Capital Budgeting for Volume Flexible Equipment

Decision Sciences Mume 27 Number 2 Spring 19% Printed in the U.S.A.

Capital Budgeting for Volume Flexible Equipment* George F. Tannous College of Commerce, University of Saskatchewan, 25 Campus Drive, Saskatoon, Canada, S7N 5A7, e-mail: tannous~commerce.usask.ca

ABSTRACT

Recent advances in technology have created oppomnities for firms to invest in expensive automated equipment designed to improve volume flexibility. Such investments are made on the basis that flexibility benefits the firm by increasing managerial control over output, reducing the risk of demand uncertainty, and improving productivity. The presumption is that these benefits will eventually translate to higher cash flows, appreciation in the firm’s market value, and better return to shareholders. Yet, there is no managerially useful analytical framework for measuring this relationship. This study develops a model that uses contingent claims analysis to evaluate the effect of volume flexibility on the firm’s value and to determine the optimal degree of automation that maximizes share value. The analysis is done by taking into consideration alternative demand characteristics, cost patterns, and the effectiveness of volume flexibility in increasing managerial control over output, reducing the risk of demand uncertainty, and improving productivity.

Subject Amas: CapW Budgeting, Corpomte Finance, Economics, Managemen! and Sbwtegy, and ProducliodOperations.

INTRODUCTION

Production volume flexibility is defined as the ability of a firm to adjust output in response to stochastic demand [16] [33]. Researchers recommend volume flexibility as a good way to counter demand fluctuations and to prevent over- or under-production [4] [6] [17] [32] [35]. They report that automated production processes that are capable of operating at various speeds and flexible run sizes will reduce inventories, increase productivity, and improve managerial control over output.

Details on the benefits of volume flexibility are provided in many recent case studies. Corbett [12] suggests that improved volume flexibility and delivery reliability has improved profitability at Formway Furniture Ltd. Culp [ 131 reports that the most compelling reason for the plastics industry’s interest in gear pumps is better metering accuracy, which improves gauge control and yields process economies. Dan Guthrie

*I am grateful to Paul Mangiameli, University of Rhode Island, for his valuable discussions and suggestions. Also, I gratefully acknowledge valuable suggestions from the editor. Lee Krajewski, the associate editor, and two anonymous referees. Any omissions or errom are mine.

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of Zenith Pumps noted that pumps also give processors more flexibility in switching materials and formulations and improve control over output. Magic Seasoning Blends Co. moved from a completely manual setup to a fully automated line to respond quickly to customer orders [l]. The move boosted the firm’s efficiency by 500%. Turiff [38] notes that Authentic Fitness of Canada Inc. was able to increase output and quality by 25% with the addition of two volume flexible machines. The improvements are the result of decreased cycle times, better repeatability, fewer rejects, and shorter setup time. Kochan [24] reports that the need for immediate response to the market for electronic fuel injection in combination with the catalytic converter has led Siemens Automotive to invest in a new manufacturing technology known as Mise En Ligne Flexible (MELF). Roe1 Hellemans of Siemens’ plant in Foix, France, indicates that the technology’s key attribute is its flexibility. The MELF employs a laser system (worth 1 million francs) to apply barcodes to boards and uses a four-armed robot (worth 4 million francs) to mount components on boards. This new technology raised output from 2,000 to 5,000 units per day. Wilson [40] studies Ikeda Hoover’s f 1.5 million investment in an ll-robot automated welding line from FANUC Robotics Ltd. of the United Kingdom to maintain output in support of the Nissan, Primera, and Micra programs. The facility has 10 robots lined in two parallel lines built close to each other to be unloaded at the end by a shared robot. For maximum flexibility, each line can operate independently. This automated facility insures product quality, reduced dedicated equipment, and almost eliminates the setup time needed for design changes. Levary [26] suggests using software to program both the processing sequence and the type of operations. The study shows that programmable automation is very efficient for production of small lots of different products and allows for product as well as volume flexibility.

The strong evidence documenting the benefits of volume flexibility has led earlier studies to suggest that firms should commit resources until full-volume flexibility is reached [17] [32]. They claim that a fully flexible manufacturer will be able to fill customer orders without lead time as they materialize, allowing the manufacturer to achieve zero inventory production. However, recent evidence suggests that the ideals of full-volume flexibility and zero inventory production are hard to accomplish. Several case studies find that inventory reductions are achieved at some firms by moving the inventory from plants to semitrailers, trucks, or suppliers’ warehouses [19] [20] [23]. Freeland [15] finds nearly half of the single source suppliers were required to hold safety stock to ensure supply. Evidently, all these cited companies have invested in flexibility but they are not operating at full-volume flexibility. Fred Boos, manufacturing operations director of Tom’s commercial equipment division in Tomah, Wisconson, states the division is looking at automation to produce as close to market demand as possible, but warns that operations should not automate simply because they can automate [2]. These comments imply that a decision model is desirable to determine the proper investment in flexibility and to avoid overinvestment.

Previous studies suggest that bener managerial control over output is a significant benefit of volume flexibility 111 [121 [13] [24] [26] [38] “1. Yet, previous models that are designed to evaluate volume flexibility have generally failed to incoprate this benefit. This failure prompted some researchers to suggest that using these models might lead to wrong conclusions and suboptimal investments. Aranoff [3]

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shows that utilizing cost minimization as basis for capital budgeting with technology choice and demand fluctuations can lead to errors if managers do not make proper allowance for expected idle capacity. Aranoff finds that the costs of wider fluctuations in output rates can be minimized by more output flexible equipment. Chung [ l l ] shows that ignoring the ability of the firm to modify plans after it learns its true demand can lead to wrong production decisions. Moreover, studies show that simple techniques, such as cost-volume-break-even analysis, are inadequate for justifying the purchase of modem automation technologies designed to provide product and volume flexibility [I81 [29]. These studies well describe and analyze the process for determining the costs or the benefits that can be achieved from volume flexibility, but do not adequately measure the financial consequences to the firm of this process, nor do they estimate the optimal investment in flexibility.

This paper helps to fill this gap by developing a model that incorporates the effectiveness of volume flexibility in reducing costs, decreasing the risk of demand uncertainty, and increasing managerial control over output in order to maximize the value of the firm. The developed model links the financial and the operations aspects of the decision to capture the full impact of volume flexibility. Specifically, the model treats expenditures on volume flexible equipment as investments rather than costs and sets the level of expenditures on flexibility after taking into consideration the return on investment. This treatment is justified on the basis that investments in volume flexibility will generate lower costs, higher sales, and better response to customer demands. These improvements will eventually translate into better profits and higher retums to shareholders. Karmarkar [22] provides excellent arguments supporting the idea of linking the operations and financial aspects of production decisions. Karmarkar suggests that the objective of the production and operations decision models should be maximizing the risk-adjusted present value of the cash flows to shareholders rather than minimizing costs or expected costs. Models that link the financial and operations aspects are particularly desirable when the decision is likely to alter the operating leverage and change the firm’s overall risk and cost of capital.

Given the desire to link the operations and financial aspects of the decision and the complex relationship between volume flexibility, productivity, managerial control over output, and the dynamics of the firm’s share value, contingent claims pricing techniques are employed to facilitate the analysis. The contingent claims framework is particularly desirable to quantify the benefits of better managerial control over output. Given that these benefits are contingent on the occurrence of certain conditions, it is not possible to measure their value without using contingent claims pricing techniques. Many studies suggest that the contingent claims framework is superior over traditional valuation methods when the objective is to evaluate managerial options such as facility operations and capacity planning [9] [28] [30] [37]. None of these studies is concerned with quantifying the benefits of volume flexibility or with the problem of finding the optimal investment in volume flexibility. However, this paper follows their assumptions to model stochastic demand and to develop a contingent claims valuation model in which volume flexibility is a decision variable and better managerial control over output is a major benefit. Also, the impact of flexibility on outputs and costs affects the firm’s pricing and other strategic decisions. The contingent claims framework allows this paper to


consider the decision to acquire volume flexible equipment along with other decisions that affect or are affected by volume flexibility.

