Human resource planning in knowledge-intensive operations: A model for learning with stochastic turnover

European Journal of Operational Research 130 (2001) 169±189

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Theory and Methodology

Human resource planning in knowledge-intensive operations: A model for learning with stochastic turnover Sanjeev K. Bordoloi a, Hirofumi Matsuo b

b,*

a School of Business, College of William and Mary, Williamsburg, VA 23187, USA Institute of Policy and Planning Sciences, University of Tsukuba, 1-1-1 Tennoudai, Tsukuba, Ibaraki 305, Japan

Received 3 November 1998; accepted 25 January 2000

Abstract Because of rapid evolution in product and process technology, many operations in the manufacturing and service industries in recent years require workers to acquire and maintain more extensive ``knowledge stock'' than before. In this paper, we address the human resource planning in these knowledge-intensive operations. We focus on the management of knowledge mix, that is, the mix of workers in di€erent knowledge levels. This research was conducted in a semiconductor equipment manufacturing plant that uses an assembly line to achieve high productivity. However, the plant also needs to increase ¯exibility to deal with a high degree of product customization, frequent technology and product changes, and relatively low volume of production, which collectively require a high level of knowledge development for each worker. We extend methodology developed for managing production and work-in-process inventory levels for a manufacturing system that is subject to random production yields. However, the structure of our problem is di€erent, and thus requires a separate mathematical development. Our results indicate that the company we studied underestimated the ideal number of workers in the higher knowledge levels in the steady state. But this problem, by itself, can be taken care of by a good intuitive heuristic. We then demonstrate that our control rule is superior to the good intuitive rule from the point of view of additional stability that is obtained from less variability in the work force levels. We o€er managerial implications of this additional stability using our computational results. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Manpower planning; Manufacturing; Control; Dynamic programming

1. Introduction Knowledge-intensive operations are characterized by extensive knowledge required for each worker. Because of rapid evolution in product and process technology, many operations in the manufacturing and

*

Corresponding author. E-mail address: [email protected] (H. Matsuo).

0377-2217/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 0 0 ) 0 0 0 4 9 - 7

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service industries in recent years require workers to acquire and maintain more extensive knowledge than before. Employees in the service sector need to know the contents of a more diverse range of services provided, while those in manufacturing industries need to keep renewing their knowledge of newer technologies that are replacing the older ones. For example, knowledge-intensive operations can be frequently experienced in manufacturing capital equipment. In the service sector, knowledge-intensive operations are observed in ``call center'' operations for companies such as high technology companies and banking and investment companies. Since such operations require workers to have an extensive ``knowledge stock'', the acquisition process must be phased into multiple stages. New recruits are assigned to relatively simple operations requiring limited knowledge, and are trained to perform more complex operations in a phased manner. We consider the situation where the knowledge is nested (i.e., the knowledge acquisition is gradual and sequential from one stage to the next). We group workers depending on their knowledge levels, which correspond to stages of their knowledge development process. It is important to note that moving from one knowledge level to the next requires time and e€ort, and we usually cannot hire workers at the intermediate or the ®nal levels. Another diculty lies in high turnover among new recruits with a lower, yet very signi®cant, turnover among more experienced workers. Many new recruits tend to quickly ®nd a mismatch between job requirements and their job aptitudes and preferences, and leave their jobs. Those who complete the training program, ®nd themselves valuable to other companies that can take advantage of the knowledge acquired by this group of workers, and therefore leave for better opportunities (Anderson, 1997). Since new recruits ®nd the training program dicult, companies tend to give them incentives such as permanent employment after the successful completion of their training, to reduce turnover. In this paper, we address human resource planning in such knowledge-intensive operations. Our focus is management of the knowledge mix, which is the mix of workers at di€erent knowledge levels. Since it takes time to develop workers to the ®nal knowledge level, our model is designed to determine the number of workers that should be maintained at the di€erent knowledge levels. Because of both high turnover of workforce and high uncertainty in the turnover rates, the knowledge-intensive industries have to invest large amounts of resources in developing knowledge in the workforce without assurance of being able to retain it. The hiring and training program should also be such that the company maintains desired output and productivity levels through sucient ¯exibility in stang to absorb the ¯uctuations and uncertainties in the number of available workers at di€erent stages of knowledge development. We also consider additional constraints for the desired promotional and assignment policies within a company that may apply to knowledge workers as a part of the companyÕs skill development practice. We have conducted this research in a semiconductor equipment manufacturing plant in which we developed a model for their human resource planning. The volume of production is relatively high for a hightech equipment manufacturer, but may be considered low as compared to the majority of manufacturers. The plant uses an assembly line to achieve high productivity, although it needs to increase ¯exibility to deal with a high degree of product customization, frequent technology and product changes, and relatively low volumes of production, which collectively require a high level of knowledge development for each worker. The plantÕs assembly operations di€er from typical assembly line operations on two counts. First, unlike typical cycle times, on the order of a few minutes, the cycle time of an assembly line in this industry can be over 1 hour. Second, since substantial amounts of knowledge are required for workers in each workstation, a systematic knowledge development and on-the-job training program must be installed. Even though we have taken an assembly line as our context, our focus, unlike most other work in the literature, is not on the ¯ow of product/material through the assembly line. Our focus is on the ¯ow of workers through di€erent skill categories. For our numerical example, we collected historical data on the turnover rates for similar knowledge workers for the entire plant. We then selected a sub-section of the plant to develop our model and verify the

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validity of our results. Our model is applicable across the entire plant. In order of magnitude, the entire plant is 10±15 times the size of the sub-section selected. We approach the proposed problem by extending the methodology developed for managing production and work-in-process levels for a manufacturing system subject to random production yields. Denardo and Tang (1992) develop a linear control policy for a single-item, multi-stage manufacturing system with random yields. Their rule restores a fraction of the gaps between the targeted production and inventory levels and their current levels. Gong and Matsuo (1997) propose a control policy for a multiple-stage production system with random yields that near-optimally assigns the ®xed capacity at each stage to multiple products. The treatment of inventory level in these papers is analogous to that of workforce level in our paper. However, the structure of our problem is di€erent, and thus requires a separate mathematical development. In our problem, the workforce at each knowledge level is subject to random losses while only the production portion of inventory in the previous two papers is subject to random losses. The workforce mix determines the capacity level in our problem, while the capacity level is given in the previous papers. Because of the high turnover and time lag in developing human resources, the management of knowledge mix is not trivial. The results of our study indicate that the company we studied evidently underestimated the ideal number of workers in the higher knowledge levels in the steady-state. This underestimation, by itself, can be taken care of by a good intuitive heuristic such as the one we have considered for comparison purpose in this paper. However, what makes our rule superior to the intuitive rule is the additional stability that our rule provides over the intuitive rule. Our results from the comparison of the two rules using simulation show that, even though the mean number of workers recommended by the two rules is the same (the heuristic may provide inferior solution in terms of mean number of workers), our approach will stabilize the hiring level of workers more than the intuitive heuristic. This stability is obtained not by employing excess number of workers, but by adopting a hiring policy that reduces the variability of the worker levels in the knowledge groups, and thereby makes them more manageable. The implications of this stability are discussed in Section 4.3 of this paper. This paper is organized as follows: In Section 2, we provide a literature survey on the current status of the available literature that is related to our chosen topic. In Section 3, we present our model and the analytical results. In Section 4, we describe an illustrative example of knowledge-intensive operations, along with the computations, analyses and the managerial implications of our results. In this section, we also provide a comparison of our rule with an intuitive rule with respect to certain performance criteria. Section 5 ®nally concludes the paper. 2. Literature survey A variety of research publications are available that address problems in human resource planning, but not many of them are directly related to hiring and training decisions in knowledge-based operations with stochastic turnover rates. The two closest areas in which some amount of literature is available are: manpower planning in aggregate planning contexts, and learning and knowledge acquisition. Holt et al. (1960) study linear decision rules that specify the output and workforce levels that minimize the long-run costs of overtime/idle time, inventory costs, and hiring/layo€ costs. This model, along with most other models in aggregate planning literature, assumes constant productivity for the workforce. Ebert (1976) incorporates the learning curve into productivity calculations for an aggregate planning in a single stage manufacturing system. This problem becomes more complex in our context, because of the progression of workers through successive knowledge levels. Charnes et al. (1969) deal with the assignment problem between personnel and positions in order to maximize the total value of assignments involving people-to-people as well as people-to-position matching. In addition, Charnes et al. (1978) utilize a goal programming approach with embedded Markov processes in modeling for a ¯exible equal employment

