Improving Manufacturing Performance Through Process Change and

Improving Manufacturing Performance Through Process Change and Knowledge Creation Janice E. Carrillo • Cheryl Gaimon John M. Olin School of Business, Washington University, St. Louis, Missouri 63130-4899 DuPree College of Management, Georgia Institute of Technology, Atlanta, Georgia 30332-0520 [email protected] • [email protected]

A

model is introduced to guide a profit maximizing firm in its quest to enhance performance through process change. The key benefit sought from process change is a long term increase in effective capacity. However, realizing success from process change is not trivial. First, while process change may increase effective capacity in the long run, the disruptions during implementation typically reduce short term capacity. Second, competitive forces such as decreasing revenue streams and shrinking product life cycles complicate the implementation of process change. Third, while knowledge may enhance the ultimate benefits derived from process change, the correct timing and means of knowledge creation are difficult to discern. Lastly, a variety of trade-offs must be evaluated when selecting the particular process change to pursue. For example, choices range from hardware and software replacements to modification of manufacturing procedures. The model introduced here explicitly considers both the short term loss due to disruption and the long term gain in effective capacity associated with the process change. In addition, investments in the accumulation of knowledge are investigated for their potential to enhance process change effectiveness. Knowledge is generated from investment in preparation and training (learning-before-doing) and as a by-product of process change (learning-by-doing). Analysis of the model provides managerial recommendations for several key decisions relating to process change implementation including: (i) the selection of an appropriate process change alternative, (ii) the rate and timing for investment in process change, and (iii) the rate and timing for investment in preparation and training. New results are reported reflecting the important relationship between process change and knowledge. For example, we show that under certain conditions, a firm should optimally delay investment in process change until sufficient accumulation of knowledge is achieved. More generally, we identify conditions whereby investment in process change occurs at an increasing rate over time. This result is particularly important since it demonstrates a limitation of the existing literature where process change always occurs at a decreasing rate. (Process Change; Knowledge Management; Optimal Control Theory)

0025-1909/00/4602/0265$05.00 1526-5501 electronic ISSN

Management Science © 2000 INFORMS Vol. 46, No. 2, February 2000 pp. 265–288

CARRILLO AND GAIMON Improving Manufacturing Performance

1. Introduction The importance of effectively aligning a firm’s manufacturing capabilities with its overall corporate strategy is well established, (see Skinner 1969). Process change is a practical means by which a firm can enhance its manufacturing capabilities to better compete during both current and future planning horizons. For example, process change may result in a decrease in manufacturing cost and an increase effective capacity. (The term effective capacity refers to the maximum volume of output realized under normal operating conditions.) A variety of process change alternatives can be utilized to achieve manufacturing goals, including hardware and software upgrades and procedural modifications of manufacturing processes. A vast body of literature attests to the difficulties associated with managing process change. Though critical for a firm’s long term competitive posture, process changes may be the source of serious short term disruptions. For instance, reduced productivity, excessive equipment downtime, and problems in scheduling, materials, quality and maintenance are common by-products of process change (see Lindberg 1992). In other empirical research, Hayes and Clark (1985) report that the short term loss in productivity from implementation of new manufacturing equipment is often more costly than the actual equipment purchase. Furthermore, these authors state that the loss in productivity may persist for up to two years. Despite the common occurrence of the short term disruptions, Goodman and Griffith (1991) state that managers tend to select and plan process change from the narrow viewpoint of the long term benefits sought. Competitive forces may exacerbate the negative impact on profit resulting from the short term disruptions during process change implementation. For example, for firms operating in environments characterized by time based competition, a short term loss in productivity may have long term repercussions. Similarly, as a result of short product life cycles, a period of disruption in production may preclude a firm from remaining in the market of an existing product or from entering a new product market (see Pisano and Wheelwright 1995). The empirical literature indicates that knowledge

266

may enhance the actual gain in effective capacity derived from process change (Adler and Clark 1991, and Bohn 1994). According to the literature, knowledge is gained in two ways: (i) explicitly through advanced preparation and work force training, and (ii) implicitly through actual experience. Pisano (1994) shows that both learning-before-doing and learningby-doing impact on the success of process change projects. In addition to increasing the ultimate gain in effective capacity obtained from process change, other benefits result from a firm’s investment in knowledge assets. First, the level of knowledge impacts a firm’s long term competitive posture. Roth and Marucheck (1994) emphasize that knowledge itself becomes an asset providing competitive advantage during subsequent planning horizons. Second, knowledge associated with process change and related activities can increase unit profits by either (i) decreasing unit costs (for example, see Fine 1986), or (ii) providing premium revenue for output (for example, see Roth and Marucheck 1994). In this paper, we introduce a normative model of a profit maximizing firm seeking performance benefits from process change. The extent and duration of process change activities are optimally determined over a finite planning horizon. The dynamic model captures both the short term loss and the long term gain in effective capacity associated with process change. In particular, we explicitly consider the impact of process change on the maximum volume of output generated (effective capacity) over time. As a result, we can explore the link between process change and the firm’s ability to generate revenue from output. The model also permits analysis of process change strategies in the context of a dynamic marketplace. Specifically, the revenue function (i.e., the value of output) as well as the costs incurred for both the process change itself and the preparation and training may vary over time. This modeling feature is important for firms operating in time based competitive markets or for firms manufacturing products with short life cycles. The model also captures the value of the firm’s level of knowledge in relation to the type of process change

Management Science/Vol. 46, No. 2, February 2000


under consideration. A firm increases its level of knowledge actively through investments in preparation and training, or passively from experience gained from implementing process change. Based on the results of the empirical literature, we assume knowledge enhances the long term gain derived from process change. Lastly, our model captures the possibility that knowledge relating to process change may be harnessed to provide premium revenue for output. Important managerial recommendations are developed based on analysis of the model. First, the model offers insights pertaining to the optimal magnitude and timing of process change over the entire planning horizon. The particular costs incurred and the level of effective capacity available over time depend on the firm’s dynamic process change strategy. For example, a relatively simple process change, such as a minor engineering change order (ECO) or a minor change in operating procedures, may affect a small portion of capacity for a short period of time. The cost and the ultimate gain in effective capacity for such a change may be modest. In contrast, suppose a firm seeking a large long term gain in effective capacity introduces a major new system of advanced manufacturing technology. If a rapid timing strategy is employed for this process change, the firm may experience a substantial reduction in effective capacity in the short term. In addition, the cost incurred at each instant of time may be relatively high. Alternatively, if the new manufacturing system is implemented slowly over time, then the magnitude of disruption and the cost incurred at any point in time may be reduced, but the overall time until the new system is fully operational may be lengthened. Clearly, both the magnitude and timing of process change must be evaluated in the context of the firm’s ability to generate revenue and incur costs over time. Second, the model provides managerial insights concerning investments in preparation and training. In particular, a firm may invest in knowledge to support its pursuit of process change. Consider the previous example in which a firm implements new advanced manufacturing technology. The firm may invest in knowledge to increase the magnitude of gain in effective capacity derived from process change.


Furthermore, our model captures the possibility that knowledge employed for process change may provide premium revenue for output. Despite these benefits, however, the firm must also consider the investment cost incurred to increase the level of knowledge and the rate of return of that investment over time. Third, the model may be employed to evaluate alternative process change projects. Process change options cover a wide spectrum including major equipment replacement, upgrade of software, or modification of manufacturing procedures. Each option is characterized by a unique combination of input parameter settings indicating the cost and impact on effective capacity over time. Therefore, cost differences among process change options are reflected in appropriate parameter settings. Similarly, parameter settings capture differences in the short and long term impact on effective capacity due to various process change alternatives. As discussed above, different levels of knowledge may be desirable for distinct process change projects. Also, particular process change projects may be more appropriate in certain competitive environments (for example, in relation to different levels of time based competition). A manager can explore these issues by resolving our model for different representations of the input parameters. Several features of our model distinguish it from the related normative literature. Despite its importance, the normative literature does not reflect the critical impact of the short term disruptions that occur when process change activities begin. Inclusion and analysis of the short term disruption is a key contribution of our research. Also, the primary focus of the normative literature is on process change that reduces a firm’s manufacturing cost. Fine (1986) introduces a dynamic model to analyze the role of quality improvement to increase the rate of learning in production processes, and thereby reduce the unit cost. Chand et al. (1996) introduce a dynamic model in which process change is driven by cost savings from increased conformance quality. However, unlike the model introduced here, these authors do not explicitly consider the possibility that process change may increase production capacity or enhance revenue. Finally, the existing normative literature on process change does not distinguish

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between activities occurring off-line (learning-beforedoing) and on-line (learning-by-doing). Li and Rajagopalan (1998) include autonomous and induced learning in their dynamic process improvement model. However, in contrast to our approach, they do not consider investments in preparation and training to enhance process change effectiveness. As a consequence of the unique features in our formulation, new results and managerial insights are obtained. By modeling the dynamics of learning in relation to the timing of process change, we are able to assess the value of knowledge generated from preparation and training versus knowledge obtained as a by-product of process change. Furthermore, we identify conditions whereby the rate of process change optimally increases over time. For example, we show that under certain conditions managers should delay investment in process change until sufficient knowledge accumulation from preparation and training has occurred. These results are particularly important since they suggest a limitation of the existing literature where process change is always advocated at a decreasing rate over time (see Li and Rajagopalan 1998, Chand et al. 1996, and Fine 1986). The remainder of the paper is organized as follows. In §2, the profit maximizing model is introduced. Optimal solutions are derived and discussed in §3. Analytic results are explored in §4 offering insights to improve managerial decision making for process change and knowledge acquisition policies. Further interpretation of results is given in §5 through analysis of numerical examples. Section 6 contains a summary of conclusions. References are given in §7.

2. The Model In this section, we present a profit maximizing model to aid decision makers’ planning for process change implementation. The model is sufficiently general to provide insights for a wide variety of process change endeavors. For example, the process change may reflect hardware, software, or procedural changes. First, we examine the impact on effective capacity over time due to knowledge. Second, we explore the relationship between the level of effective capacity over time and the magnitude and timing of process change.