Finally, previous studies have developed models to quantify the benefits of process flexibility (also known as product flexibility [16] [33]) but their models cannot determine the optimal investment in volume flexibility. Jordan and Graves [21] and Fine and Freund [14] develop models to quantify the benefits to a firm for having the flexibility to process two or more products on any one of two or more production lines. Their findings suggest that the higher the degree of process flexibility the lower will be the marginal return from increasing flexibility. Thus, firms may stop short of achieving full-process flexibility. Similarly, Triantis and Hodder [36] develop a model to evaluate process flexibility in which the f m can process two products on the same machine and can adjust periodically the volume of outputs subject to the capacity of the machine constraint. A special case of their model may be used to evaluate the flexibility of making periodic changes in the output volume of one product and may appear to be capable of accomplishing the objectives of this paper. However, Triantis and Hodder assume a manufacturing facility that is already flexible to produce any run size below the capacity constraint. In practice, small run sizes can be achieved after substantial investments. Porteus [31] and Billington [7] develop models to detennine the optimal investments in setup cost reductions which would make small run sizes economically feasible. This paper adopts their view and models the ability to produce lower run sizes as a function of volume flexibility. Indeed, the model in this paper views the one product version of the Triantis and Hodder [36] model as the ultimate goal of manufacturers, but questions the feasibility of achieving this goal for all industries. One objective of this paper is to determine the optimal investment in volume flexibility to widen the range of run sizes. In general, the optimal investment may or may not result in a fully flexible production facility capable of producing one-unit run sizes.

STOCHASTIC DEMAND AND VOLUME FLEXIBILITY: A REAL SITUATION This section presents the demand and production conditions under which a firm will benefit from volume flexibility and introduces the case of RainBow Inc. to explain the developments. The case is real, but the name of the company is disguised and the data has been modified to preserve confidentiality. The case will be used to facilitate the discussions throughout the remainder of the paper. This, however, should not imply that the model is specific only to RainBow. The case is used strictly for demonstration and can be eliminated without loss of continuity to the developments of the general model.

RainBow, established in the late 1940s, is a fabric refinisher that dyes and coats plain fabrics including silk, wool, cotton, cordura, and polyester. Customers include fashion designers, casual clothing and furniture manufacturers, and fabric retailers primarily in North America. The company sells on a smaller scale in South America and Europe and has plans to expand in the Far East markets. About 70% of the orders, mainly from US.-based fashion designers and upscale clothing manufacturers, are make-to-order where the customer provides the fabrics and the color designs and RainBow processes the order. The remaining customers, called “generic customers,”

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often purchase the treated fabrics directly from RainBow. The color designs required by these customers are simpler and more predictable than those demanded by the first group. Thus, their orders can be filled either from available inventory or by utilizing idle capacity. RainBow’s profit margin does not change whether an order is from a generic or a make-to-order customer. The company employs about 400 production workers, most of whom are highly skilled, and about 100 more employees in management, office, and sales functions. The company operates three shifts a day, 6 days a week, for about 300 days a year.

Generally, the problem of volume flexibility is real for any firm that faces uncertain demand, and the firm, either by deliberate decision or by circumstances beyond its control, commits itself to a production plan of some time T before the actual demand becomes known. An early decision is made based on a demand estimate, which may be derived from customer surveys, expressions of interest, or from past experience. Discrepancies occur between demand and output wherein some seasons’ output is higher than demand, leading to losses due to unsold inventory. Yet, in other seasons, demand is higher than output, leading to loss of sales. Also, volume flexibility is desirable when customer orders vary considerably in specifications and size, requiring the firm to process each order separately. If at the same time, the costs of setting up the equipment are high, it may not be economically feasible to process orders for amounts less than the “economic run size.” A firm facing such a situation can reject all orders below the economic run size but the rejection will lead to loss of sales. Alternatively, the firm can process a run size for every order below the economic run size, fill the required order, and store the remaining units for future sale. However, the differences in specifications between the various orders are likely to force the firm to clear these inventoried items at deep discounts from the regular price. This behavior will increase the average costs of production and depress the profits of the firm.

RainBow is an example of this type of firm. It has typically two selling seasons, with the first starting in April and the second starting in October. The production plan is generally set during the first week of the season and changes rarely occur. The company follows a level production strategy whereby the rate is determined at the beginning of the season for a total seasonal output of about 192,000 units (1 unit = 100 square yards of fabric) or an average of 1,280 units per work day.

RainBow receives orders continuously over the selling season but not evenly. Figure 1 shows a typical pattern of order arrivals in contrast to a level processing rate. As the figure suggests, orders are generally less than processing capacity in the early days of the selling season, higher than capacity during most of the third and fourth months when demand from make-to-order customers is high, and less than production capacity during the later part of the season. These later orders are often placed by generic customers.

As a consequence of the highly uncertain demand and the level processing rate, RainBow often experiences wide discrepancies between demand and output and relies on inventories to smooth such discrepancies. Figure 2 graphs the cumulative demand, output, and the size of net inventories (finished units minus back orders) for the typical season depicted in Figure 1. Figure 2 shows how RainBow uses the time when orders are slow to process and stock inventories for sale later in the season to generic customers. The figure indicates that RainBow has experienced


Figure 1: A typical pattern of order arrivals in contrast to the output rate.

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Note: A negative observation represents a day when cancellation of outstanding orders outweighed new orders.

seasons when demand from make-to-order customers was slow, which led to sizable inventories, idle capacity, or both, far above what might be sold to generic customers. Unfortunately, product specifications change from one season to the other so that any inventories remaining by the end of a season should be stored until the next season or sold at deep discounts. The company finds that discounting is more profitable than storing, as it allows the firm to recover about 50% of the unit costs of processing, while the proportion that can be recovered by storing will be 40% or less. On the other hand, RainBow has experienced seasons when demand from make-to-order customers was high, which forced the company to work on their orders throughout the remaining part of the season. Consequently, new orders were accepted until the production schedule was completely occupied and the available inventory completely sold. Afterwards, new orders were turned down and consequently lost.

The situation at RainBow is not unique. Vollman, Berry, and Whybark [39] provide other examples in which the manufacturing environment requires a lead or waiting time before production starts, thereby causing discrepancy between demand and output. For example, make-to-order manufacturers often require a lead time to design products and prepare samples or prototypes. Demand is uncextain and depends on the performance of the samples. In an assemble-teorder environment, a manufacturer

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Figure 2: The cumulative demand, output, and inventory or back orders.

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I 1 1 21 31 41 51 81 71 81 @I 101 1 1 1 121 131 141 Thnt in days

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may have essentially a single product, but with several desirable features that can be chosen by customers. The manufacturer often prepares the basic product or manufactures subassembly components waiting for orders to be confirmed. The special components that are different from one order to another will be processed and fitted as orders are confirmed. In these examples, volume flexibility would be achieved by acquiring a machine that can produce the entire product or the distin- guishing features with little lead time and in flexible run sizes to accommodate customer orders as they materialize.

For this situation, assume that the cumulative demand for the product is stochastic and can be represented by D(P)x(?), where x ( f ) is a stochastic factor that evolves over time and determines the actual cumulative (over time and markets) demand as a multiple of expected demand, x(O)=l, P is the selling price per unit of the product, and D(P) is a standard downward sloping expected demand function with D’(P)IO and D”(P)SO. D(P)x(O) represents the size of the cumulative orders at time 0 while D(P)x(T) is the size of total demand accumulated from time 0 up to time T.

Given that contingent claims pricing techniques will be used to facilitate the analysis, additional assumptions about stochastic demand are required. Following Chung [lo] and Pindyck [30], assume that stochastic changes in the cumulative demand are spanned by existing assets; that is, there is an asset or a dynamic portfolio of assets whose price is perfectly correlated with demand. Let p be the


instantaneous equilibrium rate of return on the asset or dynamic portfolio of assets. Then, x(t) may be defined by the stochastic differential equation:

where (p+6) is the instantaneous expected demand growth rate, B is the instantaneous standard deviation of the demand growth rate, and dw is the increment of a Wiener process. 6 is positive or negative depending on whether the growth rate of demand is greater or smaller than the equilibrium rate of return on the asset or dynamic portfolio of assets that has the same risk as demand. Note that (1) will allow the cumulative demand to change positively or negatively over time but the cumulative demand will never be negative.