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opportunity (EEO) program for the US Navy Oce of Civilian Manpower Management. Flexibility is desired here because elements of EEO are altered frequently to achieve day-to-day manpower goals of the organizations. Finally, continuing in this series of manpower planning studies, Charnes et al. (1970) provide further ¯exibility in the NavyÕs manpower planning model with a network formulation that allows for training as well as recruitment. Gaimon and Thompson (1994) o€er a deterministic, longitudinal personnel planning model that provides the optimal hiring, promotion, separation and retirement as a functions of time and a personÕs organizational age and grade. However, none of this literature deals with the volatility that we will treat by means of a chance-constrained programming formulation. A number of papers exist in the literature on operations management that deal with the learning aspect of manpower planning. Grinold (1976) o€ers optimal accession policies for manpower planning in a naval aviation system taking into consideration deterministic learning e€ects and retention rates. The demand for a workforce as determined by di€erent states of peaceful and con¯ict periods is represented as a Markov chain. Gerchak et al. (1990) present deterministic, discrete-time and continuous models for manpower planning, taking into account the e€ects of learning and turnover rates in production. Dorroh et al. (1994) provide an economic model that addresses the investment on acquiring knowledge in early stages of production as opposed to knowledge accumulation through learning-by-doing. Gaimon (1997) develops a model that takes account of the fact that the investment in information technology for one set of tasks contributes to the output of others through the enhancement of worker knowledge. Again, however, our model is di€erent in that our knowledge-intensive operations are subject to high workforce volatility (because of the high uncertainty in turnover rates of workers) and, therefore, the training program and the progression schedule for worker levels need to take this into account, as we do in our chance-constrained programming formulation.

3. The model 3.1. Hiring and knowledge development process with random turnover Our model is a discrete-time model in which we identify three knowledge levels of workers, 1, 2 and 3 and two production stages, A and B, in an assembly line. All new hires always start in stage A and undergo on-the-job training in stage A during the ®rst period. We call this knowledge level 1 and denote the number of workers at time t in knowledge level 1 as X …t†. When training in the ®rst stage is completed for a group of new hires, they are assigned to stage B in the next period, where they undergo training in this stage. We call this knowledge level 2 and the number of workers at this level at time t is denoted by W2 (t). After a worker has completed training in both A and B stages, the worker is promoted to knowledge level 3. We denote the number of workers at time t at this level as W3 (t). This worker assignment scheme is indicated in Fig. 1.

Fig. 1. A system with two production stages and three worker knowledge levels.

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All new employees go through this career path to gain integrated knowledge on the production at both stages. Since workers at stage B are expected to check the quality of work done at the preceding stage A, new hires must be trained ®rst at stage A. Thereafter, the new hires who have completed training at stage A is immediately assigned to stage B because the workers at stage A should know the succeeding operations at stage B to enhance quality. This promotion scheme also works as a part of their incentive policy. While we acknowledge that it may be possible to hire a worker at an intermediate level, we do not include that possibility in our model. This is because our emphasis is on the skill building practice of knowledge workers. Since this is a knowledge-intensive operation, the minimum amount of training necessary for the workers who have some experience in similar settings in other operations is large enough to warrant a complete training program. Moreover, the availability of such workers may be limited. Our assumption will also help in keeping the complexity of the formulation manageable. Some level 3 workers are assigned to training level 1 and level 2 workers on the job. These level 3 workers who are used for training new or newly promoted workers are not contributing to the production of the plant. Depending upon the demand requirement and productivity of workers in knowledge levels 2 and 3, the necessary number of workers from knowledge level 3 are assigned to stage B (denoted by W3b (t) in Fig. 1). The remaining number of workers from knowledge level 3, after adjusting for the trainers at stage B, are assigned to stage A (denoted by W3a (t) in Fig. 1). Thus, depending upon the demand requirements and productivity of workers in knowledge levels 1 and 3, the number to be hired in level 1 is determined. It is planned that all workers in knowledge level 2 are to be assigned to stage B. This is done by design in our skill building practice in order to keep the workers motivated and enhance quality. Let q1 and q2 denote the training fraction for level 1 and level 2 workers, respectively …0 6 q1 ; q2 6 1†. That is, q1 X …t† of level 3 workers are used to train level 1 workers in period t. Similarly, q2 W2 …t† of level 3 workers are used to train level 2 workers in period t. Hence, the following inequalities must be satis®ed: W3a …t† P q1 X …t†;

…1†

W3b …t† P q2 W2 …t†:

…2†

The demand for the product is denoted as d(t) which is an independent, identically distributed random variable independent of X …t† and Wi (t). We also assume that no single worker will be assigned to both stages A and B on a partial assignment basis. We denote the productivity of workers at knowledge level i as pi for i ˆ 1; 2; and 3. Since group 3 is fully trained in both the stages, its productivity will be higher than that of knowledge groups 1 and 2 (i.e. p1 6 p3 and p2 6 p3 ). Our objective is to minimize the total worker related costs for which we aim to meet demand by employing the optimal number of workers at di€erent levels. We start the assignment of workers by ful®lling the needs for stage B ®rst. We ®rst assign all level 2 workers ± level 1 in previous period ± to stage B. Then, based on the demand requirement and after adjusting for the training needs for level 2 workers at stage B, we assign the necessary number of level 3 workers to stage B. Therefore, the production function at stage B would be p2 W2 …t† ‡ p3 …W3b …t† ÿ q2 W2 …t†† ˆ d…t†:

…3†

Next, we do our assignment for stage A. First, we assign all remaining workers from level 3 to stage A. If there is insucient number of workers to meet the demand, we need to recruit new workers. While deciding the number of new recruits, we must also keep in mind the requirement of level 3 workers to train the new workers who will not be contributing to the production. Therefore, the production function at stage A becomes p1 X …t† ‡ p3 …W3a …t† ÿ q1 X …t†† P d…t†:

…4†

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The worker balance equation for level 3 workers is W3 …t† ˆ W3a …t† ‡ W3b …t†:

…5†

The non-negativity conditions are X …t†; W2 …t†; W3a …t†; W3b …t† P 0;

