Abstract
Crosstraining workers is one of the most efficient ways to achieve flexibility in manufacturing and service systems to increase responsiveness to demand variability. However, it is generally the case that crosstrained employees are not as productive as employees who are originally trained on a specific task. Also, the productivity of the crosstrained workers depend on when they are crosstrained. In this work, we consider a twostage model to analyze the affect of variations in productivity levels of workers on crosstraining policies. Our results indicate that the most important factor determining the problem structure is the consistency in productivity levels of workers trained at different times. As long as crosstraining can be done in a consistent manner, the productivity differences between crosstrained workers and workers originally trained on the task plays a minor role. We also analyze the effect of the variabilities in demand and producivity levels. We show that if the productivity levels of workers trained at different times are consistent, the decision maker is inclined to defer the crosstraining decisions as the variability of demand or productivity levels increases. However, when the productivities of workers trained at different times differ, the decision maker may prefer to invest more in crosstraining earlier as variability increases.
CrossTraining with Imperfect Training Schemes
Burak Büke
School of Mathematics, The University of Edinburgh,
Edinburgh, UK EH9 3JZ
Phone:+441316505086
Fax:+441316506553
Email: B.Buke@ed.ac.uk
Özgür M. Araz
College of Public Health,
Department of Health Promotion, Social and Behavioral Health,
University of Nebraska Medical Center,
Omaha, NE 68198,
Phone:+14025595907
Fax:+14025593773
Email: ozgur.araz@unmc.edu
John W. Fowler
W. P. Carey School of Business, Arizona State University,
Tempe, AZ 85287
Phone:+14809653727
Fax:+14809658629
Email: john.fowler@asu.edu
CrossTraining with Imperfect Training Schemes
Keywords: crosstraining, flexibility, newsvendor networks, productivity
1 Introduction
Designing flexible systems is one of the key strategies to increase responsiveness to variability in the market without sacrificing efficiency of the system. One way to achieve flexibility in a system is to crosstrain workers on several processes. Crosstraining has been proven to be highly beneficial in many different business environments, including but not limited to the semiconductor and automative industries, call centers and healthcare. For example, in the semiconductor industry, machine operators are often crosstrained to run more than one type of sophisticated equipment and technicians are often crosstrained to maintain more than one type of machine. In addition to increasing efficiency, crosstraining can help keep the budgets low, increase a company’s ability to pay more to the employees, to reduce turnover rate, and increase quality due to the workers’ ability to react to unexpected changes (see e.g., Lyons (1992), McCune (1994), Iravani et al. (2007)).
Crosstrained workers can be shifted to work on new tasks when needed, which yields a more efficient usage of the resources. However, it is generally the case that crosstrained employees do not perform equally well as employees who are originally trained on a specific task, i.e., the training schemes may be imperfect. Moreover, the productivity levels of the crosstrained workers may depend on when the crosstraining is done. In this paper, our main goal is to analyze how these differences in productivity levels affect crosstraining decisions. To analyze the effect of imperfect schemes and timing of crosstraining decisions, we consider a twostage model and study the problem in a newsvendor network setting, introduced by Van Mieghem (1998) and Van Mieghem and Rudi (2002). Our model is similar to those of Van Mieghem (1998) and Bassamboo et al. (2010). Similar to the prior work, the decision maker decides on the number of employees to crosstrain, i.e., the level of flexibility, before realizing the demand for the products. In the prior work, the first stage decisions are structural (design) decisions where the level of flexibility is fixed and does not change in the future. However, after demand reveals the decision maker may see that additional crosstraining is beneficial and may wish to increase the level of flexibility. Hence, we extend the two stage model of Van Mieghem (1998) to allow the decision maker to do online crosstraining in the second stage. As argued above, the productivities of the employees who are crosstrained before and after the demand is observed may differ. If the demand exceeds the capacity even after the second stage crosstraining, the excess demand is lost and an opportunity cost is incurred.
Our main conclusion is that the main factor which determines the structure of the crosstraining problem is whether the workers can be crosstrained with the same productivity consistently at different times. As long as the outcomes of the training schemes are consistent, the imperfections play a minor role and the crosstraining decisions with imperfect training exhibit similar properties to the case where training schemes are perfect, and the results for perfect training schemes hold with minor modifications. We show that if the online training schemes are as effective as the offline schemes, the crosstraining policies are independent of the opportunity cost. Similar to the abovementioned work, we provide newsvendortype equations to characterize the optimal crosstraining levels under different scenarios and use these equations to characterize situations when it is not profitable to invest in crosstraining. Then, we investigate how the variability of demand and productivity factors affect crosstraining policies. Under some mild conditions, we prove that the decision maker is inclined to decrease first stage crosstraining when the variance of random parameters increases and the first and second stage training schemes are consistent.