RainBow’s experience over the past 30 years suggests that orders or expressions of interest during the first week of the season are good indicators of the season’s potential demand. The company estimates its sales for a season as 25 times the first week‘s demand and allows for a growth of 9% above the total. For example, if the first week’s demand is 7,000 units, the estimate for the season’s total demand would be 190,750 units. Demand depends on price, general economic conditions, and other factors that cannot be accurately predicted. Thus, errors in the forecast have occurred and led to actual deviations of 6% or less from the forecast in almost 70% of the time. Deviations above 6% and as far as 18% from the forecast happened 30% of the time. On average, the company’s price to customers is about $200 per 100 square yards (one unit), and the average unit cost is about $125. At the $200 unit price the company expects demand per season at 187,000 units and predicts that a one dollar rise in the price might negatively affect demand by 2,400 units.

RainBow’s data suggests that the expected demand function D(P) for a season can be estimated as D(P)=(667-2.4*P) thousand units. The 9% growth rate per season suggested by the data may be represented by 8.62% continuously compounded (9=e**62) over the season, or 17.24% over a year. Management estimates the required rate of return on investments that have the same risk as demand to be approximately 14% continuously compounded. This implies that 6 is equal to 3.24% (17.24% - 14%). Furthermore, the data indicates that ts is around 12%.

THE CASH n 0 W MODEL WITHOUT VOLUME FLEXIBILITY The general cash flow model for f m s facing demand uncertainty and volume inflexible production processes is attributed to Baron [5 ] , Leland [25], and Chung [lo]. The model assumes that the firm employs a production process that uses a fixed level of labor and material to produce a constant amount, Q, per selling season. The firm that chooses this basic process decides at time 0 both the output that will be produced at time T and the price at which the product will be sold. Let q, denote the time T cash flows to the firm without volume flexibility,fo(Q) represent the total fixed costs of operating a production process of size Q and no flexibility, C(0) denote variable costs per unit, and (1-h) represent the fraction of variable costs that can be recovered by liquidating excess inventory. The cash flows to the firm with no flexibility may be represented by:

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Expression (2a) represents the cash flows when the firm exactly matches output with demand, that is, when Q=D(P)x(r). In this case, (2b) and (2c) will be null, while the first term of (2a) measures the net revenues (sales minus cost of goods sold) and the second term deducts fixed operating costs. In all other cases, demand will be lower or higher than output. If demand is higher than Q, (2a) will overestimate the actual net cash flows, as the firm cannot produce more than Q. In this case, (2c) will be null, while (2b) will be less than 0 to reduce the cash flows measured in (2a) by the net revenues that will be lost, as the firm does not have the option of satisfying demand beyond Q. On the other hand, if product demand falls short of Q, (2a) will overestimate the actual net cash flows, as the firm will produce Q units but sales at the regular price will be less than Q. In this case, (2b) will be null while (2c) will be less than 0 to reduce the cash flows measured in (2a) by the variable costs that will be lost because the firm does not have the option to put off producing the excess units.

A few comments need to be made at this point. First, the model assumes that the investment needed to acquire the inflexible production process has already been made and is no longer a decision variable. Allowances for the replacement costs plus the fixed operating costs (related to insurance, rent, and so on) are assumed constant for a given production process size and paid periodically at the end of the planning horizon. However, as the firm begins its analysis of the costs and benefits of acquiring flexibility, the size of the investment will become a decision variable. The next section will model the firm’s decision to acquire flexibility and a separate expression will be added to represent the acquisition costs. Second, variable costs are determined by multiplying the unit cost C(0) by the amount to be produced. Thus, variable costs depend on the decision of how much to produce. Under the environment of production inflexibility, this decision is made at the beginning of the decision horizon. Therefore, variable costs will be committed at that time and incurred regardless of demand. For example, the size of the labor force is determined based on planned production but once employees are hired their costs become fixed for the remainder of the season. Third, excess production will be sold at the fraction (1-h) of the variable cost of producing it.

The Case of RainBow RainBow’s dyeing and coating processes and equipment provide little or no flexibility, which results in long flow times. The normal flow time through the process for a typical order is 6 to 8 weeks. Of this amount, processing time is about 4 days. The rest of the 6 to 8 weeks is actually idle time with the order waiting to be processed. Figure 3 depicts the typical order and material flow.

?he real bottleneck is the dyeing process. ‘The Dye House” is where most of the idle work-in-process can be found. The setup time is extensive. It includes removing


Figure 3: Material flow chart at RainBow.

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Operations function

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Proccns flow

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accessories used in processing the previous order, cleaning the work station and dye kettles, installing the accessories needed to process the new order, testing the product to insure the quality of the dye mixture, and finally getting the operation under way at the appropriate quality and output level. Moreover, the dyeing process often involves complex dyes where slight changes in color or fabric would require a new setup. In order to keep the already large setup time from becoming unmanageable, the colors and fabrics have to be run in a certain sequence, light to dark colors and loose to tight weave fibers. RainBow, therefore, holds orders until it has the proper sequence of color-fabric combination. If an order arrives after the sequence has started, it must wait until the next sequence.

RainBow's dyeing equipment consist of 16 large manually operated jigs. The run size per jig is about 240 units (i.e., 24,000 square yards) with an average processing time of 3 days for an average output per day equal to 80 units. At this rate, the company is running at full capacity to produce about 192,000 units per season. Thus, the firm has no flexibility to increase production if seasonal demand happens to be higher than 192,000 units. Moreover, the firm has no downside flexibility either. Given the long lead time and the expensive setup costs involved, it is not economical to process smaller run sizes on the current jigs. For a selling season, the company processes an average of 400 different color designs, half of which are processed with an average of three different fabrics per color design. Rare are the occasions when Raidow processes, in a selling season, more than one run size of each fabric and color design combination. Generally, orders for quantities

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less than half-full run size, that is, less than 12,000 square yards, are rejected, leading to loss of sales. For orders that are less than a full run size and larger than 12,000 square yards, RainBow will process a full run size, sell the desired quantity to the customer, and store the remaining amount (odd lot) for sale later in the season. In seasons when demand is high, generic customers will clear these odd lots at the regular price. In other seasons, odd lots are cleared at the discount price.

Of course, the firm can shut off a jig until a full run size order is placed or to lower production, but the shut-off rarely saves costs. A big part of the variable costs is labor-related as the majority of the firm’s work force is highly skilled. In addition to the direct costs involved in laying off and rehiring, RainBow’s experience suggests that laying off skilled workers often results in losing them permanently and having to train new workers. Management suggests that discount sales that bring in 50% of the unit variable cost will contribute to labor costs. Thus, the company prefers to keep workers and to produce at about full capacity. The yearly fixed costs of running the current process are about $7.925 million per season or $15.85 million per year. RainBow’s data suggests that, on average, Q is about 192,000 units per season, P is $200 per unit, C(0) is about $125 per unit,fo(Q) is $7.925 million per season, and h is approximately 50%.

INTRODUCING VOLUME FLEXIBILITY

Instead of continuing with the inflexible production process, the firm can replace parts of the process (e.g., one work station, one machine, two machines, etc.) or the entire production process with advanced versions to keep the same basic capacity Q but with capabilities to produce at lower or higher rates. Alternatively, the firm may be able to modernize some or all of the current equipment by replacing the obsolete components or parts with advanced replacements. Thus, the firm with a flexible production process of nominal size Q will have the ability to adjust its output around the level Q as demand may require, where the higher the flexibility the wider will be the possible range of production outputs. Let a denote the degree of flexibility by which a firm can adjust its output, where a is a fraction that ranges from 0 to 1. A general way to model the range of output possibilities for a given level of flexibility a is to represent the maximum output by Q(l+klp) and the lowest economically feasible output by Q( 1-kLa) where k , and kL are restricted to be nonnegative constants. If the firm has k p k L ‘ O , then no adjustment in the production output is possible and the firm will produce Q units or shut off. Similarly, there should be upper bounds on both kL and k,. Note that QkL represents the maximum amount by which production can be lowered with flexibility. The constant kL can approach 1 if with flexibility the run size can be lowered down to one unit, otherwise kL must be less than 1. In general, the condition OdC,<l must be satisfied. In contrast, Qk, represents the maximum amount by which the output can be increased with maximum flexibility. In practice, the size of Qku is constrained by technological limits and will depend on the product and the process under consideration. The constant k , should be equal to 1.5 if flexibility can increase production by 150%. As an illustration, assume that k L = k F l . This specification implies that if the firm chooses the maximum investment in flexibility, a will be 1 and the firm can choose to shut down, produce one unit, or any desirable output level up to the


maximum of 2Q. When the firm chooses partial investment in flexibility, a will take a value greater than 0 but less than 1, and the firm will be able to increase or decrease production to achieve the desired output (within limits). The firm cannot produce above Q(l+krpl), which is less than 2Q. or below Q(l-kLu), which is greater than one unit.