…6†

Appendix B shows that (1)±(6) are equivalent to the following inequalities (7)±(10) that do not include W3a (t) and W3b (t): …p1 ÿ p3 q1 †X …t† ‡ …p2 ÿ p3 q2 †W2 …t† ‡ p3 W3 …t† P 2d…t†;

…7†

p2 W2 …t† 6 d…t†;

…8†

ÿp3 q1 X …t† ‡ …p2 ÿ p3 q2 †W2 …t† ‡ p3 W3 …t† P d…t†;

…9†

X …t†; W2 …t†; W3 …t† P 0:

…10†

The progression of workers along the three knowledge levels is shown in Fig. 2. The retention rates (complement of turnover rates) at the three knowledge levels are denoted as yi for i ˆ 1; 2; and 3 and are de®ned as the probability that a worker at knowledge level i will stay in the plant after a particular period. After period 1, a proportion 1 ÿ y1 of level 1 workers will on average leave the system. The bulk of this proportion is voluntary turnover, not layo€. While it is possible, we do not model layo€s to keep the model simple. We denote by Z1 …t† the remaining workers at the end of the period, who will become level 2 workers in the next period and move to stage B. After period 2, a proportion 1 ÿ y2 of level 2 workers will on average leave the system and the remainder (denoted by Z2 (t)) will become part of level 3 workers. After period 3, a proportion 1 ÿ y3 of level 3 workers will on average leave the system and the remainder (denoted by Z3 (t)) will stay in the system. It is necessary to assume y3 < 1 to attain a steady-state condition. Notice that Zi (t)'s are independent processes. The yi 's are estimated based on historical data and are estimated to be stable for a medium-term time horizon. We establish the mass balance equations for workers at the beginning and at the end of a period as: W2 …t ‡ 1† ˆ Z1 …t†;

…11†

W3 …t ‡ 1† ˆ Z2 …t† ‡ Z3 …t†:

…12†

Fig. 2. Flow of workers through knowledge levels.

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These two equations can be written in matrix form as W …t ‡ 1† ˆ BZ…t†;

…13†

where T

W …t ‡ 1† ˆ ‰W2 …t ‡ 1†; W3 …t ‡ 1†Š ; and



1 Bˆ 0

0 1

0 1

Z…t† ˆ ‰Z1 …t†; Z2 …t†; Z3 …t†Š

T



We assume that Z1 …t†…Zi …t† for i ˆ 2 and 3) follows a binomial process with parameters y1 and X …t† (yi and Wi (t) for i ˆ 2 and 3). Then, we obtain E‰Z1 …t†jX …t†Š ˆ X …t†y1 ; E‰Zi …t†jWi …t†Š ˆ Wi …t†yi

…14† for i ˆ 2; 3;

8 < X …t†y1 …1 ÿ y1 †; Cov‰Zi …t†; Zj …t†jX …t†; W2 …t†; W3 …t†Š ˆ Wi …t†yi …1 ÿ yi †; : 0;

…15† i ˆ j ˆ 1; i ˆ j ˆ 2; 3; i 6ˆ j:

…16†

We develop a hiring policy in line with Denardo and Tang (1992) in which we establish relationships between the steady-state levels of workers at the three knowledge levels. We ®rst set the target levels X  , W2 and W3 for X …t†, W2 (t) and W3 (t), respectively. We impose the following relationships between the target levels: W2 ˆ y1 X  ;

…17†

W3 ˆ y2 W2 ‡ y3 W3 ˆ …y1 y2 =…1 ÿ y3 ††X  :

…18†

Later in this paper we will show that with (17) and (18), X  , W2 and W3 will be the steady-state expected values of X …t†, W2 (t) and W3 (t) respectively. Let q denote a restoration factor for 0 6 q 6 1. This restoration factor indicates the extent to which the gap between the target workforce and the current workforce in the subsequent levels is a€ecting the decision on the recruitment level at level 1 in the very next period. A value of q ˆ 0 indicates that we disregard the gap between the current and target workforce in the upstream levels while hiring for level 1. A value of q ˆ 1 indicates that we allow the complete gap to a€ect the hiring decision as shown in Eq. (19). Using the concepts of restoration factor and target values introduced in Denardo and Tang (which is also called the proportional control in the control theory literature), we introduce the following linear control rule: X …t† ˆ X  ‡ q…W2 ÿ W2 …t††=y1 ‡ q…W3 ÿ W3 …t††=y1 y2 ˆ X  ‡ qY …W  ÿ W …t††;

…19†

where the matrix Y ˆ ‰1=y1 ; 1=y1 y2 Š. We use a single value of q across the plant instead of one per group of workers in order to maintain the mathematical complications more manageable without compromising much on accuracy. Eq. (19) is our decision rule. The decision rule implies that, at the beginning of every period, the required number of workers to be hired in knowledge level 1, X …t†, would be calculated using (19). This decision, together with the turnover rates at all knowledge levels at the end of that period, will determine the workforce levels at

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next two knowledge levels in the beginning of next period. From (19) we can see that the number of workers at a knowledge level is adjusted every period depending upon a number of factors, namely, the turnover rates, the restoration factor, the target worker levels and the current worker levels. This equation takes into consideration the uncertainty in turnover rates and regulates the number of workers at each knowledge level. To see that this is so, we write the expected values of the end-of-period worker levels by substituting (19) for X …t† in (14) to obtain: E‰Z1 …t†jX …t†Š ˆ E‰Z1 …t†jW …t†Š ˆ y1 X  ‡ q…W2 ‡ W3 =y2 † ÿ q…W2 …t† ‡ W3 …t†=y2 †;

…20†

E‰Zi …t†jW …t†Š ˆ yi Wi …t† for i ˆ 2; 3:

…21†

3.2. Steady-state results In this section, we are going to derive the steady-state results for the expected values and the covariances of X …t†, W2 (t) and W3 (t). For this purpose, we take the relationship between the target level as imposed in Eqs. (17) and (18), and the worker level equations (13) and (19); and apply to them Eqs. (14)±(16) obtained from our binomial process assumption. We derive the following lemma that leads to the steady-state equations on the expected values and covariances of worker levels. For this purpose we use   ÿq ÿq=y2 Cˆ y2 y3 in subsequent sections. Lemma 1. The linear control rule X …t† has (L1)   y1 E‰X …t†Š ‡ q…E‰W2 …t†Š ‡ E‰W3 …t†Š=y2 † ‡ CW …t†; E‰W …t ‡ 1†jW …t†Š ˆ 0 (L2) E‰W …t ‡ 1†jW …1†Š ˆ

  tÿ1 X y E‰X …t†Š ‡ q…E‰W2 …t†Š ‡ E‰W3 …t†Š=y2 † Ci 1 ‡ C t W …1†; 0 iˆ0

(L3) Cov‰W …t ‡ 1†Š ˆ C Cov‰W …t†ŠC T ‡ P …t†; where

2

P …t† ˆ 4

y1 …1 ÿ y1 †E‰X …t†Š 0

3 P iˆ2

0 yi …1 ÿ yi †E‰Wi …t†Š

3 5:

All proofs are placed in Appendix A. The equations in Lemma 1, respectively, describe the mean, variance and covariance of the worker levels at the di€erent knowledge levels in terms of their current values and their initial values. Repeated appli-