If the first and second stage training schemes are not consistent in productivity, the opportunity cost plays an important role and the structure of the problem changes significantly, i.e., the results for perfect training schemes are no longer applicable. Our findings indicate that this structural difference is mainly due to the loss/gain of “achievable capacity” as a consequence of training workers at different times. To investigate this further, we assume that first stage crosstraining is more effective on expected and test how the demand variability affects crosstraining decisions. We see that for small demand variances, the achievable capacity may be enough to satisfy the demand with high probability even when all the training is done in the second stage. Hence, the opportunity cost plays a minor role and the firststage crosstraining decreases as demand variance increases. However, when demand variance is over a certain threshold, the opportunity cost becomes an important issue and should be avoided by increasing the achievable capacity. Hence, it may be beneficial to increase firststage crosstraining levels as variance increases beyond this threshold. We also study the behavior of crosstraining policies as the variances of the firststage productivity factors change. We observe that there is a critical threshold for the variance such that the firststage productivity factor is less than the second stage productivity factor with a significant probability. Increasing variance of the firststage productivity factor below this critical threshold leads to more firststage crosstraining. On the otherhand, increasing the variance above this critical threshold leads to a decrease in the optimal firststage crosstraining.
The opportunity to benefit from flexibility without too much investment has recently accelerated research in designing efficient flexible systems and we see this work as a part of this research effort. In their seminal paper, Jordan and Graves (1995) show that almost all the benefits of a fully flexible system, where all resources can perform all tasks, can be achieved by using a moderate level of flexibility. Their results demonstrate that using a special flexibility configuration referred to as “chaining” and under certain assumptions on demand, it is possible to obtain 98% of the throughput of a fully flexible system using resources that can perform only two different tasks. Using tools from queueing theory, Jordan et al. (2004) observe that crosstraining can adversely affect performance if a poor control policy is used and demonstrate that a complete chain is robust with respect to the control policy and parameter uncertainty. In the literature, the ability of companies to achieve flexibility and efficiency while at the same time meeting customer needs is sometimes also refered as production agility (Gel et al. (2007); Hopp et al. (2004); Hopp and Van Oyen (2004)). Hopp and Van Oyen (2004) develop a framework for workforce crosstraining, provide a comprehensive review of the recent literature and suggest some future research directions. Hopp et al. (2004) and Gel et al. (2007) analyze flexibility decisions for manufacturing systems operating under CONWIP or WIPconstrained policies and conclude that a crosstrained worker should perform her original task before helping on other tasks. Pinker and Shumsky (2000) perform numerical studies to analyze the tradeoff between efficiency and quality due to crosstraining. Netessine et al. (2002) show how the crosstraining policies are affected by the demand correlation. Davis et al. (2009) indicate that under high workload imbalances, an extensive level of cross training is required to significantly improve the overall production performance. In a service environment, Gnanlet and Wendell (2009) use twostage stochastic programming model to determine optimal resource levels and demonstrate the benefits of crosstraining activities in a health care setting. In their recent papers, Bassamboo et al. (2009, 2010) define level resources to be the resources that are able to process different tasks. In Bassamboo et al. (2009), they prove that for symmetric queueing systems one only need to use dedicated resources and level2 resources. Similar to our analysis, Bassamboo et al. (2010) use the newsvendor network framework to prove that in the optimal flexibility configurations only two adjacent levels of flexibility are needed. Chou et al. (2010) discuss the effect of production efficiency comparing full flexibility with chaining structure. Another paper which is closely related to our work is Chakravarthy and Agnihothri (2005), where they study the optimum fraction of flexible servers for a two task problem with perfect training schemes. They point to the fact that crosstrained workers may not be as efficient as dedicated workers. However, they do not provide an analysis of the problem.
There have been several attempts to formulate the problem as a mathematical program. Brusco and Johns (1998) and Campbell (1999) provide integer programming models for crosstraining workers with different capability levels. Walsh et al. (2000) develop a twostage stochastic program for crosstraining problem in semiconductor industry. Tanrisever et al. (2012) propose a multistage stochastic integer programming model to design flexible systems in maketoorder environments. In the literature, methods including Markov decision processes, mathematical programming and heuristics, are used to schedule crosstrained workers (see e.g., Vairaktarakis and Winch (1999), Sayın and Karabatı (2007) and Campbell (2011)). Our primary focus is on aggregate planning of crosstraining efforts and we do not address the problem of scheduling the workers.
2 A TwoStage Model for CrossTraining
In this section, our goal is to provide a detailed analysis of the crosstraining problem for two tasks when the crosstraining schemes are imperfect. We provide a twostage model to answer how our crosstraining policies are affected by the following factors:

