will not be uti- lized. If the sales price in October goes above the expected value by more than $.04 per thousand board feet, it will pay to take up overtime production in that month. Alternatively, if the sales price in October drops more than $2 below the forecast price, 840 thousand board feet should be withheld for sale in the next month. One of the most important issues in the present planning problem is the decision on whether or not to set up a second shift. Such a decision is based on results of com- puter runs which, in turn, are based on assumptions about the market. Because these assumptions may not hold, it is im- portant to examine whether it would con- tinue to be profitable to establish a night shift if market conditions deviate by any plausible amount from the forecast con- ditions. If the sales prices and demands turn out to be generally higher than ex- pected, the two-shift plan would obviously still be the most advantageous and it would pay to extend the night shift opera- tion beyond the first week of October. However, if the market turns out to be very weak, a night shift plan may be finan- cially inferior to an operation using only one shift. One way in which management can examine this problem is to specify the minimum levels of prices and orders that it would expect under poor market condi- tions. As an example, assume that manage- ment considers the prices and demands given in table 3 to be the minimum values that would occur in a depressed market. The demand less the production of the day shift, regular time, computed month by month, can be taken to be that amount of orders to be supplied either by overtime on the day shift or by a night shift. For the months of March, April, and May these "excess orders" are 40, 1,040, and 1,040 thousand board feet, respectively. (For March the initial inventory of 800 thousand board feet is used to meet part of the new orders.) Assume, first, that a second shift is not established. The net return from the excess orders may be computed month by month as (volume produced on overtime) x (sales price less overtime production cost). Net [18] Table 3 PRICES AND NEW ORDERS IN A DEPRESSED MARKET Month Sales price New orders Dollars per thousand board feet Thousand board feet March 50 2,600 April 51 2,800 May 50 2,800 June 49 2,800 July - 47 2,600 August 48 2,700 September . 49 2,700 October .... 49 2.500 November. . 48 2,300 December.. 47 2,200 January .... 48 2,300 February. . . 49 2,500 production cost on day shift overtime was specified to be $49.50. Day shift overtime production capacity was specified to be 558 thousand board feet per month. Total excess orders for the three months are 1,120 thousand board feet. In March 520 thousand would be produced on overtime of which 40 thousand would be sold and 480 thousand held for sale in April at an inventory cost of $1 per thousand. In April 560 thousand would be produced and used with the 480 thousand carried over to satisfy all excess orders. In May only 560 thousand board feet of excess orders can be satisfied. Sales price exceeds overtime production cost by $0.50, $1.50, and $0.50 in March, April and May respec- tively. "Profit" for the three months may be computed as [(40 x $0.50) + (480 x $0.50) + (560 x $1.50) + 560 x $0.50)], that is $1,380. Because of low prices after May, overtime is not profitable and this amount represents the total net return from one shift operation for the whole planning period. Next, assume that instead of going to overtime production a second shift is estab- lished as soon as the regular day shift cannot keep up with the orders. Also, assume a low initial production efficiency of 30 per cent below normal for approxi- mately the first two weeks of operation and 15 percent below normal for the fol- lowing four weeks. The "payoff time" for the night shift may then be computed as the time it takes the profit from night shift .operation to exceed the profit from overtime operation by $4,700 which is the hiring and layoff cost specified earlier. Night shift production costs were specified as $49.50, $47.75, and $46.50 for 30 per cent below normal efficiency, 15 per cent below normal, and normal efficiency, re- spectively. For the above price and demand struc- ture the second shift would be paid off by the end of May. This conclusion may be reached as follows: Late in March the night shift is put into operation and it produces 40 thousand board feet at a cost of $49.50 per thousand. In April the pro- duction during the first two weeks is 500 thousand board feet at a cost of $49.50 per thousand. During the last half of the month the production is 670 thousand board feet at a cost of $47.75 per thousand. Inventory at the end of April is 130 thou- sand board feet. During the first two weeks of May the production is 670 thousand board feet at a cost of $47.75 per thousand. During the last two weeks of May the pro- duction efficiency has reached normal so the production volume is 790 thousand board feet at a cost of $46.50 per thousand. Inventory at the end of May (to be held for sale in June) is 550 thousand board feet. Recalling that inventory carrying cost is $1 per thousand board feet per month, the sales revenue less production and inventory