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cation of the linear control rule along with Eqs. (17) and (18) will lead geometrically to the following theorem. Theorem 1. The steady state values of W …t† and X …t† are expressed as (T1)   W2 ˆ W ; lim E‰W …t ‡ 1†jW …1†Š ˆ W3 t!1 (T2) lim E‰X …t†jW …1†Š ˆ X 

t!1

(T3) Cov‰W …t†Š converges as t goes to infinity: All proofs are placed in Appendix A. We use Cov W  to denote limt!1 Cov‰W …t†Š: Lemma 1 is necessary to establish the relationship between the expected number of workforce levels with respect to their initial and current levels. Lemma 1 indicates how the mean, variance and convariance of the three worker levels would vary recursively starting at the initial period. Applying the relationships in Lemma 1 repeatedly, we can derive the steady-state values for the three worker levels. Theorem 1 is necessary to establish that the steady-state values exist for our parameters. It is important that we establish Lemma 1 and Theorem 1 to characterize our parameters and run a numerical application. As we will see in the chance-constrained equations in later part of this paper, our formulation includes the means and variances of the workforce levels of the three knowledge groups. Lemma 1 helps in understanding how the workforce levels at a given time period are related to the initial and the previous periods. Subsequently, when we attempt to arrive at the steady-state values of the workforce levels, Theorem 1 helps, ®rst, in arriving at the steady-state values, and then, in verifying that convergence does occur as time approaches in®nity. 3.3. Chance-constrained formulation Our objective is to minimize the total worker related costs for which the production levels at the di€erent production stages are such that the demand is met with a desired level of reliability. This form of demand ful®llment is captured using chance-constrained equations. Our goal is to derive reasonably ecient workforce levels for longer-term applications. We acknowledge that any short-term ¯uctuations would be absorbed by overtime. In order not to make our formulation too complex, we do not include the overtime considerations in our formulation. We will now write the corresponding chance-constraint equations for our earlier Eqs. (7)±(10). The inequality (7) can be converted to the following chance-constraint: Prob‰…p1 ÿ p3 q1 †X …t† ‡ p2 ÿ p3 q2 †W2 …t† ‡ p3 W3 …t† ÿ 2d…t† P 0Š P b1 ;

…22†

where 0 6 b1 6 1 is the probability parameter corresponding to a ``reliability level'' (see Charnes and Cooper, 1959) so that 1 ÿ b1 is the risk of not being able to meet constraint (7). This reliability level is analogous to the typical ``service level'' in most operations management literature in this context.

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Similarly, inequalities (8)±(10) can be converted to the following chance-constraints: Prob‰ÿp3 q1 X …t† ‡ …p2 ÿ p3 q2 †W2 …t† ‡ p3 W3 …t† ÿ d…t† P 0Š P b2 ;

…23†

Prob‰p2 W2 …t† ÿ d…t† 6 0Š P b3 ;

…24†

Prob‰Wi …t† P 0Š P b4

…25†

for i ˆ 2; 3;

Prob‰X …t† P 0Š P b5 :

…26†

In our formulation X …t† and Wi …t† may take negative values. If this happens, we do not recommend laying o€ workers since this is against the company's operating policy. In such a case, for implementation in practice, we set X …t† ˆ 0 (i.e. do not hire any new recruit in period t) which will ensure non-negative values of Wi …t†. We veri®ed by simulation that the di€erence between our estimates of the ®rst two moments (mean and variance) of Wi …t† and their respective actual values will be very small and that our estimates will be close to their respective actual values. Assuming that the distribution of …p1 ÿ p3 q1 †X …t† ‡ …p2 ÿ p3 q2 † W2 …t† ‡ p3 W3 …t† ÿ 2d…t† follows a normal distribution (Charnes and Cooper, 1959), constraint (22) can be converted to an equivalent deterministic form lim E‰…p1 ÿ p3 q1 †X …t† ‡ …p2 ÿ p3 q2 †W2 …t† ‡ p3 W3 …t†Š

t!1

ÿ k1 ‰Var‰…p1 ÿ p3 q1 †X …t† ‡ …p2 ÿ p3 q2 †W2 …t† ‡ p3 W3 …t†Š ‡ Var‰2d…t†ŠŠ

1=2

P 2d  ;

…27†

where d* is the target level of demand. Here, k1 is selected so that k1 standard deviations below the mean of ‰p1 ÿ p3 q1 ŠX …t†‡ ‰p2 ÿ p3 q2 ŠW2 …t† ‡ p3 W3 …t† ÿ 2d…t† is 0. This is non-negative in (22), and Var‰…p1 ÿ p3 q1 †X …t† ‡ …p2 ÿ p3 q2 †W2 …t† ‡ p3 W3 …t†Š 2

2

ˆ …p1 ÿ p3 q1 † Var…X …t†† ‡ …p2 ÿ p3 q2 † Var…W2 …t†† ‡ p32 Var…W3 …t†† ‡ 2…p1 ÿ p3 q1 †…p2 ÿ p3 q2 † Cov‰X …t†; W2 …t†Š ‡ 2…p1 ÿ p3 q1 †p3 Cov‰X …t†; W3 …t†Š ‡ 2…p2 ÿ p3 q2 †p3 Cov‰W2 …t†; W3 …t†Š gives the remainder of the expression in (27). Using Eq. (19) which is X …t† ˆ X  ‡ q…W2 ÿ W2 …t††=y1 ‡ q…W3 ÿ W3 …t††=y1 y2 we obtain: Cov‰X …t†; W2 …t†Š ˆ ÿVar‰W2 …t†Šq=y1 ÿ Cov‰W2 …t†; W3 …t†Šq=y1 y2 ;

…28†

Cov‰X …t†; W3 …t†Š ˆ ÿVar‰W3 …t†Šq=y1 y2 ÿ Cov‰W2 …t†; W3 …t†Šq=y1 ;

…29†

Var‰X …t†Š ˆ Var‰W2 …t†Šq2 =y12 ‡ Var‰W3 …t†Šq2 =y12 y22 ‡ 2 Cov‰W2 …t†; W3 …t†Šq2 =y12 y2 :

…30†

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Therefore, by using (17), (18), (27) and Theorem 1, we can express (22) in terms of X  , as ……p1 ÿ p3 q1 † ‡ …p2 ÿ p3 q2 †y1 ‡ p3 …y1 y2 =…1 ÿ y3 †††X  ÿ k1 ‰lim ‰Var‰…p1 ÿ p3 q1 †X …t† ‡ …p2 ÿ p3 q2 †W2 …t† ‡ p3 W3 …t†Š ‡ Var‰2d…t†ŠŠ

1=2

t!1

Š P 2d  :

…31†

Similarly, Eqs. (23)±(26) can be written respectively as …ÿp3 q1 ‡ …p2 ÿ p3 q2 †y1 ‡ p3 …y1 y2 =…1 ÿ y3 †††X  ÿ k2 ‰lim ‰Var‰ÿp3 q1 X …t† ‡ …p2 ÿ p3 q2 †W2 …t† ‡ p3 W3 …t†Š ‡ Var‰d…t†ŠŠ t!1

p2 y1 X  ‡ k3 ‰lim ‰Var‰p2 W2 …t†Š ‡ Var‰d…t†ŠŠ1=2 Š 6 d  ; t!1

Wi  ÿ k4 ‰lim ‰Var Wi …t†Š

1=2

t!1

1=2

X  ÿ k5 ‰lim ‰Var X …t†Š t!1

Š P 0 for i ˆ 2 and 3;