the imperfectness in training schemes

the differences in productivity levels of workers crosstrained at different times

the variability in demand and productivity levels.
We also provide detailed numerical experiments in Section 3.
We analyze the problem of crosstraining workers between two tasks, and . We assume that the capacity is measured in time units and initially there is and units of capacity dedicated to process tasks and , respectively. The decision maker has to develop an aggregate workforce plan by crosstraining some of the available workers before observing the capacity requirements and . Before actual demand is realized, it costs to train one unit of dedicated capacity of to work on task . The crosstrained workers can still work on their original task with full capacity. However, they are not as efficient in their new skill , and their capacity needs to be adjusted by a productivity factor , where ; i.e., if a crosstrained worker spends one hour working on task , it is equivalant to hours of an original worker. The productivity factor, , can also be perceived as an indicator of the effectiveness of the training program and is assumed to be random. After the capacity requirements for tasks are revealed, we first use the dedicated workers and workers crosstrained in the first stage to satisfy the demand. If the available capacity for task is not enough to satisfy the requirement and there is excess capacity for task , additional crosstraining can be performed online at a unit cost of . The productivity factor for the workers crosstrained in the second stage is , where . If the workforce at hand cannot satisfy the capacity requirements even after the second stage crosstraining, the demand is lost incurring a unit opportunity cost of . A similar mechanism works to satisfy the demand for task interchanging the subscripts.
Without loss of generality, we assume that . To simplify our analysis and notation, we also assume that random variables and are continuous random variables with joint density function and use , whenever we do not need to address specific random variables. Throughout this work, a random variable is denoted , denotes the expected value of the random variable. Similarly, we use to denote the expectation over a scenario region .
Now, we need to write out the objective function explicitly based on the mechanism described above. The decision maker initially decides on and , which are the amount of workforce crosstrained from dedicated capacity of and , respectively. We can decompose the cost function into the first stage cost which is incurred due to initial crosstraining and the second stage cost which is revealed after the realization of random parameters. Hence,
(1) 
The second stage cost, , depends on whether new crosstraining is needed after the demand is observed. Hence, takes different forms depending on the realization of parameters. To analyze this function further and calculate the expected value for given , we first use the tower property . To calculate the conditional expectation , we first partition the support of into subsets, where in each subset the nature of the second stage decision is different. Then, we partition the support of and so that the function has a single form within a partition. This partitioning scheme is explained below and the most general graphical representation is given in Figure 1.
We start partitioning the support of productivity factors by considering whether or . This criterion determines the task to allocate a worker crosstrained to work on task , when we need the worker for both tasks. If we allocate the worker for task , our gain will be . If instead, we allocate the worker to task , our gain will be . In the first case, where , it is always more profitable to use the fully productive capacity of task to satisfy when needed. In the second case, where , even if is enough to satisfy the demand , it may be more profitable to shift some of this capacity to task and lose task demand.
2.1 Case 1: Workers Used in Their Original Tasks
Under the condition , task demand will be lost only when the demand cannot be supplied by using the initial capacity and crosstrained workers who are not allocated to task . Since , the same claim is always true for task . Now we ask whether we shall resort to second stage crosstraining. The answer depends on the value of second stage productivity factors and will be the next criterion for partitioning the support of .
If we choose to crosstrain workers in the second stage, we need to spend to satisfy unit demand of task , or we lose the demand and incur a cost of . Hence, for task , we choose to resort to second stage crosstraining first if and we never crosstrain in the second stage if . Similar logic applies to task .
2.1.1 Case 1.a: and .
For this case, losing the demand is preferable over secondstage crosstraining for both tasks. Hence, there will be no second stage crosstraining, and we can use the following partitioning to explicitly state for any given and .