Let nu denote the time T cash flows to the firm with volume flexibility s f , (a) represent the additional fixed costs of operating a production process of nominal size Q and flexibility 4 C(a) denote variable costs per unit, and G(a) represent the additional investment needed to replace the old process with the new one to accomplish 100a% flexibility. G(a) is assumed to increase in a at a non- decreasing rate, that is, G’(a)>O and G”(a)TO. Assume that the investment G(a) is made at time 0 and represents the price of the new equipment minus the salvage value of the old equipment. Given that nu represents the cash flows at time T while G(a) is paid at time 0, G(a) will not appear with the cash flows xu, but it will be subtracted later from the present value of the cash flows. Thus, xu may be defined as:

Expressions (3a). (3b), and (3c) denote, respectively, the same cash flows as in (2a). (2b). and (2c), but now the firm has volume flexibility. Expression (3a) represents the cash flows if product demand exceeds the minimum production level, Q(l-k,a), and do not surpass the maximum level, Q(l+k@). In this case, the f m will adjust output to match demand. If product demand exceeds maximum output, the firm will produce Q(l+k@) units and excess demand will be lost. Expression (3b) will be less than 0 to reduce the net revenues calculated in (3a) for the sales that will be lost as the fm does not have the option to satisfy demand beyond Q( l+k@). If product demand falls short of the minimum production level Q(l-kLa), the f m will incur the costs of producing the minimum amount but sales at the regular price will be less than production. Expression (3c) will be less than 0 to reduce the net revenues calculated in (3a) by the portion of variable costs that will be lost because the firm does not have the option of putting off producing the excess units.

The cash flows represented by (3a), (3b) and (3c) define a general model for firms facing demand uncertainty. The model represents a continuum of all possible degrees of volume flexibility in which each firm’s special situation can be obtained by setting a at the appropriate level between 0 and 1. Of course, a particular firm’s optimal a will depend on the firm’s particular circumstances.

The Case of RainBow RainBow is considering a modernization plan intended to provide flexibility whereby orders can be processed with shorter lead times and flexible run sizes. Management identified a new equipment manufacturer who is offering a new dyeing technology and is willing to custom design the equipment. The proposed technology

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and the new equipment are expected to decrease the lead and setup times significantly for the dyeing process. The lead time can be cut to 3 weeks or less while the actual refinishing time per run size can be as short as 2 days rather than the current 4 days.

Essentially, most of the current setup work is done manually, while the new process will use computer-controlled dyeing jigs. Software-driven processing equip ment will control the temperature, dye flow, mixing, and the speed of the process. These features will increase the flexibility in switching materials and dye formulations and allow for more sophisticated color designs. Consequently, the rigid color- fabric sequence required to minimize setup time on the existing manual jigs can be relaxed and the overall lead time can be reduced. Maximum output per day can roughly double to about 160 units per jig, while at the same time, it will be economically feasible to produce at the rate of 32 units per day per jig. Moreover, the new dyeing technology combined with the new machine design will provide a higher quality product that can considerably cut waste and reduce the consumption of dyes. Savings in unit costs up to $12.50 can be achieved if all the 16 manual jigs are replaced with 16 computerized jigs.

Management is enthusiastic about the prospects of the new process and the flexibility it provides. First, it gives the firm flexibility to adjust the run sizes per jig to match customer orders more closely. This flexibility will increase sales as it will allow the fm to accept orders for quantities less than the c m n t cut-off quantity of 120 units. At the same time, this flexibility will reduce the frequency and magnitude of odd lots and the possibility of discount sales. Second, the new process will provide scheduling flexibility. For example, replacing the current 16 jigs with the computerized ones, and running them at full capacity will enable the company to finish production for the entire season by the middle of that season. If needed, the remaining time can be used to double production. Alternatively, processing can be delayed up to 3 months, after which production can start at a full rate to guarantee the current level of production. This way, if demand turns out to be lower than expected, production can stop and costs can be saved. A compromise between these two extremes would, perhaps, constitute the optimal strategy. Production can start at a medium rate for 3 months to guarantee a given output as management would like to maintain the current 80 units per jig per day for an average of 192,000 units per season. By the end of the 3-month period, RainBow will have resolved much of the uncertainty regarding the demand from make-to-order customers, and the rate of production can be increased or decreased accordingly. The problem is to decide the desirable number of jigs to replace. Ideally, it is preferable to replace the entire 16 jigs, but the size of the required investment can be prohibitive.

The company and the equipment manufacturer provided Table 1, which shows the relationship between the number of jigs replaced, the required investment, the additional fixed costs per season of operating the new process, the average savings in unit costs, and the range of output possibilities consistent with the new equipment. The table assumes that the output level of 80 units per jig will be maintained for the first 3 months of the season. For RainBow, the index of flexibility a is determined as the portion of jigs to be computerized out of the existing 16. For example, if the company decides to computerize 6 jigs, then a should be equal to 6/16 or 0.375. The data suggests that the required initial investment per jig is $925,000, which makes G(a)=$14,800,000a. Also, computerizing a jig will increase fixed


lhble 1: The effects of jig replacement when the basic production capacity is 192,000 units per season.

Additional Additional Average Number Flexibility Required Fixed Costs Savings in Minimum Maximum of Jigs Index a Investment per Season Unit Costs Output' Output' Replaced (%) 6) ($) ($1 (Units) (Units)

0 0.00 0 0 0.00 192000 192000 1 6.25 925000 41000 0.78125 188400 198000 2 12.50 185oooO 82000 1.56250 184800 204000 3 18.75 2775000 123000 2.34375 181200 21oooO 4 25.00 3700000 164000 3.12500 177600 216000 5 31.25 4625000 205000 3.90625 174000 222000 6 3750 555oooO 246000 4.68750 170400 228000 7 43.75 6475000 287000 5.46875 166800 234000 8 50.00 7400000 328000 6.25000 163200 24oooO 9 56.25 8325000 369000 7.03125 159600 246000

10 62.50 925oooO 41oooO 7.81250 156000 252000 11 68.75 10175000 451000 8.59375 152400 258000 12 75.00 111OOOOO 492000 9.37500 148800 264000 13 81.25 12025000 533000 10.15625 145200 27oooO 14 87.50 1295oooO 574000 10.93750 141600 276000 15 93.75 13875000 615000 11.71875 138000 282000 16 100.00 14800000 656000 12.5oooO 134400 288000 Note: Entries are boldfaced for illustration purposes. aMinimum and maximum outputs are determined by assuming the f m will maintain output at 80 units per jig per day for the first three months (approximately 75 work days) of the season. Beginning with the fourth month, the firm can either double output per computerized jig, slow down the output rate by 60 percent, or adjust the rate to any level between the two. Thus, when six jigs are computerized, these extreme levels are determined as:

Minimum = B O X 10 x 150+ 80x 6 x 75 t 80 x (1 -0.6) x 6 x 75 = 170,400 units

Maximum= 80 x 10 x 150 +80 x 6 x 75 + 80 x 2 x 6 x 75 = 228.000units

costs per season by approximately $41,OOO, suggesting thatfI(a)=$656,000a. In contrast, a computerized jig will lead to approximately $12.50/16=$0.78125 average savings in variable costs per unit, implying that C(a) can be approximated by (125-12.5~). For example, if ~ ~ 0 . 3 7 5 , G(a)=!$5.55 million, fi(a)=$246,000, and C(a)=$120.3125, Finally, the data suggests that [email protected] and kL=0.3, which are obtained by solving Q(l+k& and Q(l-k,a) at any level of a, respectively. For example, when or=O.375,Q( l+k@)=228,OOO units and Q(l-kd=l70,400 units. Given that Q is 192,000 units, solving these equations yields the values of k, and kL. Thus, if a=O.375, it will be economically feasible for RainBow to produce a minimum of 170,400 units per season, but it can produce as much as 228,000 units. If product demand does not fall outside this range, the firm will adjust output to match demand.