268

We conclude this section by introducing the dynamic profit maximizing objective. (For a summary of the notation introduced, see Table 1.) 2.1. Knowledge As previously discussed, knowledge is a key factor that enhances the long term benefit derived from process change. Let S(t) represent the cumulative level of knowledge at time t that is relevant for the process change under consideration (state variable). The interpretation of this state variable is similar to the stock of knowledge in Dorroh et al. (1994). The level of cumulative knowledge may be modified over time through preparation and training and through experience gained during the actual implementation of the process change. This characterization follows from the empirical literature in which both learning-beforedoing and learning-by-doing are shown to contribute to knowledge (see Pisano 1994). Let s(t) denote the rate of preparation and training (learning-before-doing) at time t. This control variable reflects the efforts of line workers and supervisors who are involved with planning and who receive training to operate the process change. In addition, this variable captures the efforts expended by engineering and manufacturing specialists who develop and test the process change. An upper bound is defined to emulate a dynamic budget or labor constraints. Let p(t) represent the rate of process change implemented at time t. This control variable measures the amount of capacity affected by the process change under consideration. Learning-by-doing is embodied in the relationship between the rate of process change and the level of cumulative knowledge over time; i.e., cumulative knowledge increases from process change. To reflect dynamic resource constraints, we have an upper bound on the rate of process change. Equations (1)–(2) mathematically convey the manner in which the level of cumulative knowledge is modified over time. According to Pisano (1994), the degree to which s(t) and p(t) contribute to the firm’s level of knowledge depends on factors including the rate of technology change and the particular industry in which the firm operates. Therefore, the coefficients ␣ 1 (t) ⱖ 0 and ␣ 2 (t) ⱖ 0 represent the extent to which



Table 1

Model Notation

Variable t p(t ) s(t ) K(t ) S(t ) ␲ [S(t ), K(t ), t] c 1 [s(t ), t] c 2 [p(t ), t] V 1 [S(T)] V 2 [K(T)] ␣ 1 (t ) ␣ 2 (t ) ␺ f [p(t ), S(t )] w(t ⫺ ␶ ) x(t )

␴ (t ) ␭ (t )

Description Time; t 僆 [0, T], T is the terminal time of the planning horizon. Rate of process change at time t ; p(t ) 僆 [0, p(t )]; p(t ) is the maximum rate permitted at time t. Rate of preparation/training at time t ; s(t ) 僆 [0, s(t )]; s(t ) is the maximum rate permitted at time t. Effective level of capacity at time t ; K(0) ⫽ K 0 ⬎ 0. Cumulative level of knowledge at time t ; S(0) ⫽ S 0 ⬎ 0. Net revenue generated with K units of effective capacity and S units of knowledge at time t. Cost of s units of preparation and training at time t. Cost of p units of process change at time t. Salvage value of S units of cumulative knowledge at the terminal time. Salvage value of K units of effective capacity at the terminal time. Impact on cumulative knowledge per unit preparation/training at time t (learning-before-doing). Impact on cumulative knowledge per unit process change at time t (learning-by-doing). Extent to which a unit of process change causes an immediate loss in effective capacity. Capacity gain associated with p units of process change and S units of knowledge at time t. Time lag per unit process change at time ␶ which adds to effective capacity at time t, ␶ ⱕ t. Cumulative increase in net revenue, from time t to the end of the planning horizon, due to the lagged gain in effective capacity per unit process change at time t. Marginal value of an additional unit of cumulative knowledge at time t. Marginal value of an additional unit of effective capacity at time t.

a unit of preparation and training versus process change impact on cumulative knowledge. Lastly, note that S 0 is known at the initial time, t ⫽ 0. S˙ 共t兲 ⫽ ␣ 1 共t兲s共t兲 ⫹ ␣ 2 共t兲p共t兲, s共t兲僆关s៮ 共t兲, 0兴,

S共0兲 ⫽ S 0 ⬎ 0,

p共t兲僆关p៮ 共t兲, 0兴.

(1) (2)

2.2. Effective Capacity In this section, we discuss the dynamic nature of the firm’s level of effective capacity. Let K(t) denote the level of effective capacity at time t (state variable). At the initial time of the planning horizon, the level of effective capacity is given by K(0) ⱖ 0. The level of effective capacity is reduced due to disruptions following the introduction of process change. Ultimately, effective capacity increases as a consequence of the benefits derived from the process change. The first term in Equation (3) reflects the immediate loss in effective capacity that occurs when the process change is introduced. We assume the magnitude of immediate loss in effective capacity at a particular time is proportional to the amount of process change implemented at that time (␺ is constant). For example,


implementation of a relatively simple engineering change order may affect only a small portion of the operating capacity. Conversely, the introduction of a large new system of advanced manufacturing technology is likely to disable a substantial amount of effective capacity. Therefore, the magnitude of immediate loss in effective capacity due to a process change implemented at time t is given by the product ␺ p(t), with ␺ ⱖ 0. Lastly, note that the immediate loss in effective capacity cannot result in negative output (see Equation (6)). Next, we consider the long term gain in effective capacity from a particular type of process change. Larger rates of process change lead to larger gains in effective capacity in the long term. However, as the rate of process change at any instant of time increases, the ultimate gain in effective capacity increases at a decreasing rate (diminishing returns). Also, consistent with the empirical literature, suppose larger amounts of cumulative knowledge lead to larger long term gains in effective capacity from process change. Let f [ p( ␶ ), S( ␶ )] represent the nonnegative long term gain in effective capacity due to p( ␶ ) units of process

269


change and given S( ␶ ) units of cumulative knowledge at time ␶. From the above, we obtain the first and second order conditions in Equation (4). Empirical evidence indicates that the long term gain in effective capacity from process change is not realized instantaneously (Hayes and Clark 1985). Instead, a continuous increase in the level of effective capacity occurs over time as the manufacturing environment gains experience operating under the process change. Let w(t ⫺ ␶ ) 僆 [0, 1] represent the portion of the long term increase in effective capacity realized at time t due to process change implemented at time ␶, for ␶ ⱕ t. Therefore, w(t ⫺ ␶ ) is a continuous lag function. Eventually, the process change at time ␶ is fully effective so that 100% of the gain in effective capacity is realized (Equation (5)). However, it is possible that process change may not be fully effective in the current planning horizon. Given the definitions of f [ p( ␶ ), S( ␶ )] and w(t ⫺ ␶ ), we can complete our interpretation of the state equation for K(t ). The second term in Equation (3) denotes the portion of the long term gain in effective capacity realized at time t due to process changes that occurred prior to that time. The lower bound of the integral expression reflects the possibility that portions of process changes first implemented during the previous planning horizon become effective during the current planning horizon. (From Equation (7), process changes made in previous planning horizons are known.) K˙ 共t兲 ⫽ ⫺␺ p共t兲 ⫹

冕

t

f 关p共 ␶ 兲, S共 ␶ 兲兴w共t ⫺ ␶ 兲d ␶ ,

⫺⬁

K共0兲 ⫽ K 0 ⬎ 0, f 关p共 ␶ 兲, S共 ␶ 兲兴 ⱖ 0, ⭸ 2 f/⭸p 2 ⱕ 0, 0 ⱕ w共t ⫺ ␶ 兲 ⱕ 1,

冕

(3)

⭸f/⭸p ⱖ 0, ⭸f/⭸S ⱖ 0,

(4)

⬁

w共t ⫺ ␶ 兲dt ⫽ 1,

(5)

␶

K共t兲 ⱖ 0

for t 僆关0, T兴,

(6)

s共 ␶ 兲 ⫽ given for ␶ 僆关⫺⬁, 0兴, p共 ␶ 兲 ⫽ given for ␶ 僆关⫺⬁, 0兴.

270

(7)

2.3. The Objective The firm maximizes the profit generated from effective capacity over the planning horizon. The length of the planning horizon, T, represents the life cycle of an average product currently manufactured in the facility. First, we consider the amount of net revenue earned by the firm at time t, denoted by ␲ [K(t ), S(t ), t] ⱖ 0. This function depicts the revenue minus operating costs earned with K(t ) units of effective capacity and S(t ) units of knowledge. Therefore, while we assume all output generated from effective capacity is sold, its value may vary over time (net revenue is dynamic). As a result, we may explore the effect of dynamic demand and time based competition. While net revenue is defined in general functional form, certain conditions are intuitively clear. For example, net revenue is a nondecreasing function of both the level of effective capacity and the level of cumulative knowledge. However, as the level of effective capacity increases, the additional net revenue earned increases at a decreasing rate (i.e., decreasing returns are realized as K(t ) increases). Net revenue is defined as a function of cumulative knowledge for two reasons. First, cumulative knowledge applied to process change may have the effect of providing premium revenue. Second, the learning effects associated with process change may lead to a reduction in the unit cost (i.e., an increase in net revenue). Finally, we assume net revenue increases at a decreasing rate as S(t ) increases (diminishing returns). Next, consider the costs incurred over the planning horizon. Let c 1 [s(t ), t] ⱖ 0 denote the cost to actively increase the firm’s level of knowledge through s(t ) units of preparation and training at time t. The cost incurred for preparation/training increases at an increasing rate in relation to s(t ), reflecting diseconomies of scale at each instant of time. Similarly, let c 2 [ p(t ), t] ⱖ 0 denote the cost of p(t ) units of process change at time t. The cost of process change increases at an increasing rate reflecting diseconomies of scale associated with additional units of process change at each instant of time. The nature of these costs mirrors assumptions in the production smoothing literature. Lastly, the objective function includes expressions



depicting terminal time salvage values. Let V 1 [S(T)] denote the value of cumulative knowledge at the terminal time of the planning horizon. Similarly, let V 2 [K(T)] reflect the value of effective capacity at the terminal time. To interpret these salvage value functions, consider a firm operating in an environment characterized by rapid technology change and short product life cycles. In this situation, the magnitude of the second salvage value function may be relatively small due to the rapid obsolescence of productive capacity. In contrast, it is likely that the magnitude of the first salvage value is large reflecting the strategic benefits derived from the transfer of knowledge to the next planning horizon. The firm’s profit maximizing objective is defined in Equation (8.1). Equations (8.2)–(8.6) represent the first and second order conditions for the net revenue and cost functions. Note that the precise forms of the salvage value functions are not specified. However, since we assume the firm will manufacture similar products in subsequent planning horizons, the marginal value of an additional unit of capacity at the terminal time is nonnegative (see Equation (8.6)). In contrast, we make no assumptions concerning the marginal value of an additional unit of knowledge. This enables us to preserve the generality of the model and explore the consequences of a variety of functional forms reflecting the salvage value of knowledge. For example, a firm may perceive that the salvage value of knowledge is directly proportional to the amount of knowledge accumulated over the total planning horizon. Or, to position itself for future planning horizons, a firm may set a terminal time target level of knowledge, denoted by S T , and assign a penalty for deviation. Both approaches can be captured through the general salvage value function V 2 [K(T)].