Š P 0:

1=2

P d ;

…32†

…33†

…34†

…35†

The square roots in above equation are necessary to incorporate the standard deviation of the di€erent production functions in order to obtain the desired level of reliability. The convexity of our functions is not a major issue. We use the following worker related cost parameters for deriving our objective function: ci : wage of a worker in knowledge level i, i ˆ 1; 2; 3 ri : training cost of one worker in level i for i ˆ 1; 2, and h: hiring cost of one worker. Our focus is on long-term workforce planning of knowledge workers for which the above costs are relevant. We do not include overtime cost in our model since we consider that to be a short-term correction. Then, the objective function is Min…c1 ‡ h ‡ r1 †X  ‡ …c2 ‡ r2 †W2 ‡ c3 W3 :

…36†

Since all wages and costs are positive and constant, and using (17) and (18), this objective reduces to minimizing X  , the steady-state hiring level de®ned in (T2). Hence, under the production policy expressed in Eq. (19) the steady-state problem can be written as ‰P1Š : Min X  subject to Eqs. (31)±(35). Notice that the two decision variables for this problem are: (1) the steady-state hiring level at knowledge level 1 (i.e. X  ), and (2) the restoration factor (q). While the ®rst decision variable …X  † is explicit in the objective function, the second (q) is embedded in the constraint equations. To solve this problem, we proceed as follows: In line with the proof of Theorem 1, given in Appendix A, we ®rst calculate steady-state values of variances of W2 (t) and W3 (t), and the covariance Cov [W2 (t),W3 (t)]. Then, we use these values in Eqs. (28)±(30) to calculate Cov‰X …t†; W2 …t†Š; Cov‰X …t†; W3 …t†Š, and Var…X …t††. These values are subsequently substituted in constraint Eqs. (31)±(35) for solving the non-linear programming formulation [P1].

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4. An illustrative example: Human resource management in a knowledge-intensive assembly line 4.1. The current situation We consider a case study at a large semiconductor equipment manufacturing company having a number of plants, most of which perform assembly line operations. The assembly operations are categorized according to their complexity as ``simple build'' or ``complex build'' operation. A plant will typically employ 400±500 workers of which 200±300 will be directly relevant to production structures that ®t into out model. For our analysis, we chose a particular production line, a module of the entire plant, that is representative of the assembly operations performed across the company. Our model is applicable across the entire plant. In terms of size of employment, the entire plant is 10±15 times the magnitude of the particular production line that we have selected. The selected production line consists of 16 workstations in a tree structure as shown in Fig. 3. There exists a main assembly line and two subassembly lines that feed the main assembly line at the junctions indicated in the ®gure. Typically, each station is manned by one worker. However, a new worker is teamed up with an experienced worker for on-the-job training while assuring that the necessary output is achieved. Therefore, the line is sta€ed by more than 16 workers at any given time. We converted the 16-stage line into two major stages. The eight front-end stages (indicated in Fig. 3 as ``simple build'') comprise stage A, and the eight back-end stages (indicated in Fig. 3 as ``complex build'') comprise stage B. A new worker is assigned to the simplest of the operations among the ``simple build'' stages, which is generally the very ®rst operation. Training is mostly on-the-job: (a) in the form of initial observation, (b) teaming up with experienced workers and then (c) solo responsibility. The main issues in the stang decisions are twofold: (1) to determine how many workers to hire at the beginning of each period, and (2) to develop an appropriate training schedule for higher productivity and ¯exibility. The cycle time of the plant is 60 minutes, which means that a worker needs to learn 60 minutes worth of work content in a workstation. A worker takes a quarter (13 weeks) to complete training in either of stages A and B. The training schedule is such that each worker will be ready to move from the simple build stations to complex build stations in 13 weeks. A ``period'' as applied to our formulation will be of length one quarter (13 weeks). The demand level at the plant is considered to have a mean of 8 units per shift for an 8 hours shift length (i.e. d…t† ˆ 520 …ˆ 8  5  13† for a 5 days/week operation). It is estimated that the quarterly demand will

Fig. 3. A 16-station assembly line.

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follow a normal distribution with a mean of 520 units and a standard deviation of 130 units. The productivity of each worker at knowledge levels 1 and 2 is estimated to be 0.5 units per shift of 8 hours (i.e. p1 ˆ p2 ˆ 0:5  5  13 for the quarter). At knowledge level 3 the productivity is estimated to be 1.0 (i.e. p3 ˆ 1  5  13). Not all of the workers in knowledge level 3 are available for production since they are also engaged in training the workers in knowledge levels 1 and 2. It is estimated from experience at the given plant that from the total number of workers in knowledge level 3, 30% of number of workers in knowledge level 1 and 20% of number of workers in knowledge level 2, will be engaged in training. Therefore, q1 ˆ 0:3 and q2 ˆ 0:2. We believe that a period of 13 weeks is appropriate for this application, given that the cycle time for the products is longer and that demand does not ¯uctuate signi®cantly over a quarter. Moreover, turnover rates over a shorter time, say a month, are less meaningful. From historical data the quarterly retention rates for the three knowledge worker groups are estimated as follows: y1 ˆ 0:55; y2 ˆ 0:65; y3 ˆ 0:90. These estimates are based on the historical data on knowledge workers for the entire plant, which is in the magnitude of 200±300 workers. The particular line that we chose for our analysis currently employs seven, ®ve and eight workers at knowledge levels 1, 2 and 3, respectively. At present, the plant faces a considerable number of disruptions and therefore, delays, because of the uncertainty in turnover and the training needs. The plant manager believes that improvements can be made in the hiring and training policies that would reduce the delays and disruptions and thereby reduce overtime costs.

4.2. Computation, analysis and managerial insight We use ki ˆ 1 for i ˆ 1 to 5 in Eqs. (31)±(35) to meet demand at a required level of reliability. We calculate the Cov W  matrix utilizing our proof of Theorem 1 given in Appendix A. We can then write the constraints (31)±(33) in terms of X  and q. We then solve the problem [P1] using a non-linear programming tool with the objective function of minimizing X  subject to constraint equations (31)±(35). In our case, we obtain optimal values of q ˆ 0:958 and X  ˆ 4:83, which result in an optimal W2 of 2.66 and an optimal W3 of 17.26 ± for a total number of workers of 24.75. Our results show that the current worker ratio of 7:5:8 is not very e€ective for the stability of the plant's operation and it will not be sucient to meet the demand with the required level of reliability (Eq. (31) cannot be satis®ed with current employment). This could be the reason for the excessive overtime used. Notice that we do not consider it necessary to round o€ the optimal worker levels to integer values because fractional assignment of workers to this line is often used. Fractional assignment refers to a particular worker working part time or overtime in a given production line, and it does not refer to assigning a particular worker to both stages A and B in the same period. Our results indicate that the plant currently underestimates the number of workers required in the knowledge level 3 in steady state. An optimal q of 0.958 indicates that it is more bene®cial to have a high degree of restoration of the current-to-target workforce gaps, with the optimal level being 95.8%. We ran a simulation of our model in order to note the transient behavior of our parameters to see the rate of convergence starting with workforce levels of (7,5,8) till it reaches the steady-state values of (4.83, 2.66, 17.26). We found that the workforce levels converge to the neighborhood of their steady-state values in about three periods. Starting with (7,5,8), on average, 55% of 7 ( ˆ 3.85) will become level 2 workers in period 2 and 65% of 5 plus 90% of 8 ( ˆ 10.45) will become level 3 workers. This will necessitate the hiring of about 21 new workers in period 2 in order to meet the demand with the desired reliability level. Even though this number seems high, it can be veri®ed that sucient number of level 3 workers exists to ful®ll the requirement of trainers for both levels 1 (30%) and 2 (20%). In the similar fashion, starting with a workforce of (21, 3.85, 10.45) in period 2, adjusting for the corresponding turnover rates, it can be veri®ed that we will arrive the neighborhood of our steady-state level of