Tannous 171

If product demand exceeds the maximum output of 228,000 units, the firm will produce the maximum amount and excess demand will be lost. If product demand falls short of the minimum output of 170,400 units, the f i i will incur the costs of producing the 170,400 units, but sales at the regular price will be lower than 170,400 units. "he penalty to the f m for not having the option of putting off producing the additional units is approximately $60.16 per additional unit or 50% of the variable costs per unit.

THE INCREMENTAL VALUE FROM THE INVESTMENT IN VOLUME FLEXIBILITY Generally, investments in volume flexible manufacturing systems will produce benefits over several selling seasons. Let m represent the number of seasons that span the useful life of the equipment. This section calculates the incremental net present value that accrues to the firm over m seasons. First, the value that will be created over a typical selling season i is calculated.

Recall that (3a), (3b) and (3c) represent the cash flows expected at the end of a typical selling season i. Given the demand specifications and assumptions underlying ( I ) , the results of Black and Scholes [ 8 ] and Smith [34, p. 161 can be used to calculate the present value of the cash flows at the beginning of the season i . The calculations will yield:

N is the cumulative standard normal distribution function, kST, T=(T '~) , R=e-ff, In(.) is the natural log function, and r is the rate of return on the riskless asset. Expressions (4a), (4b), and (4c) calculate the present value at season i's beginning of the cash flows that will be received at the end of the season by the firm that has partial volume flexibility. Thus, the present value of the cash flows to the firm with no flexibility can be derived by decreasing a to 0 to obtain:


where dy and d! are d , and 4 (or d3 and d4), respectively, measured at a=O. Thus, the incremental present value that can be derived during selling season i from investing at time 0 in volume flexibility can now be calculated as the difference between Vi(P ,Qp) and Vi(P,Q,O). Let Si represent this incremental value, then:

Expressions (6a) through (6d) calculate the present values of the benefits and the costs of volume flexibility. Expression (6a) calculates the value to the firm for having the ability to increase production when demand is larger than Q. Expression (6b) represents the value to the firm for having the ability of putting off producing the amount (Q-D(P)x(T)) when demand is less than Q. Their sum represents the value to the firm generated by increasing managerial control over output. Note that if (P-C(a)), that is, the contribution margin, is positive, then (6a) and (6b) are both positive, suggesting that higher managerial control will always increase the firm’s value. Expression (6c) represents the expected change in total variable costs. It will contribute positively or negatively depending on whether variable costs will decrease or increase due to flexibility. Several studies conclude that variable costs will indeed decrease as flexibility is introduced [7] [17] [31] [32]. Incorporating this conclusion implies that (6c) is always positive. Finally, (6d) measures the change in fixed costs resulting from the increase in flexibility. Depending on the firm’s circumstances, these costs may increase or decrease with flexibility.

The sum of (6a) through (6d) calculates the net present value that will be created at the beginning of period i by acquiring volume flexibility. The investment in flexibility will last for m seasons. Thus, the incremental net present value that will be created over the m periods can be expressed as:

where Q is the firm’s discount rate per season. Assuming that the value contributed per season, Si(P,Q,a), is constant over the m seasons and equal to S(P,Q,a), then:

Tannous

where

173

The expression 4S(f,Q,a) calculates the present value that will be created during the m seasons, and G(a) represents the investment needed to create this value. Thus, if this incremental net present value is positive, the investment in flexibility will increase the firm’s value. Otherwise, no investment should be made. However, for optimal results, the firm must invest in flexibility until the marginal incremental value is equal to the marginal investment.

THE OPTIMAL INVESTMENT IN VOLUME FLEXIBILITY This section optimizes the firm’s value with respect to volume flexibility. It assumes that product price P is constant, which is a reasonable assumption when the firm is operating in a highly competitive environment prohibiting price changes. Also, it is assumed that the firm can add flexibility by modernizing existing equipment but has no plans or ability to change plant size (the number of jigs at RainBow). that is, Q is also constant. For example, RainBow can computerize the existing jigs or replace them with new ones without buying additional jigs or increasing the size per jig. These assumptions are made to simplify the presentations and to isolate the effects of volume flexibility from those attributed to other decision variables. Later, these assumptions will be relaxed to allow for changes in both price and production process size.

In reality, flexibility is introduced by taking several discrete steps, in which each step increases flexibility by a given amount. For example, RainBow can computerize one, two, or more jigs with each increasing flexibility by 6.25%. Thus, the set of feasible a values will contain a finite number. When this number is small, finding the optimal a. requires a simple routine. For every given value of a in the set, the optimal values of the other decision variables are found using traditional optimization techniques. The a that yields the highest f m value is chosen as the best or optimal. However, as the number of steps needed to achieve full flexibility gets larger, this discrete procedure will quickly become cumbersome, if not impos- sible, to implement. Given a large number of steps, it is far better to assume that a is a continuous variable. This allows traditional optimization techniques to be utilized in order to find the optimal values of a and the other decision variables in maximizing the continuous function. The optimal investment in volume flexibility would then be determined as the level that is nearest to the optimal a. Please note that in solving the case of RainBow, the assumption is that the number of jigs is large enough to warrant approximating the discrete set of a values by the continuous closed range between 0 and 1. Yet, only the values that are related to the feasible set of a values will be shown in the tables.

Assuming that the only decision variable is a, the marginal net present value from investing in flexibility is equal to the marginal investment when the first derivative of S,(f,Q,a) with respect to a is equal to 0. That is, when:


- LC(a)O[RQ( 1 - kLa)( 1 - N(d4)) - D(P)A( 1 - N(d3))I

- Rof’I(01) - G’(o~) = 0. (9)

Therefore, the value of 01 that will solve (9) is the level of flexibility that will maximize the firm’s value. The solution can be easily obtained by numerical techniques once the various parameters and functions that describe demand and costs are specified. Mathews [27] provides an excellent guide to several numerical methods such as Newton’s, Fixed-Point Iteration, Bisection, and Regula Falsi. The tables in this paper are generated by using Newton’s method.

Application at RainBow

Table 2 reports the optimal degree of flexibility under different scenarios of the investment needed to computerize a jig. The calculations are prepared using the data supplied by RainBow to estimate the values for the various parameters and functions. ’Ihus, Table 2 assumes that D(P)=1000(667-2.4P) units per season, where a unit is 100 square yards of fabric, P=$200, Q=192,000 units or 16 Jigs, C(u)=125-12.5a, k d . 5 , kL=0.3, f0(Q)=$7.925 million per season, fi(a)=$656,000a, G(a)=$14.8a million, ad.12, T d . 5 year, W.0324, pO.14, k50%, m=10 seasons, w . 1 2 5 , and 4 . 0 8 .

Column 2 shows that if the investment per jig is $925,000, the optimal degree of flexibility is 43.75%, which can be achieved by computerizing seven jigs. The firm’s optimal investment in volume flexibility would then be $6.475 million. This investment would increase the firm’s value from approximately $43.79 million to about $50.20 million.