冕

⭸c 1 /⭸s ⱖ 0,

⭸ 2 c 1 /⭸s 2 ⱖ 0,

(8.4)

⭸c 2 /⭸p ⱖ 0,

⭸ 2 c 2 /⭸p 2 ⱖ 0,

(8.5)

⭸V 2 /⭸K共T兲 ⱖ 0.

3. The Optimal Solution We solve the nonlinear dynamic model presented using optimal control methods. Pauwels (1977), Hartl and Sethi (1984), and Sethi (1974) discuss the mathematical conditions for optimal control problems with continuously distributed time lags. In this section, the optimal solution is presented under the assumption that the nonnegativity constraint on the level of effective capacity is not binding. This assumption is reasonable for three reasons. First, an upper bound on process change exists, thereby limiting the extent of instantaneous disruption to effective capacity. Second, from Equation (8.5), the cost of process change increases at an increasing rate at any instant of time (analogous to the production smoothing literature). Third, for K(t ) ⬍ 0 to occur, a firm would have to invest in process change, which in the short term generates losses (negative revenue). The Hamiltonian to be maximized at time t appears below. The adjoint variable, ␴ (t ), represents the marginal value of an additional unit of cumulative knowledge at time t. The variable ␭ (t ) represents the marginal value of an additional unit of effective capacity at time t. For simplicity, the notation depicting time is omitted in the remainder of the paper whenever possible. Finally, unless otherwise noted, mathematical proofs and nonnegativity and sufficiency conditions are omitted from the paper. These technical details are available upon request from the first author. H ⫽ ␲ 共K, S兲 ⫺ c 1 共s兲 ⫺ c 2 共p兲 ⫹ ␣ 1 ␴ s ⫹ ␣ 2 ␴ p ⫺ ␺␭ p

T

兵 ␲ 关K共t兲, S共t兲, t兴 ⫺ c 1 关s共t兲, t兴 ⫺ c 2 关p共t兲, t兴其dt

0

(8.6)

⫹ f共p共t兲, S共t兲兲

冕

T

w共 ␶ ⫺ t兲 ␭ 共 ␶ 兲d ␶ .

t

⫹ V 1 关S共T兲兴 ⫹ V 2 关K共T兲兴,

(8.1)

⭸ ␲ /⭸K ⱖ 0,

⭸ ␲ /⭸K ⱕ 0,

(8.2)

⭸ ␲ /⭸S ⱖ 0,

⭸ 2 ␲ /⭸S 2 ⱕ 0,

(8.3)

2

2


First, consider the optimal solution for the marginal value of an additional unit of effective capacity. The necessary conditions for optimality appear in Equation (9). The first equation indicates that the change in

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the marginal value of an additional unit of effective capacity at time t is directly related to marginal net revenue generated by capacity at that time. From Equation (8.6) and the second condition for optimality, we know that the terminal time value of ␭ (T) is nonnegative. In Theorem 1, the optimal solution for ␭ (t ) is characterized over the planning horizon.

␭˙ 共t兲 ⫽ ⫺⭸H/⭸K ⫽ ⫺␲ K with ␭ 共T兲 ⫽ ⭸V 2 /⭸K共T兲 ⱖ 0.

(9)

Theorem 1. The marginal value of an additional unit of effective capacity, ␭ (t ), is a nonnegative nonincreasing function of time. To simplify future discussion, we introduce the variable x共t兲 ⫽

冕

T

w共 ␶ ⫺ t兲 ␭ 共 ␶ 兲d ␶ .

t

To interpret this variable, recall that w( ␶ ⫺ t) denotes the distributed lag associated with the gain in effective capacity at time ␶ derived from a unit of process change at time t. The product w( ␶ ⫺ t) ␭ ( ␶ ) denotes the marginal value of the gain in effective capacity at time ␶ derived from a unit of process change at time t. Therefore, x(t ) represents the cumulative increase in net revenue, from time t to the end of the planning horizon, due to the lagged gain in effective capacity. In Theorem 2, it is shown that x(t ) is a nonnegative nonincreasing function of time. Intuitively, this means that greater profits are derived from a lagged gain in effective capacity that occurs earlier (rather than later) in the planning horizon. Also note that x(t ) equals zero at time t ⫽ T. Excluding the salvage value, additional net revenue cannot be derived from a gain in effective capacity at the end of the planning horizon. Theorem 2. Let x(t ) denote the cumulative increase in net revenue, from time t to the end of the planning horizon, derived from the lagged gain in effective capacity due to a unit of process change at time t. Then x(t ) is a nonnegative nonincreasing function of time, and equals zero at the terminal time. We now turn our attention to the interpretation of

272

the adjoint variable, ␴ (t ). Recall that ␴ (t ) represents the marginal value of an additional unit of cumulative knowledge at time t. The optimality conditions for ␴ (t ) appear in Equation (10). The expression ␲ S denotes the marginal increase in net revenue due to an additional unit of cumulative knowledge. Since ␲ S ⱖ 0 holds, we know that knowledge directly increases net revenue either by generating a premium price or by reducing the unit cost for output. For example, we may derive a reduction in unit cost as a result of the knowledge gleaned from process change. From Equation (10), we observe that the change in the marginal value of cumulative knowledge is a function of x(t ). Intuitively, knowledge impacts on the amount of gain in effective capacity derived from process change. Therefore, ␴ (t ) also reflects the marginal value of knowledge derived from process change (learning-bydoing) at time t over the remainder of the planning horizon. In the second optimality condition, ␴ (T) denotes the marginal salvage value of cumulative knowledge at the terminal time. Given our interpretation of Equation (10), in Theorem 3 we characterize the dynamic nature of ␴ (t ) over the entire planning horizon.

␴˙ 共t兲 ⫽ ⫺⭸H/⭸S ⫽ ⫺␲ S ⫺ 关⭸f共p, S兲/⭸S兴x共t兲 with ␴ 共T兲 ⫽ ⭸V 1 /⭸S共T兲.

(10)

Theorem 3. The marginal value of an additional unit of cumulative knowledge, ␴ (t ), is a nonincreasing function of time. 3.1. Optimal Preparation and Training Policy In this section, we explore the drivers of the optimal preparation and training policy. Let ⌽ 1 (t ) denote the derivative of the Hamiltonian with respect to s(t ) as stated in Equation (11.2). The optimal solution for s*(t ) is defined in Equation (11.1). To interpret the optimal policy, note that the first term in Equation (11.2) represents the marginal cost of an additional unit of preparation and training at time t. The second term reflects the marginal value of the increase in net revenue derived from an additional unit of preparation and training at time t. Therefore, the optimal preparation and training policy equates the marginal cost with the marginal benefit at each instant of time.



s*共t兲

such that ⌽ 1 共s, t兲 ⫽ 0 s*共t兲僆关0, s៮ 共t兲兴,

(11.1)

⌽ 1 共s, t兲 ⫽ ⭸H/⭸s ⫽ ⫺⭸c 1 /⭸s ⫹ ␣ 1 共t兲 ␴ 共t兲.

(11.2)

and where

In Corollary 1, it is shown that preparation and training optimally occurs at the beginning of the planning horizon, if at all. To prove this corollary, we assume that the marginal cost of an additional unit of preparation and training is a nondecreasing function of time. This assumption simply means that wages are nondecreasing for both production workers and engineering specialists involved in preparation and training activities. Corollary 1. Suppose the marginal cost of an additional unit of preparation and training is a nondecreasing function of time, and the marginal contribution of a unit of preparation and training to knowledge is a nonincreasing function of time. If investment in preparation and training optimally occurs at time t, then it is advocated throughout the planning horizon prior to that time. From Theorem 3 and Corollary 1, we are able to characterize the firm’s optimal strategy for preparation and training within three cases. In Theorem 4, the mathematical conditions for each case are given. Discussion of the important managerial implications of Theorem 4 follows. Theorem 4. Case 1. If ⭸c 1 (s៮ , T)/⭸s ⱕ ␣ 1 ␴ (T), then preparation and training optimally occurs at its maximum rate throughout the planning horizon. Mathematically, we have: s*(t ) ⫽ s៮ (t ), for t 僆 [0, T]. Case 2. If ⭸c 1 (s៮ , T)/⭸s ⬎ ␣ 1 ␴ (T) and ␣ 1 ␴ (0) ⬎ ⭸c 1 (s, 0)/⭸s, then preparation and training optimally occurs at the initial time and proceeds at a nonincreasing rate throughout the remainder of the planning horizon. Given t a 僆 [0, T), t b 僆 (0, T], and t a ⱕ t b , we have the following: s*共t兲 ⫽ s៮ 共t兲

for t 僆关0, t a 兲,

0 ⬍ s*共t兲 ⬍ s៮ 共t兲

for t 僆关t a , t b 兲,

s*共t兲 ⫽ 0

for t ⱖ t b .