. For the scenarios in , the initial workforce is enough to satisfy the capacity requirements. Hence, on .

. On , the initial workforce cannot satisfy . Task may need to use some workers who are crosstrained to work on , but the remaining crosstrained workforce is enough to satisfy the excess demand for . Hence, again on .

is defined similar to with and interchanged, and on .
We define . When the demand falls in this region, no recourse action is needed in the second stage and hence no cost is incurred.

in Figure 1). The demand for is low so that all the crosstrained workers can be used to work on task . However, even this is not enough to satisfy and since second stage crosstraining is not profitable, demand is lost. Hence, on .

. For scenarios in this subset of the support, some but not all of the workers who are crosstrained to work on task can be shifted to , and this is not enough to satisfy the capacity requirement. The excess demand is lost and hence,

in Figure 1). This region is defined in the same way as where and are interchanged. The second stage cost function is given by

in Figure 1). is the analog of with and interchanged. Hence, on .

in Figure 1). Finally, in this case, the initial capacity is not enough to satisfy the capacity requirement for either task. Hence, excess demand is lost for both tasks. The second stage cost is
Now we are ready to state our first result.
Proposition 2.1.
If then,

any , that satisfies the equation
(2) is an optimal crosstraining level to work on task .

if then is the optimal crosstraining level to work on task .

if then is the optimal crosstraining level to work on task .
Same result holds for when and are interchanged above.
Proof.
We derive the first order optimality conditions by setting the derivatives equal to 0. Then we need to check the second order derivatives to ensure convexity. First, we consider the partial derivative with respect to the decision variable . Since the bounds on productivity factors and do not depend on the decision variables, using Leibniz rule
The second stage cost function is constant with respect to on and and the bounds of these regions do not depend on . Also on , the second stage cost is uniformly equal to 0. Hence, the derivatives of expectation over these regions are all equal to 0. To simplify the notation, we use to denote the density function of demand vector when productivity factors are given. The derivative of expectation over is
Similarly, we can calculate the derivative of expectation over
Canceling the boundary terms, we get
(3) 
The partial derivative with respect to can be found similarly. Setting these terms to equal 0, we get equation (2). Now, we need to show that the () pairs that solve these equations, actually minimizes the expected cost by showing that expected cost function is convex in decision variables. Equation (3) suggests that the first partial derivative of the expected cost function with respect to only depends on , hence we do not need to consider the crosspartials and the hessian matrix is positive semidefinite if the second partial derivatives with respect to and are both nonnegative.
Observe that if , then
Using this relation and the fact that the density function and the productivity factor is always positive, we get
Plugging this back into the second derivative and repeating the same procedure for , we conclude that the hessian is positive semidefinite and the expected cost function is convex.
When the expectation satisfies the inequality for the second case, the derivative of the cost function is negative for any value in . Hence, to minimize the cost, we need to set to the maximum possible value. The third case can be proven similarly.
Since the distribution is assumed to be continuous, the derivative in (3) is continuous in . If the expectation takes both negative and positive values over , then intermediate value theorem ensures us that (2) will be satisfied for some . Hence, the three cases stated in the proposition covers all possible situations. ∎
Proposition 2.1 assumes continuous distributions. However, the results can be extended to discrete setting using a similar methodology to the newsvendor problem.
An important feature of Proposition 2.1 is that it suggests that the problem is separable, i.e., the optimal crosstraining levels for tasks and can be decided separately. Hence, we only state corollaries relating to task below and similar results hold for by interchanging the subscripts. An immediate consequence of the proposition is as follows.
Corollary 2.1.
If the assumption of Proposition 2.1 holds and the firststage productivity factor is deterministically equal to , then

any that satisfies the newsvendortype equation
(4) solves the crosstraining problem.

if , then is the optimal crosstraining level to work on task .