Note from column 2 that if RainBow invests in flexibility more than $6.475 million it will experience a net increase in value. For example, computerizing all 16 jigs would increase the firm’s value by $5.73 million over the value without flexibility. However, the marginal net benefits of computerizing jigs beyond seven would be less than the marginal investment, and the difference explains why firm value will decline continuously after computerizing seven jigs. This point is further demonstrated by columns 3 through 8, which are prepared by increasing the investment needed to replace one jig in steps of $1OO,OOO each. It is clear that the higher the marginal investment needed the lower will be the optimal number of jigs to be replaced. Moreover, as the investment per jig exceeds $1.225 million, full flexibility will actually decrease the firm’s value. These observations clearly indicate that optimal behavior will lead different firms to achieve different degrees of flexibility depending on the marginal investment needed to increase flexibility. Generally, the size of the marginal investment will increase with the complexity of the production process under consideration and the sophistication of the required technology.

Tannous 175

Table 2: The relationship between the initial investment required to replace one jig and the optimal degree of flexibility.

The Initial Investment Required to Replace One Jig (in Dollars) Flexibility Index (%) 925,000 1,025,000 1,125,000 1,225,000 1,325,000 1,425,000 1,525,000

0.00 6.25

12.50 18.75 25 .OO 3 1.25 37.50 43.75 50.00 56.25 62.50 68.75 75 .00 81.25 87.50 93.75

100.00

43.7893 46.0605 47.7408 48.8841 49.59 10 49.9787 50.1546 50.2023 50.1790 50.1194 50.0428 49.9589 49.8720 49.7842 49.6960 49.6078 49.5 196

43.7893 45.9605 47.5408 48.5841 49.19 10 49.4787 495546 49.5023 49.3790 49.2194 49.0428 48.8589 48.6720 48.4842 48.2960 48.1078 47.9 196

43.7893 45.8605 47.3408 48.2841 48.79 10 48.9787 48.9546 48.8023 48.5790 48.3194 48.0428 47.7589 47.4720 47.1842 46.8960 46.6078 46.3 196

43.7893 45.7605 47.1408 47.9841 48.3910 48.4787 48.3546 48.1023 47.7790 47.4194 47.0428 46.65 89 46.2720 45.8842 45.4960 45.1078 44.7 196

43.7893 45.6605 46.9408 47.684 1 47.9910 47.9787 47.7546 47.4023 46.9790 46.5 194 46.0428 45.5589 45.0720 44.5842 44.0960 43.6078 43.1196

43.7893 45.5605 46.7408 47.3841 47.5910 47.4787 47.1546 46.7023 46.1790 45.6194 45.0428 44.4589 43.8720 43.2842 42.6960 42.1078 4 1.5 196

43.7893 45.4605 46.5408 47.0841 47.1910 46.9787 46.5546 46.0023 45.3790 44.7194 44.0428 43.3589 42.6720 41.9842 4 1.2960 40.6078 39.9196

Note: The numbers shown represent firm values in millions with boldfaced enhies presenting optimal values for the particular combinations of flexibility and the initial investment needed to replace one jig. The initial investment per jig is set at $925,000, $1.025 million, ..., and $1.525 million. The values of the other parameters and functions are D(f)= lOoO(667-2.4f) units per season, where a unit is 100 square yards of fabric, f=$200, Q=192,000 units or 16 jigs, C(a)=125-12.5 a. k @ 5 , k~=03,fo(Q)=$7.!325 million per sea,wn,fl(a)=$656,0CO a, ~ ~ 0 . 1 2 . T 4 . 5 year, M.0324, p0.14, M O % . m=10 seasons, @=0.125. and e0.08.

THE BENEFITS OF FLEXIBILITY AND THE SIZE OF THE OPTIMAL INVESTMENT Previous studies suggest that a large benefit from investments in volume flexibility will be the increase in managerial control over output 141 [61 [171 [321 [351. They argue that in many industries management has no control over demand. Thus, managerial control over output becomes particularly valuable as it helps the firm to reduce the costs associated with the mismatch between demand and output. Also, these studies suggest that the benefits of volume flexibility include improving productivity and reducing variable costs. This section shows how managerial control over output and productivity improvements can affect the firm’s decision to invest in volume flexibility. Continuing with RainBow’s case, Tables 3 ,4 , and 5 illustrate the conclusions.

Managerial Control over Output and the Optimal Investment in Flexibility Intuitively, higher demand volatility implies less managerial control over demand. Thus, to keep the risk of the mismatch between demand and output constant, higher


'Igble 3: 'Ihe impact of demand volatility on the optimal choice of volume flexibility.

Volatility of the Demand Growth Rate (in Dollars) Flexibility Index (%) 0.12 0.14 0.16 0.18 0.20 0.22 0.24

0.00 6.25

12.50 18.75 25 .OO 3 1.25 37.50 43.75 50.00 56.25 62.50 68.75 75.00 81.25 87.50 93.75

100.00

43.7893 46.0605 47.7408 48.8841 49.5910 49.9787 50.1546 503023 50.1790 50.1194 50.0428 49.9589 49.8720 49.7842 49.6960 49.6078 49.5 196

42.7097 44.9796 46.7434 48.0373 48.9284 49.4995 49.8333 50.0019 50.0624 50.0560 50.0106 49.9436 49.8653 49.78 14 49.6950 49.6075 49.5 195

4 1.6203 43.8882 45.7 146 47.1248 48.1663 48.8986 49.3849 49.6847 49.8492 49.9198 49.9277 49.8956 49.8388 49.767 5 49.6880 49.6041 49.5 179

40.5252 42.7903 44.6644 46.1664 47.3306 48.20 15 48.8276 49.2571 49.5339 49.6960 49.7749 49.7950 49.7749 49.7284 49.6649 49.5909 49.5106

39.4266 41.6881 43.5993 45.1746 46.4400 47.4292 48.1802 48.7317 49.1208 49.381 1 49.5419 49.6276 49.6581 49 6492 49.6 126 49.5572 49.4894

38.3258 40.5830 42.5234 44.1583 45.5077 46.5980 47.4592 48.1226 48.6194 48.9785 49.2264 49.3863 49.4779 49.5176 49.5186 49.49 15 49.4444

37.2237 39.4762 4 1.4397 43.1234 44.5432 45.7201 46.6780 47.4427 48.0402 48.4953 48.8314 49.0697 49.2291 49.3257 49.3733 4 9 3 3 4 49.3653

Note: ?he numbers shown represent fm values in millions with boldfaced entries representing optimal values for the particular combinations of the demand volatility and flexibility. Demand volatility o is set at 0.12,O. 14, ..., 0.24. The values of the other parameters and functions are D(P)=1000(667-2.4P) units per season, where a unit is 100 square yards of fabric, P=$200, Q=192,000 units or 16 jigs, C(a)=125-12.5a, k d . 5 , k ~ 4 . 3 , fo(Q)%7.925 million per season,f,(a)=$656,0, G(a)=$14.8a million, T=0.5 year, M.0324, pd.14, h=50%, m=10 seasons, @.125, and r=O.O8.

volatility would require higher managerial control over output. Table 3 proves this argument by showing a positive relationship between demand volatility and the optimal degree of volume flexibility. As demand uncertainty increases, it becomes optimal for RainBow to commit more resources to increase the number of computerized jigs and to improve its control over output. For example, if demand volatility is 0.24 instead of 0.12, RainBow has to modernize as many as 15 jigs to achieve optimal results. That is, investment in volume flexibility should be $13.875 million instead of the $6.475 million required when volatility is 0.12. It seems that higher demand uncertainty will increase production costs as it increases the magnitude and the frequency of the mismatches between demand and output l'hus, the firm will commit additional funds to reduce these mismatches and the costs associated with them.

Table 4 demonstrates the impact of the demand growth rate on the optimal degree of volume flexibility. It shows that a higher growth rate will lead the firm to invest less in flexibility. For example, if RainBow's demand is forecasted to grow at 29.24% instead of the current forecast of 17.248, only five jigs would be computerized to achieve optimal results. It seems that flexibility gives RainBow a large benefit by reducing the costs that occur when demand is less than output. Recall that the loss per overproduced unit is estimated as 50% of the unit's variable cost.

Tannous 177

Table 4: The impact of the demand growth rate on the optimal choice of volume flexibility.