Case 3. If ␣ 1 ␴ (0) ⱕ ⭸c 1 (s, 0)/⭸s, then preparation and


training does not occur throughout the planning horizon. This optimal policy is s*(t ) ⫽ 0 for t 僆 [0, T]. First, we interpret Case 1. Suppose the marginal value of a unit of preparation and training at T exceeds (or equals) the marginal cost incurred for the maximum rate of preparation and training at T. It follows that preparation and training optimally occurs at a maximum rate throughout the planning horizon. A Case 1 solution may arise if the salvage value of knowledge is very large. In other words, the firm anticipates substantial benefits from accumulating a large level of knowledge by the end of the planning horizon. The empirical findings of Roth and Marucheck (1994) support this case. They emphasize that knowledge gained during one planning horizon is an asset for competitive advantage during the next. This is particularly relevant for firms manufacturing products with short life cycles (small T), where substantial benefits may occur from the transfer of knowledge between successive planning horizons. Conversely, in Case 3, the initial time marginal cost of preparation and training exceeds (or equals) the corresponding marginal value. Therefore, preparation and training does not occur during the planning horizon. Several factors may lead a firm to pursue this strategy. For example, suppose the process change under consideration is a routine equipment replacement for which the firm already has considerable experience. With a substantial initial level of relevant knowledge, additional investments in preparation and training are unnecessary. The Case 3 strategy may also be advocated by a firm in response to an extremely high cost for preparation and training. Finally, a firm may pursue the Case 3 strategy if preparation and training leads to a relatively small contribution to knowledge. For instance, empirical findings by Pisano (1994) indicate that certain industries benefit less from learning-before-doing than others. Case 2 represents an intermediate solution relative to Cases 1 and 3. That is, investment in preparation and training is appropriate over some portion of the planning horizon, but not at a maximal rate over the entire planning horizon. Also, the optimal rate of investment in preparation and training decreases over the planning horizon until reaching a value of zero.

273


Note that when the Hamiltonian is linear in s, then t a ⫽ t b . In this situation, the firm optimally pursues the maximum rate of preparation and training from the beginning of the planning horizon through time t ⫽ t a . Then, no further investments in preparation and training are appropriate. Based on the three cases in Theorem 4, the relative magnitude of cumulative knowledge is known over the entire planning horizon. Specifically, the level of knowledge at any instant of time in Case 1 is at least as large as the corresponding level in Case 2, which is at least as large as the corresponding level in Case 3. This result, stated in Theorem 5, has important implications in later analysis of the optimal process change policy. Theorem 5. S(t ) CASE1 ⱖ S(t ) CASE2 ⱖ S(t ) CASE3 for t 僆 [0, T]. 3.2. Optimal Process Change Policy In this section, we introduce the optimal process change policy over the planning horizon. Let ⌽ 2 ( p, t) denote the derivative of the Hamiltonian with respect to p(t ) (Equation (12.2)). Given Equation (2), the optimal rate the firm pursues process change p*(t ) appears in Equation (12.1). The first term in Equation (12.2) denotes the marginal cost of process change at time t. The second term is the marginal loss in net revenue due to the disruption in effective capacity caused by process change implementation at time t. The sum of the first and second terms represents the total marginal cost incurred per unit process change pursued at time t. The third term in Equation (12.2) is the marginal value of the gain in cumulative knowledge derived from learning-by-doing, (i.e., per unit process change implemented at time t). To interpret the fourth term, consider each component of the product separately. ⭸f/⭸p is the increase in the long term gain in effective capacity from an additional unit of process change at time t. x(t ) is the cumulative increase in net revenue, from time t to the end of the planning horizon, derived from the lagged gain in effective capacity. Therefore, the last two terms in Equation (12.2) reflect the marginal benefit to the objective from an additional unit of process change as measured by the impact on both knowledge and effective capacity.

274

p*共t兲 and

such that ⌽ 2 共p, t兲 ⫽ 0 p*共t兲僆关0, p៮ 共t兲兴,

(12.1)

where ⌽ 2 共p, t兲 ⫽ ⭸H/⭸p ⫽ ⫺⭸c 2 /⭸p ⫺ ␺␭ 共t兲 ⫹ ␣ 2 ␴ 共t兲 ⫹ 关⭸f/⭸p兴x共t兲. (12.2) To fully explore the drivers of process change, we must delve more deeply into analysis of Equation (12). This is the focus of §4, where we present the results of analytic sensitivity analysis. The remainder of this section is devoted to analysis of how the level of knowledge is itself a driver of process change, and how the optimal level of process change evolves over time. The effect of the level of knowledge on the optimal process change policy is observed by examining Equation (12.2) in relation to the variable S(t ). The first three terms are functions of time, but are independent of S(t ). Therefore, the variable S(t ) impacts on p*(t ) solely through the function ⭸f/⭸p. In addition, we assume that as the level of knowledge increases, the marginal long term gain in effective capacity per unit process change increases. Specifically, the second derivative of f with respect to p and S is nonnegative (i.e., ⭸ 2 f/⭸p⭸S ⱖ 0.) This assumption is reasonable, as f [ p(t ), S(t ), t] is a nondecreasing function of p and S from Equation (4). To illustrate, suppose that f [ p(t ), S(t ), t] is a separable multiplicative function of p and S (i.e., f [ p(t ), S(t ), t] ⫽ f 1 [ p(t )] f 2 [S(t )]). Then it follows directly from Equation (4) that ⭸ 2 f/⭸p⭸S ⱖ 0. The insights pertaining to the effect of S(t ) on p*(t ) are quite important. Specifically, we can identify the relationship between the level of knowledge and both the rate and timing of process change. Theorems 6 – 8 contain these results. Before stating these theorems, we introduce the following notation. Let t piBegin denote the time process change begins in Case i, for i ⫽ 1, 2, and 3. Similarly, t piEnd denotes the time process change ends in Case i, for i ⫽ 1, 2, and 3. Theorem 6. (1) Process change optimally begins first Begin Begin Begin in Case 1 and last in Case 3, so that t p1 ⱕ t p2 ⱕ t p3 .



(2) Process change optimally concludes last in Case 1 and End End End ⱖ t p2 ⱖ t p3 . first in Case 3, so that t p1 Theorem 7. A firm with a higher level of cumulative knowledge at time t optimally pursues more process change at that time (process change is maximal in Case 1 and minimal in Case 3). Theorem 8. The amount of process change pursued over the planning horizon is greater for a firm with a higher level of cumulative knowledge (the total amount of process change is maximal in Case 1 and minimal in Case 3). The results in Theorems 6 – 8 demonstrate the crucial link between knowledge and a firm’s process change strategy. From Theorem 6, process change starts earlier and ends later due to a higher level of cumulative knowledge. From Theorem 7, the rate process change is optimally pursued at any time increases in relation to the firm’s level of cumulative knowledge. Furthermore, from Theorem 8, the total amount of process change pursued over the planning horizon is greater for firms with larger amounts of knowledge. Clearly, it is imperative for managers to recognize the key role of cumulative knowledge on the optimal process change policy. We conclude this section with analysis of the dynamic nature of process change. In Corollary 2, conditions are stated characterizing situations where a firm’s process change strategy increases versus decreases over time. It is important to recognize that these results are in direct contrast with those in Chand et al. (1996) and Fine (1986) where process change always decreases over time. The proof of Corollary 2 is omitted but follows from the fact that ⭸[⌽ 2 ( p*, t)]/⭸t ⫽ 0 holds over the interval of time where an interior solution exists for p*(t ) (i.e., ⌽ 2 ( p*, t) ⫽ 0 is satisfied during the time interval where p*(t ) 僆 (0, p៮ (t ))). The interpretation of Corollary 2 follows. Corollary 2. The optimal rate of process change at time t is an increasing (decreasing) function of time if the right side of Equation (12.3) is positive (negative). p˙ *关⭸ 2 c 2 /⭸p 2 ⫺ 共⭸ 2 f/⭸p 2 兲x共t兲兴 ⫽ ⫺⭸ 2 c 2 /⭸p⭸t ⫺ ␺␭˙ 共t兲 ⫹ ⭸关 ␣ 2 共t兲 ␴ 共t兲兴/⭸t ⫹ 关⭸f/⭸p兴x˙ 共t兲 ⫹ 关⭸ 2 f/⭸p⭸S兴S˙ 共t兲x共t兲.

(12.3)


From Equations (4) and (8.5), we know that the bracketed expression on the left side of Equation (12.3) is nonnegative. Therefore, if the right side of Equation (12.3) is positive (negative), we know p˙ *(t ) ⬎ 0 ( p˙ *(t ) ⬍ 0) so that p*(t ) increases (decreases). Next, consider the terms on the right side of Equation (12.3). Since ␺ ⱖ 0, ␭˙ (t ) ⱕ 0, ⭸ 2 f/⭸p⭸S ⱖ 0, S˙ (t ) ⱖ 0, and x(t ) ⱖ 0, we know the second and last terms on the right side of Equation (12.3) are nonnegative. Given ⭸f/⭸p ⱖ 0 and x˙ (t ) ⱕ 0, the fourth term on the right side is nonpositive. Analysis of the third term on the right side requires that we apply the product rule of calculus. Although we know ␴˙ (t ) ⱕ 0 and ␣ 2 (t ) ⱖ 0, the sign of this term is unclear. Corollary 2 provides important insights on factors that drive process change to occur at an increasing rate over time. For example, if technological improvement is anticipated such that the marginal cost of process change declines over time, then the first term on the right side of Equation (12.3) is positive (⭸ 2 c 2 /⭸p⭸t ⬍ 0). Therefore, in response to a declining marginal cost, investment in process change may be delayed or may optimally occur at an increasing rate over time. Next, suppose the second term in the right side of Equation (12.3) is relatively large. Here, either process change initially causes a substantial disruption to capacity, or the marginal contribution to net revenue from capacity is large. Since ␺ is fixed over time, a reduction in the value ⫺ ␭˙ (t ) is needed to lessen the impact of this term on Equation (12.3). Therefore, if ⫺ ␺␭˙ (t ) is large, the firm delays the pursuit of process change until a time later in the planning horizon when a smaller loss in net revenue occurs from production disruption. For example, if the firm experiences seasonal demand, then process change may delayed from a time when peak demand occurs (⫺ ␭˙ (t ) is large) to a later time when excess capacity exists (⫺ ␭˙ (t ) is small). Finally, consider the last term in Equation (12.3). Recall that ([⭸f/⭸p] x(t )) represents the cumulative increase in net revenue, from time t to the end of the planning horizon, due to an additional unit of process change at time t. Suppose the level of cumulative knowledge significantly enhances the long term gain in net revenue from process change (i.e., [⭸ 2 f/ ⭸p⭸S] x(t ) is positive and relatively large). Since S˙ (t )

275


ⱖ 0, we know the last term in the right side of Equation (12.3) is a substantial positive value. In other words, if knowledge significantly impacts the long term gain in net revenue derived from process change, a firm optimally delays its pursuit of process change until sufficient knowledge accumulation has occurred. Alternatively said, as additional knowledge is accumulated over time, the pursuit of process change is more desirable. To summarize, we have demonstrated that a firm may optimally delay its investment in process change until later in the planning horizon when greater benefits or lesser costs are available. These results are particularly notable since they extend the current literature that identifies only those situations where process change optimally decreases over time. Above, three factors are shown that drive a firm to pursue process change at an increasing rate over time: (i) the marginal cost of process change decreases over time, (ii) substantial production disruption occurs when process change is implemented, or (iii) knowledge substantially enhances the gain in effective capacity derived from process change. Therefore, under any of these circumstances, managers should carefully assess the rate of investment in process change.