costs from the night shift opera- tion in March, April, and May is therefore $[(40 x (50.00-49.50) + 500 x (51.00-49.50) + 540 x (51.00-47.75) + 130 x (50.00-1.00- 47.75) + 670 x (50.00-47.75) + 240 x (50.00- 46.50) + 550 x (49.00-1.00-46.50)], that is $5,860. Because of low prices in July, inventory carrying charges, and the requirement that the night shift work at full capacity or not at all, night shift production is not profit- able beyond the first six or seven working days in June. June production amounts to 490 thousand which gives a return of $[490 x (49.00-46.50)], that is $1,225. Total revenue less production and inventory costs for the night shift operation is there- fore $7,085. After allowing for a hiring and lay-off cost of $4,700, the net return is $2,385 which is only $1,005 above the net return from the day shift overtime operation. This suggests that under these [19] particular market conditions it would scarcely be worthwhile for management to undertake the job of setting up a second shift. Though this particular analysis has been carried out without the aid of a computer or linear programming it should be noted that it was practicable only because of the characteristics of the particular market data. It was feasible largely because the price and demand structure made it neces- sary to focus only on the first few months of the planning period. If prices and de- mands had been higher later in the year it would have been profitable to build up inventories for sale in later months and the analysis would have ballooned into a very tedious and difficult computing prob- lem. Furthermore, this simplified approach requires that none of the constraints be binding. As soon as limitations with re- gard to log supply, storage space, and credit availability have to be considered, this type of analysis becomes infeasible and more refined techniques have to be resorted to. Nevertheless the point should not be overlooked that under certain cir- cumstances management can derive an optimum production schedule by a rela- tively simple technique. Under certain conditions the linear programming trans- portation model may be utilized as a rapid and simple means of solving scheduling problems. The interested reader is referred to Bowman (1956). Other aspects. Numerous aspects of the firm's operation can be examined through sensitivity analyses. Information can be obtained on the effect of changes in almost any parameter of the model whether it be inventory carrying charges, space for log storage, prices, or costs. In addition, par- tial answers can be given to certain invest- ment problems of the firm. For instance, a mill may be contemplating installing a debarker and chipping equipment. An- alyses carried out with and without the assumption of chip sales can yield informa- tion on the profitability of such installa- tion. One type of sensitivity analysis that is readily carried out is the introduction or removal of constraints. Removal of cer- tain constraints may increase the value of the solution. New constraints may or may not reduce this value. In either case man- agement has a highly flexible means of reviewing the structure and operation of the firm. CONCLUSION None of the companies in this study used, or had ever used, a production plan that called for night shift operation only part of the year. For mills whose total orders for the year were below a two-shift produc- tion capacity, the policy was to operate one shift only and make overtime adjustments as the need arose. It is an open question whether any company faced with the situ- ation presented in this report would ac- cept the two-shift plan. It may be argued by management that although analysis indicates establishing of a second shift to be the most profitable solution — even if operated only for two and a half months — practical difficulties make it unrealistic to have a two-shift operation for less than 12 months a year. It may be claimed that it is almost impossible to hire a crew for summer employment only. Also, it may be argued that community relations, which have not been taken into account in this study, constitute an issue that may have considerable influence on the company's decisions. If these assertions are correct, skilled management's periodical produc- tion adjustments may closely approach quantitative decision-making in opera- tional performance. However, if the asser- tions are only expressions of conventional thinking, the quantitative analysis has indicated how management can best assess the feasibility of a two-shift operation for part of the year. The analysis based on a 12-month plan- ning period need only be a first step in scheduling production. If desired, detailed planning on a week-to-week basis can be [20] carried out with essentially the same model using, say, a 12- week planning horizon. The alterations that would have to be per- formed in the objective function would be to substitute forecasts of weekly sales prices for the expected monthly prices. The al- terations in the constraints would involve substituting weekly production capacities for monthly production capacities, weekly demands for monthly demands, and weekly log supply for monthly log supply. In the matrix, the