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(4.83, 2.66, 17.26) in period 4. This analysis validates that the transient behavior of our model is quite satisfactory for medium-term planning horizon. It is also important to analyze the signi®cance of the restoration factor (q) and the e€ect of the variances of the number of workers in the three knowledge groups on the optimal solution. The optimal workforce level …X  † is determined for the requirement that a certain level of reliability is maintained. If the variances of the number of workers in di€erent worker groups increase, this optimal workforce level also increases in order to maintain that reliability level. The restoration factor, q, a€ects the variability of the three workforce groups in di€erent ways. When restoration factor approaches 0, Var‰X …t†Š also tends to 0 and Var ‰W3 …t†Š becomes very high, resulting in a need to hire more workers (as compared to the optimal X  at the optimal q) in order to achieve the desired reliability level. When q approaches 1, then the variance, Var‰X …t†Š, becomes very high as compared to Var[W2 (t)] and Var[W3 (t)], thereby resulting in a greater number of workers recruited once again as compared to the optimal X  . Fig. 4 delineates the behavior of the three variances when optimal solution was calculated with given values of restoration factor, q. The relative magnitude of the values of the variances plays an important role in determining the optimal value of q. Thus, in our model, a non-optimal q leads to a higher overall workforce level and a higher expected production expense. This should be contrasted to the case in Denardo and Tang (1992), where the expected production levels remain constant for di€erent values of q. This di€erence results from the di€erent mathematical structures of the two problems caused by our use of chance-constraints to deal with volatility in the work force. Another managerial insight can be derived from our computation by looking at the slack of the nonbinding constraint Eqs. (32) and (33). The binding constraint Eq. (31) indicates the production levels

Fig. 4. Behavior of variances with restoration factor.

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required to meet demand with the desired level of reliability. As expected, there is no slack for this constraint equation. The other two equations (32) and (33), indicate the amount of slack available before we may have to violate some of the other constraints such as: (i) assigning level 1 workers directly at stage B, or (ii) holding back level 2 workers at stage A despite completing training. The non-binding nature of our speci®c optimal solution does not mean that these two constraint equations are redundant. At the current optimal operating point, these two equations happen to be non-binding. We have observed that for di€erent values of ki and also for di€erent values of yi , some of those equations become binding too. The slack for Eq. (32) indicates the amount of cushion that the plant has in its skill accumulation (``skill cushion''). The left-hand side of Eq. (32) re¯ects the capability of the workers at levels 2 and 3 together to assemble the required number of products at stage B. Carrying the terms involving variances over to the right-hand side of (32) we then see that this is an additive amount in the form of a safety factor to allow for the risks of not being able to meet demand by having insucient number of workers in knowledge levels 2 and 3. (Notice that it is important to have sucient amount of cushion here because a new hire cannot be directly employed at stage B without being ®rst trained at stage A.) The need for skill cushion is driven by our policy requirement that no worker from knowledge level 1 will be assigned to stage B directly. As explained earlier, this policy is a part of the skill building practice of knowledge workers. Similarly, an interpretation of the slack in Eq. (33) provides a margin (``safety margin'') in which the plant can avoid a situation in which workers are asked to stay back in stage A after being trained in that stage. This situation would operate against worker motivation and defeat the purpose of our prescribed training scheme. It is important for the manager not to lose sight of the skill cushion or the safety margin derived from these equations because of the uncertainty in the turnover rates of the workers. In our case, the value of ki ˆ 1 is the most appropriate level of reliability. The optimal recruitment level …X  † increases linearly with ki while the optimal restoration factor q increases with ki . However, for ki P 1, the optimal q becomes close to 1. On the other hand, a lower value of ki results in a higher overtime requirement for short-term correction and our estimates for the parameters become less reliable. In this case, around ki ˆ 1, a good trade-o€ between quick restoration and the cost of high protection is achieved. The appropriate reliability level could be di€erent depending on situations, but it can be found easily by simulation. A key result of this paper is the decision rule that determines the ``necessary'' number of workers at each knowledge level for every period in order to meet the ``reliability level'' selected for the chance-constrained equations, while at the same time, reducing the variability of the three worker levels. The stochastic nature of the situation is rooted at the inherent variability of the demand and worker turnover. The ``excess'' workforce at each knowledge level is analogous to the notion of ``safety stock'' in inventory management. As we go for more protection, the optimal workforce levels for the three knowledge groups also go higher. 4.3. Comparison of our rule with an intuitive rule using simulation A question may now arise: can the same results be obtained if an intuitive rule is derived based on a manager's past experience and good judgment without having to calculate the optimal solution as suggested by our model? It may be argued that if overtime was the only concern, it is possible to derive an appropriate intuitive rule which will result in workforce levels at the three knowledge groups such that the same steadystate values of workforce levels, as in our rule, can be obtained, resulting in the same overtime costs. In this section, we will take up such an intuitive rule and with the help of simulation, we will compare the results from our rule with this those from the intuitive rule in terms of selected performance criteria. The intuitive rule that we have considered attempts to maintain a steady overall productivity of a target value (N) in each period, taking all three knowledge groups together. Depending upon the number of workers that stayed back from one period to next ± and hence were promoted from one knowledge level to next ± this intuitive rule will decide, for every period, the number of new recruits necessary at knowledge

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level 1 to maintain this target productivity level of N. The value of N may be determined from past experience or may be calculated using certain basic formula. The intuitive rule for hiring new recruits that we have considered, can be expressed as follows: X …t† ˆ …N ÿ p2 W2 …t† ÿ p3 W3 …t††=p1 :