4. Analytic Sensitivity Results In this section, analytic sensitivity results are presented providing managerial insights on a firm’s pursuit of preparation/training and process change. Corollaries 3 and 4 list those factors that directly affect the optimal rates of preparation/training and process change at a particular time. A discussion of the managerial implications of the key results given in these corollaries follows. First, however, it is important to recognize the different nature of the results previously given in Corollary 2 as compared with those below, in Corollaries 3 and 4. The focus of Corollary 2 is on factors that drive a firm’s process change strategy to occur at an increasing (decreasing) rate over time (i.e., the sign of p˙ *(t )). In contrast, the focus of Corollary 4 is to identify those factors that drive a higher level of process change to be undertaken at a particular time (i.e., the magnitude of p*(t )). For example, a greater investment in process change at a particular time

276

( p*(t ) increases), does not preclude the possibility that process change optimally decreases over time ( p˙ *(t ) ⬍ 0). Corollary 3. An increase in the optimal rate of preparation and training at time t is advocated in response to the conditions below. Also, the interval of time over which preparation and training occurs is extended, and the cumulative knowledge is greater throughout the planning horizon. (1) The marginal contribution of preparation/training to cumulative knowledge increases, ␣ 1. (2) The marginal contribution of process change to cumulative knowledge increases, ␣ 2. (3) The marginal loss in effective capacity at the start of process change decreases, ␺. (4) The long term marginal gain in effective capacity from cumulative knowledge increases, ⭸f/⭸S. (5) The long term marginal gain in effective capacity from process change increases, ⭸f/⭸p. (6) The length of time necessary to realize the full gain in effective capacity from process change decreases, (measured by w(t ⫺ ␶ )). (7) The marginal cost of preparation and training decreases, ⭸c 1 /⭸s. (8) The marginal cost of process change decreases, ⭸c 2 /⭸p. (9) The terminal time of the planning horizon is delayed, (T increases). (10) The marginal salvage value of cumulative knowledge increases, ⭸V 1 /⭸S(T). (11) The marginal salvage value of effective capacity increases, ⭸V 2 /⭸K(T). (12) The marginal revenue associated with cumulative knowledge increases, ␲ S . (13) The marginal revenue associated with effective capacity increases, ␲ K . Corollary 4. In response to conditions (1)–(12) in Corollary 3, an increase in the optimal rate of process change is advocated at time t. Also, the pursuit of process change starts earlier in the planning horizon and lasts longer. (Note that to interpret conditions (1)–(3) Corollary 4, we assume the optimal rate of process change is positive at some time during the planning horizon.) Condition (9) in Corollaries 3 and 4 provides a key



result linking the length of the product life cycle and the strategic benefits of process change for future planning horizons. Specifically, we show that process change is less desirable in environments with short product life cycles. The implications of this result are twofold. First, managers should pursue process change projects that offer effective capacity gains in a timely manner. This result concurs with the empirical evidence of Pisano and Wheelwright (1995) where the authors argue that short product life cycles necessitate investment in process change before a product goes to market, and aggressive pursuit of process changes at the beginning of the planning horizon. Second, managers should carefully analyze the salvage value functions associated with both cumulative knowledge and effective capacity. Indeed, the experience derived from manufacturing process change in the current planning horizon may provide competitive advantage in future time horizons. In addition, from conditions (10) and (11) of Corollaries 3 and 4, we know that an increase in the marginal salvage values of either cumulative knowledge or effective capacity leads to an increase in both the rates and duration of preparation/training and process change. Therefore, in environments with short product life cycles, it is particularly important for managers to carefully assess the future strategic benefits derived in relation to the terminal time levels of effective capacity and knowledge. Another key result of the analysis highlights the complementary nature of the relationship between the optimal preparation/training and process change efforts. From condition (8) in Corollary 3, a decrease in the marginal cost of process change leads to an increase in the optimal rate of preparation and training. Likewise, from condition (7) in Corollary 4, a decrease in the marginal cost of preparation and training leads to an increase in the optimal rate of process change. Similarly, an increase in the marginal gain in effective capacity associated with either effective capacity or cumulative knowledge leads to an increase in both the rates of preparation and training and process change (see conditions (4) and (5), Corollaries 3 and 4). Note that these results concur with the empirical literature pointing to the crucial role of advanced preparation


and training on the ultimate success of process changes (see Adler and Clark 1991, and Galbraith 1990). In conclusion, the synergetic nature of the preparation/training and process change efforts is clear. Managers should carefully assess the relationship between preparation/training activities and process change to fully realize the potential gain in effective capacity and profit. Condition (13) of Corollary 3 states the effect of an increase in the marginal net revenue of effective capacity on the optimal rate of preparation and training. However, discerning the impact of ␲ K on the firm’s optimal process change strategy is not straightforward. An increase in ␲ K increases both the marginal value of capacity, ␭ (t ), and the marginal value of cumulative knowledge, ␴ (t ). Therefore, the last three terms in Equation (12.2) increase. Ultimately, as ␲ K increases, the effect on p*(t ) reflects the following trade off: the increase in the immediate loss in net revenue when process change is initiated (second term) versus the increase in the immediate benefit from knowledge creation via learning-by-doing (third term) plus the increase in the long term benefit from process change (fourth term). This trade off is characterized in Theorem 9. Specifically, we analyze the impact in changes in the marginal net revenue over time (i.e., ␲ Kt ) on the optimal process change policy. The proof of Theorem 9 is shown in Appendix 2. Theorem 9. Consider the situation where p*(t ) ⬎ 0 holds over some nonzero interval of time, and the marginal net revenue from effective capacity decreases over time (i.e., ␲ Kt ⬍ 0). Corresponding to a faster rate of decline in the marginal net revenue from effective capacity at time t (as ␲ Kt becomes more negative), we have the following. Case 1. If ␺ is zero or relatively small, then process change is optimally pursued to a lesser extent at time t( p*(t ) decreases). In addition, the interval of time where process change optimally occurs over the planning horizon is shorter. Case 2. If ␺ is positive and relatively large, then process change is optimally pursued to a greater extent at time t( p*(t ) increases). In addition, the interval of time where process change optimally occurs over the planning horizon is extended.

277


The results of Theorem 9 have implications for firms operating in highly competitive markets where the marginal net revenue from effective capacity is a declining function of time ( ␲ Kt ⬍ 0). Suppose the initial loss in effective capacity, ␺, is moderate or zero (Case 1). As the rate of decline in the marginal net revenue from effective capacity increases, a firm optimally reduces its pursuit of process change. In addition, from Corollary 3, in response to a faster rate of decline in the marginal net revenue from effective capacity, the firm optimally undertakes less preparation and training over time. Therefore, the firm has a lower level of effective capacity, a lower level of knowledge, and earns less net revenue over the entire planning horizon. These analytic results demonstrate the importance of having a sufficient level of effective capacity at the start of the planning horizon if a firm’s net revenue declines over time. Next, consider Case 2 of Theorem 9. For this case to occur, the firm cannot recoup sufficient benefits over the remainder of the planning horizon to compensate for the immediate loss in effective capacity when process change is initiated. For example, this situation may arise if the production disruption costs associated with process change are very high, or if the planning horizon (product life cycle) is very short. However, if the rate of decline over time in net revenue is sufficient (i.e., if ␲ Kt ⬍ 0 is large in magnitude), the disruption cost from process change (the loss in revenue) is diminished. Therefore, as the rate of decline in the marginal net revenue of effective capacity increases, a firm optimally increases its pursuit of process change. Note, however, that Case 2 is unlikely to occur. Specifically, if a firm’s disruption cost due to process change is large (i.e., ␺ is large), then the value of Equation (12.2) is negative and process change does not optimally occur (i.e., p*(t ) ⫽ 0 for t 僆 [0, T]). In the next section, the effect of the dynamic revenue is explored further in the context of numerical examples. To conclude our presentation of analytic results, suppose the benefits from process change in relation to knowledge accumulation are subject to depreciation. The phenomena of work force forgetting or knowledge depreciation has been observed empirically in both industrial organizations (Argote et al.

278

1990, and Bailey 1989) and service organizations (Ingram and Baum 1997, and Darr et al. 1995). In particular, Argote et al. (1990) find that knowledge acquired through production depreciates rapidly. Bailey (1989) reports that the rate of depreciation is a function of the amount learned and the passage of time. For our model, it is of interest to examine the impact of forgetting on the firm’s pursuit of both process change and preparation and training over time. As defined here, the level of knowledge, S(t ), measures the accumulation of learning-before-doing and learning-by-doing activities, and therefore remains a nondecreasing function of time. However, due to forgetting, the ability of knowledge at time t to increase effective capacity at time ␶ ⬎ t deteriorates as ␶ increases. In particular, in environments where knowledge depreciation occurs, the gain in effective capacity from process change is diminished. Recall that the product f( p(t ), S(t ))w( ␶ ⫺ t) measures the lagged gain in effective capacity at time ␶ due to knowledge and process change initiated at time t ⱕ ␶ . Therefore, mathematically, knowledge depreciation is easily captured through the function w( ␶ ⫺ t), as described below. Let w d ( ␶ ⫺ t) and w( ␶ ⫺ t) denote the lagged effects of process change if knowledge depreciation does and does not occur, respectively. Clearly, w d (0) ⫽ w(0) for ␶ ⫽ t; 0 ⱕ w d ( ␶ ⫺ t) ⱕ w( ␶ ⫺ t) ⱕ 1 for ␶ ⬎ t; and the value of w( ␶ ⫺ t) ⫺ w d ( ␶ ⫺ t) increases as ␶ increases. In addition, as the rate of forgetting increases, the difference between w( ␶ ⫺ t) and w d ( ␶ ⫺ t) increases, for ␶ ⬎ t. Lastly, as a consequence of forgetting, the full potential gain in effective capacity from process change and knowledge does not occur so the value of integral expression in Equation (5) is less than 1. With this interpretation of the continuous lag and from condition (6) in Corollaries 3 and 4, we have the following. Theorem 10. A firm whose knowledge enhancing benefits to process change (i.e., the gain in effective capacity) are subject to depreciation, optimally pursues less preparation/ training and process change throughout the planning horizon. In addition, the intervals of time over which preparation/training and process change optimally occur are shorter. Lastly, as the rate of knowledge depreciation in-