SI(t) and LI(t) variables (levels of stud and log inventories) may be "fro- zen," using an option of the M3LP code to make sure that inventories are built up and reduced in accordance with the over- all 12-month plan. The optimal solution would then show how much to produce and sell in each of the coming 12 weeks. Only the production plan for the first week might be put into effect, however, because circumstances seldom develop ex- actly as assumed. When the first week is over, some parameters may have changed. Sales and prices may not have developed as expected, and the production output may have been higher or lower than calcu- lated. Therefore, the problem may be re- solved using the new knowledge accumu- lated during the first week and a new optimal 12-week production plan derived. In this way, management can make a week-by-week re-evaluation of its produc- tion scheduling problem, implementing — if it desires — only the first week or two of each plan. The planning procedures described in this bulletin require that management understand how a linear programming model is constructed and how to manipu- late a punched card deck. In addition, it requires access to an electronic computer. The latter necessity will probably be easy to meet in the future as more and more computer centers become available for use by business firms. The cost will be reason- able. For example, the four computer runs referred to in this report used only seven minutes of execution time on an IBM 7094 computer. A weekly re-solving of the 12-week problem, as suggested above, would probably take about one and a half minutes of computer time on each occa- sion. The stud manufacturing process exam- ined in this report was selected for its simplicity. Great potential exists for ex- tending this type of study to more complex situations in lumber manufacturing. For instance, the model may be extended to cover multiple product operations, hand- ling a number of species, grades and sizes of lumber, each with its particular market conditions. In addition, more involved production processes can be examined. For example, the necessity to air dry certain types of lumber for lengthy periods of time creates a special planning problem that can be handled by the use of the type of technique employed here. In conclud- ing it should be emphasized that this study was an exploratory venture. If the lumber industry is to improve its competitive posi- tion it must be ready to use any technique that allows it to make the most of the very difficult market conditions it faces. Though this study represents only a first step it is hoped that its imperfections will not obscure the value of the techniques employed. LITERATURE CITED Bowman, E. H. 1956. Production scheduling by the transportation method of linear programming. Operations Research 4, Baltimore, Md. Dantzig, George B. 1963. Linear programming and extensions. Princeton, N. J.: Princeton University Press. Directory of the Forest Products Industry. 1966. Portland, Ore.: Miller-Freeman Publishing Co. [21] Hansmann, Fred 1962. Operations research in production and inventory control. New York-London: John Wiley and Sons, Inc. Holt, Charles C, Franco Modigliani, John F. Muth, and Herbert A. Simon 1960. Planning production, inventories, and work force. Englewood Cliffs, N. J.: Prentice-Hall, Inc. Standard Oil Company 1962. M3 linear and separable system. San Francisco, California. In cooperation with the Rand Corp., Santa Monica, California. APPENDIX A: OUTCOME OF THE INTERVIEWS The data each mill was able to provide during a short interview varied consider- ably in quantity and reliability. General characteristics of the data are given below. GENERAL INFORMATION Types of mills. Of the 13 mills in the sample, nine were basically stud mills, with studs making up between 85 and 98 per cent of the output by volume. Three were mills with dimension lumber as their main product, with studs making up 2 to 5 per cent of the total output. One was a plywood mill, with sawmilling a side- operation to convert peeler cores into studs. Some of the data furnished by the non- stud mills had no immediate relevancy for the present study but information col- lected suggested that the planning prob- lems faced by them are much the same as those at stud mills and that a decision- making system similar to the one devel- oped in this study might well be appli- cable in many sectors of the lumber in- dustry. Crew sizes and production rates. The number of men in the sawmill crew (in- cluding log and lumber yards) and the average production rates are given in table A-l. The large spread in productivity per man hour reflects some significant differ- ences in the types of sawmill equipment, in the age of this equipment, and in the quality of the logs that each company had available. It did not appear that this vari- ation was due to differences in quality of labor. It was clearly pointed out by all mills that because stud manufacturing is a routinized operation from headrig to [22 green chain — with a fixed number of manned positions for the particular mill layout — it is not feasible to vary produc- tion by increasing or reducing the number of men in the mill crew. Shipping may be speeded up by