Flexibility Index (%) 0.1724 0.1924 0.2124 0.2324 0.2524 0.2724 0.2924

The Growth Rate of Demand (in Dollars)

0.00 6.25

12.50 18.75 25 .OO 3 1.25 37.50 43.75 50.00 56.25 62.50 68.75 75 .OO 8 1.25 87.50 93.75

100.00

43.7893 46.0605 47.7408 48.8841 49.59 10 49.9787 50.1546 50.2023 50.1790 50.1 194 50.0428 49.9589 49.8720 49.7842 49.6960 49.6078 49.5 196

44.0973 46.3844 48.0580 49.1699 49.8270 50.1572 50.2787 50.2819 50.2263 50.1455 50.0563 49.9654 49.8750 49.7855 49.6966 49.6080 49.5 196

44.4935 46.8 12 1 48.4943 49.5824 50.1854 50.4428 50.4878 50.423 1 50.3 146 50.1970 50.0844 49.9798 49.8 820 49.7887 49.6980 49.6086 49.5199

44.9389 47.3069 49.0226 50.1096 50.6700 50.8515 50.8043 50.6489 50.4637 50.2885 50.1369 50.008 1 49.8964 49.7956 49.701 1 49.6 100 49.5204

45.3974 47.8323 49.6124 50.7330 5 1.2782 513954 5 1.2502 50.9850 50.6977 50.4398 50.228 1 50.0596 49.9238 49.8094 49.7077 49.6130 49.52 1 8

45.8393 48.3556 50.2321 5 1.4289 5 1.9998 52.0799 5 1.8438 5 1.4574 5 1.0442 50.6753 50.377 1 50.1479 49.9730 49.8353 49.7206 49.6 19 1 49.5245

46.2425 48.8499 50.8516 52.1693 52.8159 52.9006 52.5964 52.0890 5 1.53 18 5 1.0235 50.6083 50.2914 50.0566 49.8813 49.7446 49.6309 49.5301

Note: The numbers shown represent fm values in millions with boldfaced entries representing optimal values for the particular combinations of the demand growth rate and flexibility. The demand growth rate (p+6) is set at 0.1724,0.1924, ..., 0.2924 by increasing 6 from 0.0324 while leaving I.( constant at 14%. The values of the other parameters and functions are D(P)=I000(667-2.4P) units per season, where a unit is 100 square yards of fabric, P=$200. Q=192,000 units or 16 jigs, C(a)=l25-12.5a, k d . 5 , kL=0.3, f0(Q)=$7.!X25 million per season, fi(a)=$656,000a, G(a)=$14.8a million, d . 1 2 , T=0.5 year, k 5 0 % , m=10 seasons, e0.125, and eO.08.

A higher growth rate of demand will reduce the risk of this loss as demand will fall short of output less frequently and the magnitude of the shortfall will be lower. Thus, the higher growth rate of demand will reduce the benefits that RainBow can derive, thereby eliminating justification for high investments in flexibility.

The above observations have important managerial implications. If a firm is facing highly volatile demand, it should invest to achieve high levels of volume flexibility for better control over output. In contrast, if demand is forecasted to grow at a high rate that decreases the risk of overproduction, small investments in flexibility might achieve the desired results.

Unit Cost Reductions and the Optimal Investment in Flexibility Productivity improvements may result in a decrease in fixed or in variable costs per unit, or both. Table 5 shows the relationship between the size of the unit cost reductions and the optimal investment in volume flexibility. As the magnitude of the reductions decreases, the firm will invest less and less in flexibility. Intuitively, the reduction in variable costs resulting from improvements in flexibility magnifies the benefits.


’Igble 5: The average savings in unit costs and the optimal degree of flexibilitySa

Flexibility The Average Savings in Unit Costs if All Jigs Are Replaced (in dollar^)^

Index (’KO) 12.50 10.00 7.50 5 .00 2.50 0.00 -2.50

0.00 43.7893 43.7893 43.7893 43.7893 43.7893 43.7893 43.7893 6.25 46.0605 45.8385 45.6165 45.3945 45.1725 44.9504 44.7284

12.50 47.7408 47.2922 46.8435 46.3949 45.9462 45.4975 45.0489 18.75 48.8841 48.2064 47.5288 46.8512 46.1735 45.4959 44.8182 25.00 49.5910 48.6837 47.7764 46.8691 45.9618 45.0545 44.1472 31.25 49.9787 48.8419 47.7052 46.5685 45.4317 44.2950 43.1583 37.50 50.1546 48.7889 47.4233 46.0577 44.6920 43.3264 41.9608 43.75 503023 48.6083 47.0142 45.4202 43.8261 42.2321 40.6380 50.00 50.1790 48.3569 46.5347 44.7126 42.8905 41.0683 39.2462 56.25 50.1194 48.0694 46.0193 43.9693 41.9193 39.8692 37.8192 62.50 50.0428 47.7649 45.4871 43.2092 40.9313 38.6535 36.3756 68.75 49.9589 47.4532 44.9475 42.4419 39.9362 37.4306 34.9249 75.00 49.8720 47.1386 44.4052 41.6717 38.9383 36.2048 33.4714 81.25 49.7842 46.8230 43.8617 40.9005 37.9393 34.9781 32.0169 87.50 49.6960 46.5070 43.3180 40.1290 36.9400 33.7510 30.5620 93.75 49.6078 46.1910 42.7742 39.3574 35.9406 32.5239 29.1071

100.00 49.5196 45.8750 42.2304 38.5858 34.9413 31.2967 27.6521 aThe numbers represent firm values in millions (boldfaced entries represent optimal values) for the particular combinations of the savings in variable costs per unit and flexibility. The variable cost savings per unit are set at $12.50, $10, ..., 0, and -$2.5, which correspond with C(a)=125-12.5a. C(a)=125-10a, ..., C(a)=125, and C(a)=125+2.5a. The values of the other parameters and functions are D(P)=1000(667-2.4P) units per season, where a unit is 100 square yards of fabric, P=$200, Q=192,000 units or 16 jigs, k @ S , k ~ d . 3 ,

fo(QH7.925 million per season,fi(a)=$656,0, G(a)=$14.8a million, m.0324, ~ 4 . 1 4 , -0.12, T=OS year, b50’KO. m=10 seasons, $=0.125, and 4 . 0 8 .

bA negative value represents an increase in unit variable costs.

Thus, the lower the magnitude of the reductions the lower will be the firm’s desire to invest in volume flexibility. However, Table 5 shows that the lack of productivity improvements does not necessarily mean zero investments in flexibility. For example, column 7 shows that it would be optimal for RainBow to computerize two jigs even in the case when computerization is not likely to reduce variable costs. Column 8 shows that investments in flexibility would be desirable in circumstances in which the new equipment would increase variable costs. These observations suggest that better managerial control over output can provide enough benefits to justify investing in volume flexibility.

IMPLICATIONS OF FLEXIBILITY FOR PRICE AND PRODUCTION PROCESS SIZE The analysis so far assumes that the selling price and the production process size (the number of jigs at RainBow) are constant. For example, the assumptions are that RainBow’s product price is about $200 per unit and that its production process has 16 jigs. However, volume flexibility will affect the cost structure of the firm and its

Tannous 179

ability to process large and small run sizes. Thus, it is reasonable to expect that flexibility would affect product price P and the production process size Q. In many situations, the firm would be able to change these variables when it adds flexibility, This section demonstrates how (8) can be used to determine the optimal investment in volume flexibility while considering its impact on product price and production process size.

In the presence of the interaction between the decisions involving product price, the production process size, and the degree of volume flexibility, equation (8) must be solved for the optimal values of the three decision vatiables. Assuming that these variables are continuous and the value of a is limited to the range O l c c l , we can optimize the firm’s value described in (8) by forming the Lagrangian function f=S,(P,Q,a)+O( 1-a), and satisfying the Kuhn-Tucker conditions d€&a= O,O( 1 -a)=O,O20, and

af/aP = @D(P)A( 1 - N(dl)) + @RQN(d2)( 1 + &@)

+ @D’(P)A((P - C(a))( 1 - N(d1)) + hC(a)(l - N(d3))) = 0, (10)

where aSfla is calculated in (9) and 0 is a multiplier. The solution will produce the optimal values P*, Q*, and a* to optimize the firm’s value.