creases, less knowledge acquisition and process change are undertaken throughout the planning horizon. The results of Theorem 10 are intuitively appealing. Specifically, if the benefits from a given level of knowledge deteriorate over time (forgetting), the long term value of investing in knowledge creation is reduced. Therefore, the desirability of increasing knowledge through preparation and training (learning-before doing) and process change (learning-bydoing) is reduced. Since knowledge enhances the firm’s ability to increase capacity from process change, the smaller level of knowledge further reduces the long term benefits from process change. From Theorem 10, it is clear that a firm capable of sustaining knowledge enhancing benefits over time has a competitive advantage when undertaking process change to increase effective capacity. Furthermore, we have demonstrated the importance of adopting various mechanisms to preserve knowledge obtained from process change. For example, in his article on learning organizations, Garvin (1993) suggests that a firm can effectively capture and transfer knowledge through a variety of methods such as reports, site visits, personnel rotation, and standardization programs. In addition, when selecting the best process change to pursue from a set of alternatives, a manager should consider the associated rates of knowledge depreciation. For example, Darr et al. (1995) suggest that the technological sophistication of the production process may be a key factor affecting the rate of knowledge depreciation.

5. Numerical Analysis Analytic results have been presented offering insights concerning the effect of various factors on the optimal process change and knowledge acquisition strategies. However, the impact of these factors on firm performance measures such as profit cannot be derived analytically. This section contains results derived from extensive numerical analysis. Solutions are obtained using an ordinary shooting method for a discrete approximation of the continuous model (see Sethi and Thompson 1981). (Details of the numerical analysis are available upon request to the first author.) To proceed with the numerical investigation, spe-


cific functions and parameter settings must be given which reflect careful consideration. In part, the examples are motivated by the empirical and case oriented literature. For example, Chew et al. (1991) discuss various costs incurred in manufacturing plants due to production disruption when new technology is introduced. The authors describe actual hardware and software installations in the furniture, aluminum processing, and electronics industries. Our numerical examples are also motivated by extensive discussions with a high level manufacturing manager in the micro-electronics industry (Kempf 1994). In these discussions, we were provided with first hand insights concerning the effects of various hardware, software, and procedural types of process changes undertaken in dynamic manufacturing environments. Lastly, motivated by the aggregate planning literature, the costs of both process change and preparation/training are expressed as quadratic functions of p and s respectively (see Hax and Candea 1984). Highlights of nine numerical examples are presented. The functional forms used in the numerical examples appear in Equations (13)–(16) of Appendix 1. Also, the parameter settings used in the first (Base) Example are given in Appendix 1. From Table 2, we see how the input functions and parameters settings are varied in Examples 2–9 relative to the Base Example. Specifically, the first row of Table 2 indicates the functional form (by reference to an equation in Appendix 1) and parameter settings of the Base Example. Blank cells in the rows corresponding to Examples 2–9 indicate the function and parameter settings are identical to the base case. The diagonal entries in Table 2 show how the input functions or parameter settings are modified in Examples 2–9 relative to the Base Example. Table 3 contains a summary of the numerical results. While the results reported correspond to particular input parameter settings, it is important to recognize that consistent results were obtained over a relatively wide range of input parameter settings. In particular, sensitivity analysis was performed for the input parameters over the range of values given in the square brackets in Table 2. Thus, our numerical results are robust over a reasonably wide range of parameter settings.

279


Table 2 Experiment Base Case

2 3

Numerical Experiment Design Revenue Function

Learning Variables

Unit Cost Function

Gain Function

␣1 ⫽ 1 ␣ 2 ⫽ 0.001

Independent From S: ␦ ⫽ 0 (13.5)

Decreasing Returns to S: ␥ 2 ⫽ 0.5 (15.1)

Decreasing Over Time (13.3)

Salvage Function Linear in S: V 2 ⫽ 100 (14.2)

Gamma With: ␤ 1 ⫽ 1; ␤ 2 ⫽ 2 (16.1)

Constant Over Time Avg ⫽ 17 (13.2) Seasonal in Time (13.4)

␣1 ⫽ 0 ␣ 1 僆 [0, 2] ␣2 ⫽ 0 ␣2 僆 [0, 0.002]

4 5 6

Decreasing in S: ␦ ⫽ .01 ␦ 僆 [0, 0.02]

7

No Effect from S: ␥2 ⫽ 0 ␥ 2 僆 [0, 1]

8

No Salvage in S: V2 ⫽ 0 V 2 僆 [0, 200]

␤ 1 ⫽ 7; ␤ 2 ⫽ 1 ␤ 1 僆 [0, 8] ␤ 2 僆 [1, 8]

9

Table 3

Lag Variables

Numerical Experiment Results

Experiment

Change in Cumulative Knowledge

Change in Effective Capacity

Total Cost Preparation and Training

Total Cost Process Change

Cumulative Knowledge Salvage Value

Capacity Salvage Value

Total Net Revenue

Profit

Base Case 2 3 4 5 6 7 8 9

S(T ) ⫺ S(0) 2.79 8.54 8.06 0.05 2.06 3.75 0.64 1.89 0.60

K(T ) ⫺ K(0) 47 332 307 3 30 70 2 33 0

¥c 1 (s) 878 9,311 8,931 0 583 1,575 30 511 30

¥c 2 (p) 1,250 16,408 15,526 39 689 2,038 28 848 0

SV 1 [S(T )] 379 954 966 105 306 475 164 0 160

SV 2 [K(T )] 4,404 12,957 12,207 3,080 3,893 5,102 3,064 3,994 3,000

¥␲ 17,868 38,602 37,846 16,793 17,395 19,118 16,790 17,515 16,789

20,523 26,794 26,562 19,940 20,324 21,081 19,961 20,149 19,919

5.1. Analysis of the Base Example We assume that the firm under consideration is operating in a highly competitive market such that revenues earned from sales are declining over time.

280

In particular, the net revenue function used for the Base Example is given by Equation (13.3). Later, in §5.2, we analyze the impact of alternate net revenue functions.



Figure 1

Optimal Rate of Preparation and Training

Figure 2

Cumulative Knowledge

The optimal rate of preparation and training derived in the Base Example is positive and decreasing throughout the planning horizon (in concurrence with Corollary 1), as illustrated in Figure 1. As a consequence, the level of cumulative knowledge increases at a decreasing rate throughout the planning horizon, as shown in Figure 2. The Base Example optimal process change solution is illustrated in Figure 3. Note that the dynamic process change strategy obtained reflects the analytic insights as given in Corollary 2. Specifically, we have: (i) p*(0) ⬎ 0, (ii) p*(t ) increases until reaching a peak at t ⫽ 5, (iii) p*(t ) decreases until t ⫽ 21, and (iv) p*(t ) ⫽ 0 over the remainder of the planning horizon. From Figure 4, we observe that the level of effective capacity: (i) initially decreases due to process change implementation, (ii) then in-


creases reflecting the lagged gain from process change, and (iii) ultimately reaches a terminal time value of 147 units. Next, consider the managerial implications of the Base Example. First, the firm optimally pursues preparation and training throughout the planning horizon (see Theorem 4). Here, the terminal time strategic value of knowledge is sufficiently large such that managers optimally invest in preparation and training even at the end of the planning horizon. Second, the firm’s investment in preparation and training directly impacts its process change strategy, as described below. At the initial time, both the level of cumulative knowledge and the rate of process change are small. Over time, as the magnitude of cumulative knowledge increases, the rate of process change increases, as well. Figure 3

Optimal Rate of Process Change

Figure 4

Effective Capacity

281


Therefore, the firm optimally delays substantial investment in process change until a sufficient level of knowledge has been accumulated, thereby ensuring a greater long term gain in effective capacity (see Corollary 2). After reaching its peak, the rate of process change declines until reaching a value of zero. However, even during the period of declining (and zero) process change, the level of effective capacity increases as a result of the lagged benefits from prior process changes as well as the additional accumulation of knowledge. In conclusion, the Base Example illustrates key elements of the relationship between knowledge and process change. 5.2. Impact of Dynamic Demand Various dynamic demand patterns may be encountered by the firm as depicted in Equations (13.2)– (13.4). Example 1 corresponds to a firm operating in a competitive environment where net revenue rapidly declines over time (Equation 13.3). In Example 2, the firm earns average or constant net revenue throughout the planning horizon (Equation 13.2). In Example 3, the firm’s revenue stream is initially low, peaks in the middle of the planning horizon, then decreases over the remainder of the planning horizon (seasonality) (Equation 13.4). To facilitate comparison, the parameter settings for Examples 2 and 3 are defined such that the average net revenue over the planning horizon equals that of the Base Example. Dramatic differences are obtained for the optimal process change strategies in response to variations in the dynamic revenue function (see Figure 5). As compared with the situation where net revenue declines over time, the firm operating under stable (constant) net revenue: (i) undertakes far more preparation and training, (ii) pursues far more process change, (iii) realizes a substantially greater accumulation of knowledge, (iv) achieves a substantially greater level of effective capacity, and (v) earns considerably more profit, as measured over the entire planning horizon. Also, as compared with the situation where net revenue declines over time, a firm experiencing demand seasonality: (i) undertakes far more preparation and training, (ii) pursues far more process change, (iii) realizes a substantially greater accumulation of knowledge, (iv) achieves a substantially greater level