putting one or two extra men in the lumber yard, but this appeared to be more of an exception than a rule. Production rates were reported as being sufficiently stable figures for planning pur- poses. At times, log quality or machine stoppages may cause production to fall below normal levels but subsequent suc- cessful runs usually make up for the defi- cit. These fluctuations in production rates have the character of random variables, but for any length of planning period that might be used for production smooth- ing, the average rates appear to be suffi- ciently reliable. It should be understood, however, that the use of these average figures in planning is not the same as ignoring the unplanned short-run varia- tions in productivity that almost always occur when a plan is being implemented. Species handled. The species that the sample mills used for studs were redwood, Douglas fir, and whitewoods, with red- wood and Douglas fir being by far the most important. "Whitewoods" refers to white fir, hemlock, and spruce. For the mills in this study, white fir constituted about 95 per cent of the whitewoods. Grading and seasoning. Companies in the sample sorted the studs into the fol- lowing grades (listed from lower to better grades): in the case of fir, "economy," "utility and better," and "standard and better"; in the case of redwood, "one star and better," and "two star and better." ] Table A-l PRODUCTION RATES AND LABOR REQUIREMENTS FOR SAMPLE MILLS Crew size Average Production Rates Number of man hours Code number of mill Board feet per hour (8 hour shift) Board feet per man hour (rounded) required to produce 10,000 board feet 1 38 19 20 45 22 9 27 85 22 18 36 26 * 10,000 6,900 9,500 18,750 10,000 4,400 13,000 18,750 9,600 5,000 6,500 260 360 480 420 460 490 480 220 440 280 180 38 2 27 3 : 21 4 24 5 22 6 21 7 21 8 45 9 23 10 36 11 12 13 55 * 'Information not available. Some mills did not sort all lumber but lumped two or more grades together. Most studs from the sample mills were sold green though some companies, ship- ping long distances, airseasoned or kiln- dried a small proportion of their redwood studs in order to save on freight charges. A couple of the mills also airseasoned a small fraction of their fir studs for special cus- tomers. Some companies gave rough estimates of their average grade distribution as shown in table A-2. Variation among the com- panies with regard to species distribution was primarily due to differences among the companies' log supplies, but it also reflected some differences in the type of customer. Variation in grade distribution was also primarily due to differences in log supply, but differences in interpreta- Table A-2 TYPES OF LUMBER PRODUCED BY SAMPLE MILLS Code number of mill Species Red- wood Douglas fir White fir Grades Douglas fir and White fir Econ- omy and better Utility and better Stand. and better Redwood One star and better Two star and better Percent* age shipped green 1 2. 3. 4. 5. 6. 7. 8. 9 10. 11 12 13 Per cent of output by volume 10 18 80 30 90 30 50 33.3 100 90 30 20 50 * 80 2 20 45 35 * 15 5 10 15 75 * 50 20 * * * * 10 100 100 10 90 100 25 60 15 55 15 20 70 10 * 40 10 2 98 * * 33.3 33.3 * * * * 100 * * * * 100 * * * * * * * • 100 100 98 32 90 33 100 100 100 100 70 100 100 100 "Information not available. [23 tion of grading rules also appeared to play a role. Ofher products. For the nine stud mills, other lumber products being manufac- tured were redwood fence material, cross arms, 4x4 clears, 2x3 shorts, and 2 x 1 shorts. These are all products that are close to studs in dimensions and have basically the same production process. Ex- cept for the redwood fence material — that is usually cut in the early spring — all have the character of a by-product cut from the same logs and produced simultaneously with studs. These other products ac- counted for 2 to 15 percent of the total volume output for each stud mill in the sample. As a pure by-product, some mills sold chips and planer shavings to pulp manu- facturers. Whether this was the case or not depended on the mill's proximity to a pulp mill and on whether or not it had a debarker. PRODUCTION COSTS Production costs per thousand board feet for individual mills varied as the type of "activity" changed. Activities that were relevant for the sample mills were: Day shift regular time, day shift overtime, night shift regular time, and night shift overtime. The questionnaire considered both di- rect production costs and overhead costs. Only costs that were obviously indepen- dent of the production level, i.e. strictly "fixed costs," such as fire insurance on buildings, were disregarded. The infor- mation consisted only of approximate fig- ures based on estimates of the mill man- agement. Some mills had little experience in working at more than one level, so less weight was given to their statements. With- in each individual activity level, all costs were found to be more or less linearly related to output. Log costs. This cost varied a good deal from mill to mill for three major reasons. Firstly, the quality of logs was not con- stant. Normally, logs used for studs were of lower grade and small size, often second growth but some mills had available to them only timber of higher quality and hoped to have their higher log price off- set by better lumber and higher sales price. Secondly, the location of the mill influ- enced the delivered log cost. Some mills had to go some distance to buy their tim- ber, and this affected transportation costs. Three of the mills had their own timber land. Thirdly, the amount of overrun that the mill was able to obtain affected the log cost per thousand board feet of lumber. Mills did not furnish detailed information Table A-3 DELIVERED PRICES FOR LOGS Code number of mill Species Average for all species combined Douglas fir Redwood White woods Dollars per thousand board feet log scale 1 45 53 49 * 60 53 47 55 53 60 t 38 45 40 * t t 38 40 t t 45 45 38 * * t t * 34 40 t t t „ 2 • 3 * 4 * 5 45 6 7 .. * 8 . 