…37†

For comparison purpose, we ran the simulation using our control rule ®rst, calculated the mean total productivity and equated this to the total target productivity (N) for the simulation using the intuitive rule. The rationale behind using this value of N is to allow the manager to have perfect information, so that we can compare the two rules on ®ner issues such as stability. In reality, the manager will have less than perfect information in deciding the value of N, which will make our model even more valuable and superior. For each rule, we considered a time horizon of 10 years (40 periods), started with initial worker levels of 7, 5 and 8 at the three knowledge levels, respectively, and ran necessary number of simulation runs to tabulate the 95% con®dence intervals for selected performance criteria for comparing the two rules. The results are tabulated in Table 1. As we mentioned in Section 3.3, in the simulation we did not allow negative values of workforce levels, and as stated, we ®nd that the means and variances from the simulation output, while simulating our rule, turned out to be very close to our optimization results. While simulating the two rules, we ®nd that each rule reached its steady-state values within four to ®ve periods (rounds of simulation). Since the smoothing factor is absent in the intuitive rule, the intuitive rule makes bigger jumps towards the direction of the optimal values in every round than our rule does. While doing so, the intuitive rule mostly ends up overcompensating for the gap between the demand and the actual production. Overall, the performances of the two rules in terms of the duration to reach steady-state values are comparable. The results indicate that while the means of all criteria are comparable for the two rules, the variability in the two rules is not. This sheds additional insights on the applicability of the two rules. The stability in the mix of workers at di€erent knowledge levels is an important consideration by the following two reasons. First, the standard deviation of the number of workers at the ®rst knowledge level is important for stability in the recruitment process. If it is very high, as it is in the intuitive rule, some of the prescribed recruitment numbers may even be impractical. This is because the prescribed number of workers with necessary skill requirements may not be available in the job market for hiring. Our proposed rule calculates the recruitment level for the next period keeping in mind a long-term view in terms of the restoration factor and the target worker levels. On the other hand, the intuitive rule calculates the recruitment levels only from a single-period perspective. Second, the stability in the mix of knowledge groups is an important consideration because it maintains both productivity and quality of the production process and eciency in on-the-job training. Although ineciencies of new workers are factored in their productivity in our formulation, a very high percentage of new workers may result in disproportional deterioration in both productivity and quality. Similarly, a very high percentage of new workers tend to result in a disproportional decrease in training eciency. Therefore, Table 1 Comparison of simulation results Performance criteria

Our rule

Intuitive rule

Mean X …t† S.D. ‰X …t†Š Mean W2 …t† S.D. ‰W2 …t†Š Mean W3 …t† S.D. ‰W3 …t†Š

4.905 ‹ 0.016 2.53 ‹ 0.06 2.698 ‹ 0.014 1.42 ‹ 0.04 17.208 ‹ 0.014 1.16 ‹ 0.03

4.908 ‹ 0.016 4.22 ‹ 0.08 2.691 ‹ 0.022 2.31 ‹ 0.04 17.334 ‹ 0.014 1.26 ‹ 0.03

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we emphasize that our control rule is superior to the intuitive rule from the point of view of stability in the mix of workers at di€erent knowledge levels. This is important to be emphasized because, on a simple outlook, the two rules may appear to be similar, looking at the ®rst moments of the performance criteria only. 5. Discussions and conclusions In this paper we have combined control theory and chance-constrained programming to deal with workforce planning which allows for worker learning and controls for the risks involved in a high labor turnover context. We derive steady-state workforce levels for di€erent knowledge groups within the plant in order to minimize total labor related costs. For simplicity, we did not include overtime costs, which are deemed to be short-term adjustments. Our attempt was to meet a stochastic demand with a desired level of reliability. We also introduced certain additional constrains to re¯ect the skill building policy of the plant keeping in mind behavioral factors such as motivation. We then applied our model to an actual problem in an appropriate plant to indicate its applicability in the high-tech industry. This was accompanied by considerations that deal with the risk involved in uncertainties. An actual application of our linear control rule showed that the optimal mix of workers at di€erent knowledge levels ([4.83, 2.66, 17.26] for a total of 24.75) is quite di€erent from those currently applied in the plant ([7, 5, 8] for a total of 20). We also discussed managerial signi®cance of concepts such as restoration factor, skill cushion and safety margin for operating leverages. We then derived a good intuitive rule that would provide the same mean number of workers at the three knowledge groups as our rule, thereby resulting in same labor and overtime costs. We showed, using simulation, that our control rule is superior to the intuitive rule when we bring our attention to the variability in the system. Our rule results in a hiring policy and a knowledge mix that has much less variability than the intuitive rule. This makes the human resource levels derived from our rule more appropriate. In this paper, we did not consider layo€ of experienced workers or hiring knowledge workers at intermediate stages. Nor did we consider ®ner time periods shorter than 3 months. While possible, these would make the model considerably complex to analyze mathematically. We consider only the primary factors of recruitment and learning of knowledge-intensive operations while keeping the required mathematical development manageable. We believe that the proposed model is applicable in many other knowledge-intensive operations with minor modi®cations. Generalizing our model and analysis to accommodate some of the other elements was deemed beyond the scope of this paper. Acknowledgements The authors are grateful to Professor William W. Cooper for his extensive comments on an earlier draft of this paper. Appendix A. Proofs of Lemma 1 and Theorem 1 Proof of Lemma 1. From (13), (20) and (21) we obtain

2 3  E‰Z1 …t†jW …t†Š 1 0 0 4 E‰Z2 …t†jW …t†Š 5: E‰W …t ‡ 1†jW …t†Š ˆ E‰BZ…t†jW …t†Š ˆ BE‰Z…t†jW …t†Š ˆ 0 1 1 E‰Z3 …t†jW …t†Š     ÿq ÿq=y2 y X  ‡ q…W2 ‡ W3 =y2 † ‡ ˆ 1 W …t† ˆ D ‡ CW …t†; y2 y3 0 

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where



 y1 X  ‡ q…W2 ‡ W3 =y2 † : Dˆ 0

This proves (L1). Applying the relationship (L1) recursively, we obtain E‰W …t ‡ 1†jW …1†Š ˆ

tÿ1 X

C i D ‡ C t W …1†:

iˆ0

This proves (L2). Note that Cov‰W …t ‡ 1†Š ˆ Cov‰E‰W …t ‡ 1†jW …t†Š ‡E‰Cov‰W …t ‡ 1†jW …t†ŠŠ:The ®rst term of the righthand side equals to Cov‰E‰W …t ‡ 1†jW …t†ŠŠ ˆ Cov‰D ‡ CW …t†Š ˆ C Cov…W …t††C T : The second term equals to E‰Cov…W …t ‡ 1†jW …t††Š ˆ E‰Cov…BZ…t†jW …t††Š ˆ E‰B Cov…Z…t†jW …t††BT Š 2 32 3 1 0 0 0 " # y1 …1 ÿ y1 †E‰X …t†Š 1 0 0 6 76 7 6 76 0 1 7 0 0 y2 …1 ÿ y2 †E‰W2 …t†Š ˆ 4 54 5 0 1 1 0 1 0 0 y3 …1 ÿ y3 †E‰W3 …t†Š 2 3 0 y1 …1 ÿ y1 †E‰X …t†Š 6 7 3 ˆ4 5; P yi …1 ÿ yi †E‰Wi …t†Š 0 iˆ2

Hence, Cov‰W …t ‡ 1†Š ˆ C Cov‰W …t†ŠC T ‡E‰Cov…BZ…t†jW …t†; X …t†BT †Š which is (L3).