282

Figure 5

Comparison of Optimal Process Change Policies

of effective capacity, and (v) earns considerably more profit, as measured over the entire planning horizon. In addition, in response to seasonality, the firm optimally shifts the majority of its process change efforts earlier in the planning horizon (Figure 5). Intuitively, the firm’s behavior reflects its desire to take advantage of peak revenue available in the middle of the planning horizon. The managerial implications of these results cannot be overlooked. In particular, managers operating in fiercely competitive environments in which revenues are declining (Base Example) should pursue less aggressive process change strategies once production begins. Instead, it is imperative that managers in these environments ready the manufacturing facility with the necessary capacity at the start of the planning horizon. These results concur with the intuition developed by Pisano and Wheelwright (1995), as well as insights from a high tech manufacturing manager (Kempf 1994). Furthermore, these results demonstrate that profit suffers if managers operating in environments with declining revenue streams ignore the dynamics of the marketplace by simply assuming an average net revenue function. Conversely, if the competitive environment leads to either stable or seasonal demand, then a more aggressive process change strategy is appropriate. Lastly, consider the relationship between these numerical examples and our analytic results. In §4, analytic results were given contrasting the optimal process change strategy for different representations



of the marginal net revenue from effective capacity. Specifically, we compared the solutions corresponding to a constant versus declining marginal net revenue, where both functions were equal at the initial time. In contrast, in the numerical experiment, the marginal net revenue at the initial time for the firm in the Base Example (declining marginal net revenue) is higher than that in the Example 2 (constant marginal net revenue). Yet the average net revenue over the planning horizon is the same for both Example 2 and Base Example. Despite this difference, the analytic and numerical results are the same. The firm anticipating a declining marginal net revenue from effective capacity undertakes relatively less process change over a smaller portion of the planning horizon than the firm operating in a stable (constant marginal net revenue) environment. Therefore, our numerical results extend those obtained analytically regarding the effect of declining marginal net revenue. 5.3. Impact of Knowledge and Learning In this section, we explore the impact on performance due to both learning and the accumulation of knowledge. First, we examine the benefits obtained from learning-before-doing (preparation and training) versus learning-by-doing (process change). Empirical findings by Pisano (1994) indicate that certain industries benefit less from learning-before-doing than others. To assess the impact of this observation, we compare the Base Example and Example 4. In the Base Example, both process change and preparation and training contribute to knowledge. In Example 4, the firm benefits solely from learning-by-doing (preparation and training has no impact on increasing the level of knowledge so that ␣ 1 ⫽ 0). However, when ␣ 1 was varied over the range [0, 2], consistent results were obtained relative to the Base Example (␣ 1 ⫽ 1). The numerical results confirm condition (1) in Corollaries 3 and 4. Specifically, when learning-before-doing does not contribute to the accumulation of knowledge, the firm: (i) greatly reduces its pursuit of preparation and training, (ii) greatly reduces its pursuit of process change, (iii) accumulates far less knowledge, (iv) accumulates far less effective capacity, and (v) earns less profit. To assess the importance of learning-by-doing, we


perform the following comparison. In Example 5, the firm benefits solely from learning-before-doing (process change has no impact on increasing the level of knowledge so that ␣ 2 ⫽ 0). Relative to the Base Example (␣ 2 ⫽ 0.001), the numerical results obtained for Example 5 are analogous to those of Example 4. Moreover, consistent results were obtained relative to the Base Example when ␣ 2 was varied over the range [0, 0.002]. Therefore, we have shown that both sources of learning significantly impact the extent of process change advocated, the amount of gain in effective capacity, and profit. Second, we explore the direct impact of knowledge on net revenue. Suppose knowledge associated with certain types of process change either enhances revenue or decreases unit cost (see Fine 1986). In the Base Example, knowledge has no direct effect on net revenue (␦ ⫽ 0 in Equation (13.5)). In contrast, net revenue in Example 6 is a nondecreasing function of cumulative knowledge (␦ ⫽ 0.01). The comparative results for Examples 1 and 6 concur with condition (12) in Corollaries 3 and 4. The insights derived from Example 6 are robust since varying ␦ over the range [0, 0.02] led to similar solutions. The results indicate that a firm capable of using knowledge to directly enhance net revenue optimally: (i) pursues far more preparation and training, (ii) undertakes considerably more process change, (iii) accumulates more knowledge, (iv) accumulates more effective capacity, and (v) earns higher profit. These results illustrate the competitive advantage realized by a firm capable of harnessing knowledge to directly increase net revenue. Third, we continue our analysis of knowledge by exploring its relationship with the gain in effective capacity from process change. In Equation (15.1), the long term gain in effective capacity from process change may exhibit increasing, constant, or decreasing returns to knowledge as measured by ␥ 2. In the Base Example, the capacity gain function assumes decreasing returns to S ( ␥ 2 ⫽ 0.5). In contrast, in Example 7, the gain function is independent of the amount of knowledge accumulated (␥ 2 ⫽ 0). As compared with the Base Example, if knowledge has no effect on the gain in effective capacity from process change, a firm optimally: (i) significantly decreases its pursuit of

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preparation and training, (ii) decreases the extent of process change pursued, (iii) accumulates far less knowledge, (iv) accumulates far less effective capacity, and (v) earns less profit. Note that the above insights are robust since consistent results were obtained by varying ␥ 2 over the range [0, 1]. We conclude this section by examining the impact of the strategic value of knowledge in future planning horizons. In the Base Example, we employ a standard approach in the literature: the terminal time salvage value is proportional to the level of cumulative knowl៮ 2 ⫽ 100). In Example edge at T (Equation (14.2) with V 8, we assume knowledge obtained during the current ៮ 2 ⫽ 0). The planning horizon has no salvage value (V numerical results confirm condition (10) in Corollaries 3 and 4. If cumulative knowledge has no salvage value, the firm optimally: (i) pursues less preparation and training, (ii) undertakes less process change, (iii) accumulates less knowledge, (iv) accumulates less effective capacity, and (v) earns less profit. Again, these insights are robust since consistent results were ៮ 2 was varied over the range [0, 200]. obtained when V In conclusion, we have numerically shown that managers should optimally pursue more process change and preparation and training if: (i) the creation of knowledge is enhanced by learning-before-doing, (ii) the creation of knowledge is enhanced through learning-by-doing, (iii) knowledge contributes to net revenue either through premium price or reduced unit cost, (iv) knowledge enhances the gain in effective capacity from process change, or (v) knowledge is strategically valuable to compete in future planning horizons. Our numerical insights are consistent with and extend the analytical results given earlier in the paper. Interpretation of the results offers direction for managers selecting a particular type of process change from several available. For example, suppose the benefits from learning-by-doing differ among various process change options. A firm capable of deriving greater value from knowledge may optimally select that process change activity associated with the maximum rate of learning-by-doing. Finally, note that while our numerical results assume net revenue decreases over time, the same qualitative results were

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obtained for environments with constant or peaking revenue functions. 5.4. Impact of the Lag Distribution In this section, we explore the impact of a firm’s characterization of the time lag associated with the gain in effective capacity due to process change (Equation (16)). Recall that the function w(t ⫺ ␶ ) defines the portion of the long term gain in effective capacity realized at time t due to a unit of process change implemented at time ␶, for ␶ ⱕ t. In our numerical examples, the lag satisfies the Gamma distribution. Mathematically, the distribution appears in Equation (16.1), its mean and variance appear in Equations (16.2) and (16.3), respectively. Pauwels (1977) and Gaimon (1997) have illustrated the flexibility of employing this type of dynamic lag in optimal control models. Note that different types of lag distributions are suitable for different process changes. For example, the Gamma distribution with ␤ 1 ⬎ 0 is appropriate if the gain in effective capacity from process change is relatively small initially, increases over time until reaching a peak, and then gradually tapers off to zero. This form of lag may reflect a situation where process change is complex and requires extensive adjustments in the manufacturing system to ensure compatibility. In contrast, if the lag is given by the exponential function (␤ 1 ⫽ 0), the gain in effective capacity from process change is immediate and substantial but rapidly diminishes over time. This form of lag may reflect a relatively simple and narrowly focused process change. In Example 9, the mean time that transpires until a unit of process change is effective at increasing capacity is twice the mean assumed in the Base Example; the variance is the same. As a consequence, no process change is advocated in Example 9. In addition, consistent with our analytic results, given its synergy with the process change strategy, the optimal rate of preparation and training is minimal throughout the planning horizon. Specifically, if a greater amount of time is required to obtain an increase in effective capacity from process change, the firm optimally: (i) undertakes far less preparation and training, (ii) reduces the extent of process change pursued, (iii) accumulates far



less knowledge, (iv) accumulates far less effective capacity, and (v) earns less profit, as compared to the Base Example. Furthermore, similar results were obtained by varying the two lag parameters ␤ 1 and ␤ 2 within the ranges given in Table 2. These results demonstrate the importance of carefully assessing the nature of the lag between the time process change begins and the time it becomes effective at increasing capacity.