36 9 * 10 * 11 * 12 • 13 * "Information not available. tSpecies not purchased. [24 The general opinion was that produc- tivity was lower for hours worked on over- time than for regular time, because the crew is less alert during overtime. Produc- tivity was also considered to be lower on night shifts than on day shifts so that labor costs per thousand board feet were higher for these activity levels. Efficiency decreases as roughly estimated by the mills are shown in table 5. Table A-5 DECREASE IN PRODUCTIVITY COMPARED TO REGULAR TIME, DAY SHIFT about the amount of their overrun. Some stated that the overrun tended to be lower during overtime operation and on the night shifts. Decrease in alertness of the crew was given as the reason, but no esti- mate of the likely difference was provided. Average log prices (delivered in the yard) during the past year are shown in table A-3. Mills with limited opportunity for purchasing logs stated that their log cost would go up if they increased produc- tion significantly over some period of time, simply because the only way to exceed their normal log purchases would be to outbid their competitors .No estimate of the cost increase was obtained. Labor costs. Labor cost per thousand board feet of lumber also varied consider- ably from mill to mill. One reason was differences in the average wage per hour, but the main reason was differences in productivity per man hour. All mills stated that wage rates were the same on day shift as on night shift. Overtime rates were stated to be 150 per cent of the regu- lar rates, regardless of the number of overtime hours worked per week and re- gardless of the day on which overtime was worked. Ten mills gave information on their wage per regular man hour. These 'information not available. rates, including; unemployment compen- _. ■ ■ . i , \ a Power costs. Six mills gave estimates of sation, are Given in table A-4. , . & . their average power cost per thousand board feet. They were: $ .60, $ .80, $1.00, $1.25, $1.50, $2.00. These mills stated that the power cost per thousand board feet dropped if production increased signifi- cantly because of changes in rate schedules with increased use. There was, however, no correlation between the power costs listed above and the corresponding mill's level of production. Saws and saw maintenance. Four mills gave this cost per thousand board feet as | .75, $1.25, $1.50, $2.25. The cost of saws was estimated to be independent of the activity level though it was mentioned that the maintenance cost was higher for studs produced on overtime simply because the maintenance crew were paid at the over- time rate. Depreciation and maintenance. Though machinery depreciation is posted as an overhead in almost any company's account- Code number of mill Overtime, day shift Regular time, night shift 1 30 per cent no decrease "some decrease" no decrease "some decrease" 20 per cent "a little" 20 per cent "a little" * "some decrease" 10 per cent 2 3 4 5 6 10 per cent • "some decrease" 7 8 "a little" 9 • 10 * 11. 12. 13 * • • Table A-4 HOURLY LABOR RATES IN SAMPLE MILLS Code number of mill Green Chain Headrig Weighted Dollars 1 2.40 * * 2.70 2.60 2.60 2.30 3.85 * 4.60 3.50 * * 3.25 2 75 2 2 60 3 2 95 4 2.80 5 3 00 6 2 95 7 2 60 8 * 9 2.75 10 11 * 12 * 13 * "Information not available. [25] ing record, it was treated in this study as a direct production cost because it depends to a large extent on the level of production. None of the companies stated this cost ex- plicitly — probably because mill manage- ment usually thinks of depreciation as a yearly overhead cost computed in accord- ance with tax regulations. The cost they were questioned about was the cost per thousand board feet of keeping machinery constantly in good operating condition. Hence the cost included spare parts and machine replacements but not installation of additional equipment. A general feature of this cost was pointed out by several mills. When a sawmill is operated more than eight hours per day, the cost per unit of production of keeping the machinery in good shape starts to in- crease. During overtime operation the cost of maintenance naturally goes up because the maintenance crew draws overtime wages. If the activity level is two shifts per day, machinery fatigue starts to speed up wear and tear. Saw milling equipment cannot be operated continously without pauses for greasing, adjustments, replace- ment of parts, etc. If the operating time goes