if , then is the optimal crosstraining level to work on task .
Proposition 2.1 also helps us understand when it is not profitable to crosstrain.
Corollary 2.2.
It is not profitable to crosstrain for task , if the assumption of Proposition 2.1 holds and at least one of , or is less than .
Proof.
Using the fact that with probability one, we get
Using the third part of Proposition 2.1 the result follows. To prove the second and third parts of the corollary, we use the same methodology realizing with probability one. ∎
Intuitively, one should not resort to crosstraining if the effectiveness of training programs is not good enough. The first part of the corollary quantifies how “good enough” should be understood. The second part says that before crosstraining one should make sure that there is a solid chance that extra capacity will be needed. Even when extra capacity is needed, demand for the other task may make it impossible to utilize the crosstrained workers.
Under some mild conditions, we can use Proposition 2.1 to infer how the variance of the demand for different tasks affect the crosstraining decisions.
Corollary 2.3.
Suppose and are symmetric random variables around zero, i.e.,
for all values of and . If is deterministically equal to , , the demands for different tasks, and , are independent, continuous and can be written as
then the optimal crosstraining level is nonincreasing in both and .
Proof.
Using equation (4) and independence, we get
Now, we can infer that the probabilities on the lefthand side of the inequality should be greater than 0.5. Then, using the fact that and are symmetric random variables
(5) 
The lefthand side of equation (4) decreases as increases. Hence, if statement 1 of Corollary 2.1 is true, we need to decrease to recover the equality. If statement 2 is true, we may either wish to stay at or we may wish to decrease . For the third statement, we do not need to take any action. Hence, this proves that the optimal crosstraining level is nonincreasing in . Similar arguments show that is nonincreasing in . ∎
Corollary 2.3 essentially states that if demands for products are independent and follow a symmetric distribution, e.g. normal or uniform distributon, and if the costs and productivity factors satisfy the conditions above, the optimal crosstraining level for task is decreasing with the variances of demands. Unfortunately, when is not deterministic, we can construct counterexamples where the monotonicity result does not hold. When is random, the first inequality of (5) may fail to hold for some realizations of and these may force us to increase the crosstraining level as the variance increases. In Section 3, we provide examples to show that we can lose monotonicity, if . The next corollary shows how the crosstraining policies are affected by the variances of firststage productivity factors.
Corollary 2.4.
Suppose is a symmetric random variable around zero with support , and . If , and and are independent, then the optimal crosstraining level is nonincreasing in .
Proof.
Using independence, we can write (2) as
(6) 
First, we note that both the first and second multiplier on the lefthand side are nonincreasing in for any value of . Taking the derivative of the first multiplier on the lefthand side with respect to , we get
Using , we can infer that the first term is nonpositive. Also, as all the integrands in the second term are nonnegative, we can conclude that the derivative is nonpositive. Thus, the first multiplier in (6) is nonincreasing with respect to and if it decreases as increases, we need to decrease to restore the equality in (6). ∎
We construct the proof based on the assumption that crosstraining will not be beneficial in the second stage. However, if the secondstage productivity is preferable, then we might lose the monotonicity of the firststage crosstraining with respect to the variance of the productivity factor. Examples of such cases are demonstrated in Section 3.
2.1.2 Case 1.b: and .
Now, we consider the situation where it is profitable to crosstrain in the second stage. The demand is lost only when the available workforce is not able to satisfy the demand even after crosstraining all the free workers. Hence, the cost function will differ only when there is a tradeoff between the second stage crosstraining and losing demand, i.e., the cost function does not change for and and we only need to consider and further.

. If , both the initial workforce and the crosstrained workforce are used in performing task . If units of the workforce are crosstrained from task , the remaining demand can be satisfied. Hence, the second stage crosstraining cost is

. For the scenarios in , it is not possible to satisfy the demand for even after all the crosstrained workforce work on task . The decision maker crosstrains the idle workforce of and then excess demand is lost. Hence, over
The structure for and is similar and we are ready to state the allocations for this case.
Proposition 2.2.
If and the training effectiveness for programs before and after demand is realized are the same, i.e., with probability one, then optimal crosstraining decisions do not depend on the opportunity cost and

any solution that satisfies the newsvendortype equation
(7) is an optimal crosstraining level to work on task .

if , then is the optimal crosstraining level to work on task .

if