q 2 Proof of Theorem 1. The eigenvalues of C are k ˆ f…y3 ÿ q†  …y3 ‡ q† ÿ 4q†g=2. Case 1: …q ‡ y3 †2 P 4q. Then, …y3 ÿ q†2 ÿ f…y3 ‡ q†2 ÿ 4qg ˆ 4q…1 ÿ y3 † > 0 q 2 Therefore, jy3 ÿ qj > …y3 ‡ q† ÿ 4q. q Then, jkj < jy3 ÿ qj 6 1 2 2 Case 2: …q ‡ y3 † < 4q. In this case, k ˆ …y3 ÿ q†=2  i ÿ…y3 ‡ q† ‡ 4q=2. Then, the square of the norm of k is kkk2 ˆ …y3 ÿ q†2 =4 ‡Pfÿ…y3 ‡ q†2 ‡ 4qg=4 ˆ q…1 ÿ y3 † < 1 Hence, the spectrum radius of C is less than 1. Therefore, tiˆ0 C i converges as t goes to in®nity. Hence, from (L2) E‰W …t ‡ 1†jW …1†Š converges as t ! 1 to lim E‰W …t ‡ 1†jW …1†Š ˆ …I ÿ C†ÿ1 D:

t!1

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Note that



ÿq I ÿC ˆI ÿ y2 …I ÿ C†

ÿ1

ˆ

ÿ yq2 y3





1‡q ˆ ÿy2



q y2

1 ÿ y3

 1 1 ÿ y3 y2 1 ‡ 2q ÿ …1 ‡ q†y3

187

;

ÿq y2



1‡q

:

Therefore, we can con®rm for (17) and (18) that lim E‰W …t ‡ 1†jW …1†Š ˆ …I ÿ C†ÿ1 D ˆ W  :

t!1

This proves (T1). Substituting (T1) in (19), we obtain lim E‰X …t†Š ˆ X  :

t!1

This proves (T2). Applying (L3) repeatedly along with Cov‰W …1†Š ˆ 0, we obtain T

Cov‰W …t†Š ˆ C tÿ1 W …1†…C tÿ1 † ‡

tÿ2 tÿ2 X X C i P …t ÿ i ÿ 1†C i T ˆ C i …t ÿ i ÿ 1†C i T : iˆ0

iˆ0

When k1 6ˆ k2 , the matrix C can be diagonalized with a symmetric matrix A as   k1 0 ÿ1 : C ˆ AKA ; where K ˆ 0 k2 Let qij …t† denote that …i; j† element of Aÿ1 P …t†AÿT where AÿT denotes the transpose of Aÿ1 , which is the inverse of A. ! tÿ2 tÿ2 tÿ2 X X X i ÿ1 i iT ÿ1 i ÿ1 i T ÿT i C P …t ÿ i ÿ 1†C ˆ …AKA † P …t ÿ i ÿ 1†…AKA † ˆ A K A P …t ÿ i ÿ 1†A K AT iˆ0

iˆ0

2

iˆ0 tÿ2 P

k21 q11 …t ÿ i ÿ 1†

6 6 iˆ0 ˆ A6 6 tÿ2 4P k1 k2 q21 …t ÿ i ÿ 1† iˆ0

3 k1 k2 q12 …t ÿ i ÿ 1† 7 iˆ0 7 T 7A : 7 tÿ2 5 P 2 k2 q22 …t ÿ i ÿ 1†

tÿ2 P

iˆ0

Since E‰X …t†Š and E‰Wi …t†Š converge to X  and Wi  , respectively, as t goes to in®nity, Aÿ1 P …t†AÿT converges to 2 3 0 y1 …1 ÿ y1 †X  3 4 5: P yi …1 ÿ yi †Wi  0 iˆ2

Hence, qij …t† converges as t goes to in®nity.

…A:1†

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Let qij denotes limt!1 qij …t†. Since kk1 k; kk2 k < 1, then it can be shown that (A1) converges as t goes to in®nity, and   1 X q =…1 ÿ k21 † q12 =…1 ÿ k1 k2 † T A : C i P …t ÿ i ÿ 1†C i T ˆ A  11 q21 =…1 ÿ k1 k2 † q22 =…1 ÿ k22 † iˆ0 This proves (T3). For the case k1 ˆ k2 , even though the matrix C may not be diagonalized, (T3) can be proved similarly. In this case, K above becomes a Jordan Block and there exists a non-singlular matrix A such that C ˆ AKAÿ1 . Although (A.1) above needs to be modi®ed accordingly, the corresponding proof is similar to the one above.  Appendix B. Proof that Eqs. (1)±(6) and Eqs. (7)±(10) are equivalent First, assume that (1)±(6) hold. (i) Add (3) and (4): p1 X …t† ÿ p3 q1 X …t† ‡ p2 W2 …t† ÿ p3 q2 W2 …t† ‡ p3 W3a …t† ‡ p3 W3b …t† P 2d…t†. Then, using (5) we obtain (7). (ii) Multiply both sides of (2) with p3 and subtract from (3): p2 W2 …t† ÿ p3 q2 W2 …t† ‡ p3 W3b …t† ÿp3 W3b …t† 6 d…t† ÿ p3 q2 W2 …t†. This results in (8). (iii) Multiply both sides of (1) by p3 and add to (3): p2 W2 …t† ÿ p3 q2 W2 …t† ‡ p3 W3b …t† ‡ p3 W3a …t† P d…t† ‡ p3 q1 X …t†. Using (5) we obtain (9) (iv) (5) and (6) together result in (10). Next, assume that (7)±(10) hold and de®ne W3a and W3b by (3) and (5). (v) Subtract (3) from (7): …p1 ÿ p3 q1 †X …t† ‡ …p2 ÿ p3 q2 †W2 …t† ‡ p3 W3 …t† ÿ p2 W2 …t† ‡ p3 q2 W2 …t† ÿp3 W3b …t† P d…t†: Using (5), this results in (4). (vi) Subtract (8) from (3): p2 W2 …t† ÿ p3 q2 W2 …t† ‡ p3 W3b …t† ÿ p2 W2 …t† P 0. We obtain (2). (vii) Substitute for W3 …t† using (5) in (9): ÿp3 q1 X …t† ‡ …p2 ÿ p3 q2 †W2 …t† ‡ p3 W3a …t† ‡ p3 W3b …t† P d…t†. Subtract (3) from the above: ÿp3 q1 X …t† ‡ p3 W3a …t† P 0. We obtain (1). (viii) (1), (2), (5) and (10) together result in (6).  References Anderson, E.G., 1997. Managing software implementers in the information services industry: An example of the impact of market growth on knowledge worker productivity and quality. Working paper, Management Department, University of Texas at Austin. Charnes, A., Cooper, W.W., 1959. Chance-constrained programming. Management Science 6, 73±79. Charnes, A., Cooper, W.W., Lewis, K.A., 1978. A multi-level coherence model for EEO planning. TIMS Studies in Management Sciences 8, 13±29. Charnes, A., Cooper, W.W., Niehaus, R.J., Stedry, A., 1969. Static and dynamic assignment models with multiple objectives and some remarks on organizational design. Management Science 15 (8), B365±B375. Charnes, A., Cooper, W.W., Niehaus, R.J., 1970. A generalized network model for training and recruiting decisions in manpower planning. In: Manpower and Management Science. The English Universities Press, London. Denardo, E., Tang, C.S., 1992. Linear control of a markov production system. Operations Research 40 (2), 259±278. Dorroh, J.R., Gulledge, T.R., Womer, N.K., 1994. Investment in knowledge: A generalization of learning by experience. Management Science 40 (8), 947±958. Ebert, R.J., 1976. Aggregate planning with learning curve productivity. Management Science 23 (2), 171±182. Gaimon, C., 1997. Planning information technology ± knowledge worker system. Management Science 43 (9), 1308±1328. Gaimon, C., Thompson, G.L., 1994. A distributed parameter cohort personnel planning model that uses cross-sectional data. Management Science 30 (6), 750±764.

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