beyond two nine-hour shifts per day, the machinery simply gets "stress" and breaks down frequently. This is particu- larly true for older mills. Rough estimates of the average depreci- ation and maintenance cost per thousand board feet were given by five mills as fol- lows: $2.50, $2.75, $3.50, $5.70, $10.00. These are average figures relating to the particular pattern of activity levels char- acterizing the individual mill. Overhead costs. As previously mentioned, the study was concerned only with so- called overhead costs that were likely to be influenced by the activity level. The inter- views confirmed that costs of administra- tion and supervision were relevant with respect to this. The primary consideration was to determine differences between the expenses incurred when the mill operated at its normal activity level and those at other activity levels. If, for example, a mill normally worked 40 hours per week, the information sought was the increase in ad- ministration and supervision costs when the mill went to overtime and two-shift operation. [ Administration cost may be expected to change activity level mainly because de- mands on administrative personnel may change. The interviews indicated that the sample mills had one person in adminis- tration for about every ten men in the mill crew, regardless of whether the mill was working on a 40-hour week or on a two- shift basis. It was also the general opinion that whether a mill worked on regular time or overtime made no difference to the composition and size of the administrative staff. However, it appeared that if a mill went from one-shift to two-shift operation, the increase in office work required an additional office helper on a half-time basis. Supervision cost, on the other hand, was closely related to a mill's activity level. If the mill were to start producing on over- time, the supervision bill would be in- creased by the foreman's overtime wages. If the mill changed from a one-shift opera- tion to a two-shift operation, it was noted that supervision expenses would go up by the amount of the salary of at least one additional foreman. In fact, because the office would generally be closed during night shifts, it appeared that two addi- tional supervisors would be required in larger mills. INVENTORY CARRYING COST Interest charges. Interest on the capital tied up in studs and logs appeared to be the most important inventory carrying cost, although several of the mills did not seem to recognize this. Three companies stated that they were self-financing. In such cases it seemed that the inventory investment should be charged an interest rate corresponding to the yield which the company could obtain in alternative in- vestments. Two mills stated that the appro- priate rate of interest for them was 6]/ 2 per cent per annum. One stated that it was 2 per cent per month. Seven com- panies gave no information relating to this part of the questionnaire. The rate of 6i/4 per cent was the corresponding mills' interest rate on their long-term bank loans. The 2 per cent rate was the rate paid by the particular mill on its short term loans, and appeared to be the more realistic fig- 26] ure because stud mills tend to operate under very tight capital restrictions and many lending institutions consider fi- nancing of sawmilling operations a risky business. Handling costs. These costs were not given explicitly by any mill. A cost of about $.20 per thousand board feet per month for lumber and 1.5 times as much for logs appeared to be realistic. For studs the cost covers moving and restacking of occasion- ally misplaced and overturned piles, and for logs the cost covers mainly stacking and watering of the log deck. Property tax. Property tax is paid once a year on the value of inventories on hand on March 1. Information provided by the California State Chamber of Commerce indicated that in the North Coast counties of California the assessed value generally is between 22 and 23 per cent of the market value, and that the tax rate per $100 of assessed value varies between $8.00 and $11.00. time over 40 hours per week per shift must be paid as overtime, the only feasible division of weekly capacity for the mills in the sample appeared to be: • Day shift, regular time, 40 hours • Day shift, overtime, 14 hours (five hours during the week, nine hours on Saturday) • Night shift, regular time, 40 hours • Night shift, overtime, 14 hours. Inventory capacity limits. Mills were asked about limits on storage space and credit, for both lumber and log invento- ries. The information obtained is shown in table A-6. No specific data on credit availability were obtained, but the im- pression was formed that insufficient credit is an important constraint at many mills and prevents building up inventories when it is desirable. The constraint appeared to be a joint limit for both log and lumber inventories with the mill being able to hold less lumber when log inventories are high, and vice versa. Deterioration. Deterioration in value of the lumber and logs in inventory was con- sidered by all the mills to be negligible for SPACE LIMITS ON LUMBER any storage period encountered in prac- tice, say, up to three months for lumber and up to one year for