6. Conclusions A profit maximizing model is introduced to aid a firm planning its process change strategy. The model captures both the short term loss and the long term gain in effective capacity derived from process change. The two key decision variables are the rate of process change and the rate of preparation/training derived over the finite planning horizon. Whereas investment in preparation and training creates knowledge actively (learning-before-doing), process change creates knowledge passively as a by-product of the experience gained from implementation (learning-by-doing). The model incorporates the possibility of various benefits derived from knowledge including a greater long term gain in effective capacity from process change, and a greater value of net revenue earned from output. Overall, the firm’s profit maximizing objective embodies the following: (i) the net revenue earned from effective capacity, (ii) the costs of process change and preparation/training, and (iii) the strategic value of the terminal time levels effective capacity and knowledge (salvage functions) for future planning horizons. 6.1. Importance of Knowledge A significant contribution of the model is its broad portrayal of the impact of knowledge on the effectiveness of a firm’s process change strategy. Analytical results show that managers should optimally pursue more process change and preparation/training if any of the following holds: (i) the firm is capable of creating knowledge through learning-before-doing; (ii) the firm is capable of creating knowledge through learning-by-doing; (iii) the firm can harness knowledge such that it contributes directly to net revenue through premium price or reduced unit cost; (iv) the


firm is able to employ knowledge to enhance the gain in effective capacity from process change; (v) the firm is able to preserve knowledge obtained (less knowledge depreciation); or (vi) knowledge is strategically valuable to compete in future planning horizons. Furthermore, the numerical results demonstrate that process change profitability is enhanced when any of the aforementioned conditions hold. These results support the importance of understanding how knowledge is accumulated over time as well as understanding the mechanisms by which knowledge can be applied in practice for competitive advantage. (See Dorroh et al. 1994, and Bohn 1994.) A key analytic result derived is the characterization of the complementary relationship between preparation/training activities and the process change policy. Optimal solutions are derived demonstrating that process change begins earlier and lasts longer as a direct consequence of increasing the level of cumulative knowledge through preparation and training. Therefore, preparation and training enhances the value of process change and leads to a greater investment in process change over the planning horizon. Finally, the results of analytic sensitivity analysis further highlight the synergistic relationship between preparation/training and process change. We show that corresponding to a decrease in the marginal cost of process change, the rate of preparation/training increases. Similarly, corresponding to a decrease in the marginal cost of preparation/training, the rate of process change increases. Analytic results also indicate that an increase in the marginal contribution to knowledge from process change (learning-by-doing), leads to an increase in the extent of investment in preparation/training. Similarly, an increase in the marginal contribution to knowledge from preparation/training (learning-before-doing) leads to an increase in the optimal process change strategy. To summarize, managers should increase their investment in both preparation/training and process change if either becomes more effective or less expensive. 6.2. Dynamics of Process Change A key result of this research is its characterization of factors impacting the rates of process change and preparation and training over time. First, we find that

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preparation and training activities (learning-beforedoing) optimally occur at a nonincreasing rate over the planning horizon. Second, in contrast to previous research indicating that process change optimally decreases over time, we identify scenarios where the pursuit of process change occurs at an increasing rate over the planning horizon. Essentially, the firm optimally delays investment in process change until later in the planning horizon when greater benefits or lesser costs are available. For example, a declining marginal cost of process change over time may drive the firm to pursue process change at an increasing rate. Similarly, if knowledge significantly enhances the long term gain in net revenue from process change, the pursuit of process change may occur at an increasing rate. Note that many of the above dynamics might occur as a result of technological innovation as well as managerial efforts aimed at improving the firm’s accumulation of knowledge. 6.3. Impact of Competitive Factors Various external forces can impede a firm’s realization of enhanced performance through process change. For example, in highly competitive markets, product life cycles may be short (the terminal time T is small). In addition, intense competition often results in net revenue streams that rapidly decline over a product’s life cycle. This situation may reflect the firm’s need to lower prices as new competitors enter the marketplace over time. We derive analytic results showing that in response to either of these competitive conditions, a firm optimally reduces the rates of process change and preparation/training throughout the planning horizon. In other words, under these competitive forces, a firm should shift its emphasis from activities undertaken during the product’s life cycle to activities that ensure the correct specification of the manufacturing process at the initial time (see Pisano and Wheelwright 1995, and Kempf 1994). Also, under short product life cycles, the salvage function associated with knowledge is particularly important since it connotes the value of experience that can be transferred to manufacturing in future planning horizons. Another exogenous force impacting on a firm’s realization of enhanced performance from process change is the dynamic nature of demand. Analytic

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and numerical results are developed illustrating the impact of various demand patterns on the optimal solution. We have already discussed the fact that if net revenues steadily decline over time, the firm optimally invests in less process change and less preparation/ training throughout the planning horizon. Next, we compare environments in which net revenue is static (or is approximated by a static average value) versus seasonal. In particular, seasonality is reflected by net revenues that initially increase, reach a peak, and then decrease over time (for the purpose of comparison, the average net revenue in the case of seasonal demand is set equal to the static value). Based on extensive numerical analysis, we show that the firm experiencing seasonality implements more process change earlier in the planning horizon so that a greater level of effective capacity is available during the period of peak revenue. The managerial implications of these results are twofold. First, it is crucial that managers understand the dynamic nature of net revenues to realize success through process change. For example, numerical results demonstrate that total profit suffers if managers assume a constant average net revenue instead of the actual dynamic demand pattern. Second, in highly competitive markets with short product life cycles or declining net revenues, the firm must carefully assess the marginal salvage value associated with knowledge. The heightened importance placed on the level of knowledge at the terminal time reflects the value of learning derived from process change in the current planning period to future manufacturing endeavors. 6.4. Process Change Selection Analysis of the model provides insights to aid managers responsible for selecting an appropriate process change alternative. For example, the model can be re-solved with different input parameters characterizing various process change options including equipment replacement, software upgrade, and procedural change. The value of attributes (input parameters) that differ for each alternative include: the cost of process change, the long term gain in effective capacity, the impact of knowledge on the long term gain, the extent of knowledge increase from process change (learningby-doing), the impact of knowledge on increasing net



revenue, the amount of loss in effective capacity when process change begins, and the nature of the lag depicting the continuous gain in effective capacity over time from process change. The results of analytic sensitivity analysis (Corollaries 3 and 4) offer guidance for project selection by describing the impact of each of the above attributes on the optimal pursuit of both preparation/training and process change, respectively. Furthermore, numerical results are given comparing alternative process change attributes and highlighting the impact of project specific factors on firm profitability. 1 1

We gratefully acknowledge the comments of two anonymous referees and the area and department editor. This research was funded in part by a fellowship from the Intel Foundation.

Appendix 1 The functional forms employed in the numerical experiments are as follows:

␲ 共K共t兲, S共t兲, t兲 ⫽ 关r共t兲 ⫺ c u 共S共t兲, t兲兴K共t兲, r共t兲 ⫽ r៮

(13.1)

where r៮ ⱖ 0,

r共t兲 ⫽ r˜ e ⫺ ␳ t

(13.2)

where r˜ ⱖ 0,

r共t兲 ⫽ 关r 1 共tˆ 2 ⫺ 共t ⫺ tˆ 兲 2 ⫹ r 2 兴 ⱖ 0

(13.3)

for t 僆关0, T兴,

(13.4)

c u 共S共t兲, t兲 ⫽ c a ⫹ c b e ⫺ ␦ S ,

(13.5)

V 1 关K共T兲兴 ⫽ V 1 K共T兲,

(14.1)

V 2 关S共T兲兴 ⫽ V 2 S共T兲,

(14.2)

f关p共 ␶ 兲, S共 ␶ 兲兴 ⫽ ␥ 1 p共 ␶ 兲S共 ␶ 兲 ␥ 2 ,

(15.1)

w共z兲 ⫽ 关1/共 ␤ 1 ! ␤ 2␤ 1⫺1 兲兴z ␤ 1e 共⫺z/ ␤ 2兲 ,

(16.1)

mean ⫽ 共 ␤ 1 ⫹ 1兲 ␤ 2 ,

(16.2)

variance ⫽ 共 ␤ 1 ⫹ 1兲 ␤ 22 .

(16.3)

The numerical input parameters for the Base Example are as follows: T ⫽ 24 r 1 ⫽ 0.05 ␣ 1 ⫽ 1.0 ca ⫽ 5

V 1 ⫽ 30 r 2 ⫽ 12 ␣ 2 ⫽ 0.001 cb ⫽ 5

V 2 ⫽ 100 ␤1 ⫽ 1 ␺ ⫽ 0.1 ␦⫽0

r˜ ⫽ 25 ␤2 ⫽ 2 K 0 ⫽ 100 c 1 ⫽ 2,000

␳ ⫽ 0.035 ␥ 1 ⫽ 0.15 S0 ⫽ 1 c 2 ⫽ 0.2

r៮ ⫽ 17 ␥ 2 ⫽ 0.5 tˆ ⫽ 12

Appendix 2 Proof of Theorem 9. Let the optimal solution corresponding to ␲ 1 [K(t ), S(t ), t] with ␲ 1Kt ⫽ 0 be denoted by ␭ 1 (t ), ␴ 1 (t ), x 1 (t ), and p 1 (t ). Let the optimal solution corresponding to ␲ 2 [K(t ), S(t ),


t] with ␲ 2Kt ⬍ 0 be denoted by ␭ 2 (t ), ␴ 2 (t ), x 2 (t ), and p 2 (t ), such that ␲ 1K [K(0), S(0), 0] ⫽ ␲ 2K [K(0), S(0), 0]. Therefore, while the marginal net revenues of effective capacity are the same at the initial time, ␲ 1K remains constant and ␲ 2K decreases over the planning horizon. Since the terminal time value is fixed, we obtain ␭ 1 (t ) ⬎ ␭ 2 (t ), x 1 (t ) ⬎ x 2 (t ) and ␴ 1 (t ) for ⬎ ␴ 2(t) for t 僆 [0, T). If ␺ ⫽ 0, the second term in Equation (12.2) is zero. Given the above adjoint variable relationships, we know ⭸H/⭸p 1 (t ) ⬎ ⭸H/ ⭸p 2 (t ). Furthermore, as ␲ 2Kt ⬍ 0 increases in magnitude (faster rate of decline), the difference p 1 (t ) ⫺ p 2 (t ) ⬎ 0 increases. Extending this logic, if ␺ is positive but relatively small in value, the magnitudes of the third and fourth terms of Equation (12.2) dominate the second term (this relationship characterizes Case 1). Therefore, ⭸H/⭸p 1 (t ) ⬎ ⭸H/⭸p 2 (t ) still holds, but the difference p 1 (t ) ⫺ p 2 (t ) ⬎ 0 is smaller than when ␺ ⫽ 0. Eventually, for sufficiently large ␺, we obtain p 1 (t ) ⫽ p 2 (t ). Let ␺ denote that value of ␺ such that p 1 (t ) ⫽ p 2 (t ). Therefore, for 0 ⱕ ␺ ⬍ ␺ (i.e., Case 1), as ␲ 2Kt ⬍ 0 increases in magnitude, we know p 1 (t ) ⫺ p 2 (t ) ⬎ 0. Lastly, suppose ␺ ⬎ ␺ so that the second term of Equation (12.2) dominates the sum of the third and fourth terms. (This relationship characterizes Case 2.) This gives ⭸H/⭸p 1 (t ) ⬍ ⭸H/⭸p 2 (t ) and p 2 (t ) ⫺ p 1 (t ) ⬎ 0. Q.E.D.

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Accepted by Linda Argote; received September 1997. This paper has been with the authors 10 months for 2 revisions.

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