Depending upon the functional relations between volume flexibility, price, and production process size, general strategic implications can be considered for manufacturing firms. By letting a be an exogenous parameter, (10) and (1 1) can be solved for the optimum P’ and Q* as functions of a. The derivatives of these functions with respect to a will determine the overall strategic implications. These derivatives along with their signs can be determined under fairly general conditions, and several conclusions can be derived. The conclusions are reported below, but the derivations and proofs are omitted to simplify the presentations. Detailed derivations and proofs can be obtained from the author upon request.

First, it can be shown that dP*/da is negative, indicating that introducing flexibility will lead the firm to decrease product price. Flexibility will enable the firm to satisfy higher demand through larger run sizes and lowering the price is optimal as it promotes higher demand, giving the firm an opportunity to capitalize on its new capabilities. Second, di?/da will be positive for a firm whose demand function is strongly price sensitive. Such a firm will decrease product price as suggested above and simultaneously increase production process size to increase further its ability to satisfy higher demand. In contrast, dp/da is negative for a firm whose demand function is weakly price sensitive. This firm will decrease product price and simultaneously decrease the size of the production process as it introduces flexibility. ’Ihis is optimal because a weakly price sensitive demand will limit the increase in sales as price drops. This makes the additional output that can be accommodated


by larger run sizes more than enough to satisfy the additional demand that can be created by the lower price.

Depending upon the operating environment of the firm, other cases may be considered for general conclusions. First, consider the case when the firm is a price leader that may change product price but cannot change the size of its production process as it invests in flexibility. This possibility exists when the firm can modify its existing production process to install flexibility or when the firm can choose between several processes with the same basic size but with various degrees of flexibility. In this case, the firm will increase price as flexibility is introduced if demand is weakly price sensitive. As flexibility protects the firm from the risk of low demand, increasing price is optimal because it allows the firm to increase the contribution margin. On the other hand, investments in volume flexibility will lead the firm to decrease product price when demand is highly price sensitive. In this case, a slight decrease in price will lead to a significant increase in demand, and the combination of a slightly lower contribution margin and significantly higher demand will increase the firm’s value.

Another interesting situation is when the firm can change the size of the production process as it acquires flexibility, but product price is market-detemined and no longer a decision variable; that is, the firm is a price-taker in a highly competitive market. In this case, the firm will increase the size of the production process as it invests in flexibility if demand is highly volatile. Now that the risk of facing a low demand is reduced by acquiring the ability to produce small run sizes, the firm can justify a larger production process. In contrast, when demand volatility is relatively mild, investments in volume flexibility will enable the firm to decrease the size of its production process. For this firm, a highly flexible small production process is more desirable than a large inflexible process.

Numerical Illustrations: The Case of RainBow The implications of flexibility for RainBow’s product price and the number of jigs in operation are illustrated by Table 6. The table is prepared in two steps. First, equations (10) and (1 1) are solved to derive the optimal P and Q for the values reported in column 1. The results of this optimization are reported in columns 2 and 3, and they assume that the production process size Q can be increased by steps of one unit each. While this might be possible at some firms, it does not hold in the case of RainBow, where the size of the production process must be increased by a minimum of one jig or 12,000 units. The second step is designed to adjust the optimization process to allow size increments of 12,000 units each. This step assumes that it is reasonable to add a new jig whenever 55% of the new jig’s capacity is likely to be utilized; that is, the optimal Q exceeds the existing capacity by more than 6600 units, or when Q is greater than 198,600 units. This happens when a exceeds 12.5% or when the number of computerized jigs exceeds 2. Thus, the next feasible c1 value is 3/17=17.65%, as at this point, the number of jigs increases from 16 to 17 and the reported values of a beyond 17.65% are increased by increments of 1/17. or 5.88%. Columns 4,5, and 6 report the results after adjusting for discrete changes in the number of jigs.

Table 6 suggests that volume flexibility will allow the firm to decrease product price. It seems that dropping the price is optimal for RainBow as the lower price

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Table 6: The implications of flexibility for product price and production process size.

Optimization Results Assuming Optimization Results Assuming Continuous Qa Discrete Qb

Flexibility Product Production Production Product Firm Index a Price P Process Size Process Size Price Value (7%) ($1 Q (Units) Q (Jigs) p ($1 ($)

O.oo00 0.0625 0.1250 0.1765' 0.2353 0.2941 0.3529 0.4118 0.4706 0.5294 0.5882 0.647 1 0.7059 0.7647 0.8235 0.8824 0.94 12 1 .m

203.10 202.20 201.43 200.87 200.31 199.81 199.36 198.94 198.55 198.16 197.79 197.42 1 97.05 196.68 196.3 1 195.94 195.58 195.21

196025 197199 198148 198798 199437 200010 200561 201 127 20 1736 202409 203 162 204004 204940 205975 207087 208328 209677 211122

16 16 16 17 17 17 17 17 17 17 17 17 17 17 17 17 17 18

204.41 203.88 203.35 199.34 199.06 198.84 198.65 198.48 198.29 198.05 197.75 197.42 197.06 196.69 196.32 195.95 195.58 195.21

44.3 194 46.6046 48.3514 49.3027 50.2726 50.9272 5 1.3425 5 1 S802 51.6858 51.6416 5 1.5 194 51.3341 51.0951 50.8073 50.523 1 50.2435 49.9688 49.699 1

Note: Boldfaced entries represent optimal values. aFor a given value of a. equations (10) and (1 1) are solved to obtain the values of P and Q. %he number of jigs is derived by dividing Q as reported in Column 3 by 12,000 and rounding off decimals of 0.55 or less to 0 and decimals higher than 0.55 to 1. Product price is then obtained by solving (10) for the given a (Column 1) and Q (Column 4).

'This number is 3/17 and represents the third jig to be computerized from a total of 17 jigs.

will stimulate demand, while acquiring flexibility will allow the firm to process more fabrics and capitalize on the higher demand. At the same time, it would be optimal for RainBow to increase the number of jigs to 17. Increasing the basic capacity along with dropping the price is optimal as the lower price would stimulate demand, suggesting a higher capacity to capitalize on the possible increase in demand. For example, computerizing eight jigs (d.4706) would make it optimal for RainBow to drop its price slightly to $198.29 per unit and to increase the number of jigs by one. The price drop and the additional capacity will maximize the firm's value as it will lead to higher sales.

MANAGERIAL IMPLICATIONS AND CONCLUSIONS

This paper provides a managerial tool that can be used, along with other tools, in determining the optimal budget that must be allocated to purchase volume flexible equipment. The model is applied to a real situation to demonstrate data needs and


the ability of the model to generate information for the decision-making process. Several useful results are derived.

The analysis indicates that the firm’s value will increase as a function of the degree of flexibility but the optimal investment is likely to be reached before achieving full flexibility. It is shown that the industry under consideration, the complexity of the technology required for flexibility, the firm’s desire to improve managerial control over output, the growth rate and the volatility of product demand, and the effectiveness of flexibility to reduce costs will all affect the pattern of spending on volume flexibility. The optimal degree of flexibility will decrease with the complexity of the required technology as complexity would mean higher costs of flexibility. Fast-growing firms will spend less on flexibility compared to slow-growing firms. In contrast, the level of spending on flexibility would increase as demand uncertainty increases. Moreover, it is shown that better managerial control over output can provide enough benefits to justify investing in volume flexibility under circumstances in which flexibility would increase variable costs. Finally, it is shown that investments in flexibility directly affect product price and production process size. The implications, however, will vary depending upon the market environment in which the firm operates. For the illustrative case solved in this paper, flexibility will allow the fm to decrease product price and increase the size of its production process as flexibility is introduced. [Received: September 19, 1994. Accepted: November 30, 1995.)

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George E 'Ignnous is an associate professor of finance at the College of Commerce, University of Saskatchewan, Canada. Dr. Tannous has published in such journals as International Review of Economics and Finance, Decision Sciences, Interna- tional Journal of Bank Marketing, and Managerial and Decision Economics. His current research interests include pricing of managerial options, integration of financial and operating decisions, effective executive compensation, financing foreign trade, pricing interest rate collars, and use of futures and options to hedge risk.

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Capital Budgeting for Volume Flexible Equipment