logs. In the case of studs the argument was that it is a low- value product and is used in places that are not exposed to view so that a little weathering makes no difference to its mar- ketability. It was mentioned that if it was necessary to store studs for longer periods of time, the mill could stock them unsur- faced and run them through the planer when the shipping date comes up. For logs the argument was that proper watering of the deck prevents any deterioration. The value of the airseasoning that auto- matically follows as a result of Storing Studs *Information not available. was considered to be negligible except for some freight saving on the small quan- Constraints on log supply. The inter- tities of redwood studs that were shipped views revealed that two major factors regu- to the east coast. lated mills' log supplies. The first was that mills had what they regarded as a "normal quota" of logs available for purchase each year. For the individual mill this was a conditional constraint which could be re- moved if it were profitable to bid against its competitors. The second factor was the amount of logs which could be harvested Code number of mill Maximum inventory 1 Weeks of production 6.5 2 3.5 3 4.0 4 5.0 5 10.0 6 2.5 7 5.5 8 2.0 9 20.0 10 15 11 12 « 13 * CONSTRAINTS ON PLANNING Product/on capacity limits. As mentioned, it is practically infeasible to operate the mill machinery more than two nine-hour shifts per day, 6 days a week. Because all [27] during the winter. This quantity is en- tirely determined by the weather, with logging completely stopped by rain for several months in some areas. In such cases, if the weather is bad in the spring, the mills' cold deck may become depleted before logging can be resumed and a tem- porary closing down is inevitable. MARKET CONDITIONS Orders and prices. Because new orders and prices for studs are, by and large, mar- ket determined, the individual company has little influence on the orders it receives and on the sales prices it can obtain. The existence of a broad seasonal pat- tern in both sales and prices is clear. All 13 mills stated that their sales and prices start increasing in early spring and reach peaks for the year at the beginning of May. A more or less pronounced slowdown in sales and a drop in prices is encountered in July and August. In September and Octo- ber the sales and prices usually rise slightly but a "dead" period is common from De- cember through January, when both ship- ments and prices hit bottom for the year. The reason given for this concurrent sales and price pattern was that the stud is a product whose demand and price are strongly affected by activity in the construc- tion industry. When the home building season starts in spring demand and prices for studs go up. Vacations in the middle of the summer cause a slight slowdown in the building industry and a reduction in the demand for construction lumber. When the construction season ends late in No- vember, the stud market drops off very markedly. Although this broad seasonal pattern seems to be characteristic of the stud mar- ket, all mills emphasized that no two years are exactly alike. This was said to be due partly to instability in building activity, and partly to other factors. A wet spring can delay the start of construction activity, or a tight money market with scarcity of mortgage loans can depress the whole con- struction business. However, large govern- ment purchases, a shortage of rail cars, or fear of strikes as union negotiation dead- lines approached may temporarily more than offset such a depression. An example of a completely unpredictable situation is found in the flood in Northwest Cali- fornia in December 1964, which immedi- ately forced up lumber prices for the whole redwood region. Most companies could give approximate figures for the highest and lowest sales vol- umes that they encounter in a single week. Table A-7 gives this information both as volume in board feet and as proportions of the normal weekly output of the mill. It can be seen that the range from maxi- mum sales per week to minimum sales per week at most mills is from 5, 4 or ^ 3 of nor- mal weekly production to about ]/ 2 of nor- mal weekly production. It should be noted, however, that sales at mills do not neces- sarily correspond to orders received. When a mill states its highest weekly sales volume to be about 1.3 times normal weekly pro- duction, this must not immediately be in- terpreted as the highest level that can be achieved under any conditions. The maxi- mum sales figure may be an upper limit caused by capacity limits on shipping facil- ities (such as space in the yard for truck loading). On the other hand, it may simply be caused by stockouts so that the mill's production and inventory policy is the actual reason. None of the mills had re- cords conveniently available on customer orders, as distinguished from actual sales but some stated that stockouts occurred "now and then during spring" and others stated that orders were turned down frequently. All mills gave estimates of the normal yearly oscillation in their average sales prices. If one disregards price differences among mills because of differences in lum- ber quality, and also the latest year's devel- opments on the market that have forced prices to swing outside the usual intervals, these estimates suggest a general price fluctuation for redwood and white fir studs from about $55 per thousand board feet in the spring to about $48 in December, and for Douglas fir from about $65 in spring to about $55 in December. Customers. Nine of the sample mills ship- ped their production to California custom- ers only and in particular to wholesalers in the San Francisco Bay Area and Los Angeles. Two mills stated that their cus- tomers were located all over the nation 28 Table A-7 MAXIMUM AND MINIMUM QUANTITIES SOLD IN ONE WEEK Maximum quantities Minimum quantities Code number of mill Volume Proportion of normal weekly production Volume Proportion of normal weekly production 1 * Thousand board feet 700 1,000 1,000 500 210 * 1,000 500 400 • 1.3 1.3 1.0 1.25 1.2 1.3 1.3 2.0 * • * Thousand board feet 300 500 500 100 105 * 500 200 * • • 2 .55 3 .66 4 .50 5 .25 6 .50 7 8 .66 9 .53 10 11 * • 13 • information not available. and two gave no information. In addition, most mills were reported as occasionally selling minor quantities directly to con- struction firms. CURRENT PLANNING PRACTICES The interviews showed that almost all stud production was "to order," with most mills operating on a "back-ordering prin- ciple" that works as follows. In periods of market improvement, or whenever mill management expects sales prices to in- crease, mills do not book orders totalling more that one or two weeks' production. When this policy results in large quantities of new orders being turned down, the mills may go to overtime or a two-shift oper- ation if the management judges this to be profitable. Conversely, when the manage- ment expects sales prices to decline, mills are eager to book as long an order file as the customers wish. During decreasing de- mand when the order file keeps shrinking, a point may be reached where the excess production accumulated in inventory be- comes so large that management finds it best to reduce production or perhaps even close down temporarily. APPENDIX B: COMPUTER OUTPUT j OBJFN FIT F12 F13 F14 F21 F22 F23 F24 F31 F32 F33 X(J) 218597.109375 1759.999969 0. 651.683167 0. 1760.000046 0. 1584.158401 0. 1760.000000 0. 1584.158401 DELTA(J) POSITION 0. 1 -0. 301 3.000000 0.000000 101 5.500000 -0. 302 2.000001 -0. 107 4.500001 -0. 303 1.000001 -0. 111 29] J F34 F41 F42 F43 F44 F51 F52 F53 F54 F61 F62 F63 F64 F71 F72 F73 F74 F81 F82 F83 F84 X(J) 0. 1759.999985 0. 1584.158401 0. 1360.000000 71.719193 1224.122391 0. 1760.000000 0. 1575.805725 0. 1759.999817 0. 1584.158401 0. 1759.999939 0. 360.035999 0. POSITION 3.500001 0.000000 304 1.000001 0.000000 115 3.500001 0.000000 305 0.000001 117 0.000000 119 2.500001 0.000000 306 3.000001 0.000001 121 5.500001 0.000000 307 2.770001 0.000000 127 5.270001 0.000000 308 2.540001 0.000000 131 5.040001 F91 F92 F93 F94 F101 F102 F103 F104 Fill F112 Fl 13 F114 F121 F122 F123 F124 FS11 FS12 1760.000031 0. 0. 0. 1360.000000 0. 0. 0. 1 760.000000 0. 0. 0. 1760.000092 0. 0. 0. 0. 62.000000 0.000000 309 2.580000 0.000000 135 5.080000 0.000000 310 4.350000 1.350000 6.850000 0.000000 311 4.120000 1.120000 6.620000 0.000000 312 3.890000 0.890000 6.390000 20.000000 0. 102 FS62 FS63 FS64 62.000000 0.927998 62.000000 122 123 124 FS124 SA1 62.000000 3100.000000 [30] 0. -0.000000 148 201 J SA2 X(J) 3400.000000 DELTA(J) -0.000000 POSITION 202 SA12 DS1 DS2 DS3 DS4 DS5 DS6 DS7 DS8 DS9 DS10 DS11 DS12 Sll S12 1760.000092 0. 0. 0. 0. 0. 0. 0. 0. 839.999969 1140.000000 840.000000 1039.999908 111.683167 55.841614 -0.000000 6.500000 7.500000 5.500001 3.500000 0.500000 2.730000 2.730001 1.730001 0. 0. 0. 0. 0.000000 -0.000000 412 209 210 211 212 501 502 SI12 SIS1 SIS2 1688.316833 1744.158386 -0.000000 0. 0. 313 314 SIS12 1800.000000 0. 324 \ LIT LI2 LI3 LI4 LI5 LI6 LI7 LI8 LI9 LI10 Llll LI12 FLPOl FLP02 FLP03 FLP04 FLP05 FLP06 FLP07 FLP08 LIS! LIS2 0. 0. 0. 0. 888 2320 3745 5112 3757 2710 1355 143 1424 1424 1424 1066 6000 6000 799927 229431 227539 799988 599976 400024 200043 003937 998032 998077 998093 202057 000000 000000 1 .000000 1 .000000 1.000000 1 .000000 -0. -0.000000 0.000000 0.000000 0.000000 -0. 0.000000 9.000000 0. 0. 0. 0.000000 -0.000000 1 .000000 2.000000 3.000000 0. 0. 505 418 419 508 509 510 511 401 402 403 145 113 413 414 [31] X(J) DELTA(J) POSITION LIS3 LIS4 LIS5 LIS6 LIS7 LIS8 LIS9 LIS10 LIS11 LIS12 CRS1 CRS2 CRS3 CRS4 CRS5 CRS6 CRS7 CRS8 CRS9 CRS10 CRS11 CRS12 FLPOS 6000 6000 5111 3679 2254 887 2242 3289 4644 6000 244415 247207 250000 242792 210003 128799 47466 19923 80907 128031 189015 250000 3484 000000 000000 200073 770447 772308 200012 399963 599945 799927 .000000 839844 .919922 000000 078125 998047 378906 .549805 997070 997070 996094 996094 000000 200073 0. 0. 0. 0. -0.000000 0. 0. 0. 0. 0. -0.000000 -0.000000 ot 0.000000 0. -0.000000 -0.000000 0.000000 -0.000000 0. 0. 0. 0. 415 416 417 406 407 420 421 422 423 424 105 109 503 404 405 125 129 408 133 137 141 512 350 J F13 Fll FLPOl COST RANGE -1.000000 0. -1.298703 1.000001 2.000000 1 .000000 J(-) FS23 000000 F32 J(+) F32 FS11 Lll SA8 0. 1.730001 000000 DS8 DS9 -0.420000 0.269999 FS93 SI8 DS10 -0. 1.350000 SI10 F103 DS11 -0. 0. Sill SI10 1 195 18 7|rn-3,'68(H7531s)J.F.