Impact of productivity on cross-training configurations and optimal staffing decisions in hospitals

European Journal of Operational Research 238 (2014) 254–269

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Decision Support

Impact of productivity on cross-training configurations and optimal staffing decisions in hospitals Adelina Gnanlet a,⇑, Wendell G. Gilland b,1 a

Department of Management, Mihaylo College of Business and Economics, California State University Fullerton, 800 N State College Blvd., Fullerton, CA 92834, United States Department of Operations, Technology, and Innovation, Kenan-Flagler Business School, University of North Carolina at Chapel Hill, CB 3490, McColl Building, Chapel Hill, NC 27599, United States b

a r t i c l e

i n f o

Article history: Received 26 March 2013 Accepted 21 March 2014 Available online 3 April 2014 Keywords: Cross-training Productivity Chaining Healthcare Stochastic programming

a b s t r a c t Cross-training of nursing staff has been used in hospitals to reduce labor cost, provide scheduling flexibility, and meet patient demand effectively. However, cross-trained nurses may not be as productive as regular nurses in carrying out their tasks because of a new work environment and unfamiliar protocols in the new unit. This leads to the research question: What is the impact of productivity on optimal staffing decisions (both regular and cross-trained) in a two-unit and multi-unit system. We investigate the effect of mean demand, cross-training cost, contract nurse cost, and productivity, on a two-unit, full-flexibility configuration and a three-unit, partial flexibility and chaining (minimal complete chain) configurations under centralized and decentralized decision making. Under centralized decision making, the optimal staffing and cross-training levels are determined simultaneously, while under decentralized decision making, the optimal staffing levels are determined without any knowledge of future cross-training programs. We use two-stage stochastic programming to derive closed form equations and determine the optimal number of cross-trained nurses for two units facing stochastic demand following general, continuous distributions. We find that there exists a productivity level (threshold) beyond which the optimal number of cross-trained nurses declines, as fewer cross-trained nurses are sufficient to obtain the benefit of staffing flexibility. When we account for productivity variations, chaining configuration provides on average 1.20% cost savings over partial flexibility configuration, while centralized decision making averages 1.13% cost savings over decentralized decision making. Ó 2014 Elsevier B.V. All rights reserved.

1. Introduction The Bureau of Labor Statistics (BLS) in their 2010–2020 employment projections estimated that there will be 1.2 million job openings for nurses by 2020 and Buerhaus, Auerbach, and Staiger (2009) found that the US nursing shortage is projected to grow to 265,000 registered nurses by 2025. Nursing shortages in US hospitals are typically alleviated using cross-trained nurses floating between units, and/or contract nurses hired from an external agency on an ad hoc basis (Hassmiller & Cozine, 2006). Wright and Bretthauer (2010) found a 16.3% reduction in labor costs by coordinating float pool, agency staff, and regular nurses. Gnanlet and Gilland (2009) analyzed the financial benefit in using two

⇑ Corresponding author. Tel.: +1 657 278 4555; fax: +1 657 278 2645. E-mail addresses: [email protected] (A. Gnanlet), Wendell_Gilland@unc. edu (W.G. Gilland). 1 Tel.: +1 919 962 8465; fax: +1 919 962 4266. http://dx.doi.org/10.1016/j.ejor.2014.03.033 0377-2217/Ó 2014 Elsevier B.V. All rights reserved.

resource flexibilities (staffing flexibility and patient upgrades) under different types of decision making. These benefits vary based on the type and breadth of crosstraining. The benefit of using various cross-training configurations, such as partial cross-training, chaining, cherry-picking, and total cross-training, has widely been evaluated in service and manufacturing applications (Brusco & Johns, 1998; Brusco, 2008; Easton, 2011; Hopp, Tekin, & Oyen, 2004; Iravani, Oyen, & Sims, 2005; Jordan & Graves, 1995). However, as cross-training breadth increases (number of additional skill sets a worker is trained for), cross-trained workers may be less efficient than specialists. Pinker and Shumsky (2000) analyze a system with specialist and flexible servers when there is a trade-off between the efficiency of the specialist and the quality of the flexible servers. Increasing flexibility (cross-training breadth) may have significant impact on productivity and costs (Campbell, 1999; Karuppan, 2006; Pinker, Lee, & Berman, 2010; Stratman, Roth, & Gilland, 2004). Schultz, McClain, and Thomas (2003) experimentally show the negative effects in worker productivity because of work

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interruptions when using flexible workers. Despite the potential benefits of cross-training, the changing work environment can create stress, tension, negative attitude towards floating, and dissatisfaction. These arise from the fact that cross-trained nurses are not familiar with the physical layout, unit culture, and work relationship with unit nurses. In some cases float nurses are not confident in their skills and are apprehensive in taking on the responsibility for a patient on a different unit (Dzuiba-Ellis, 2006). Karuppan (2006) shows that too much cross-training may not be good for complex tasks. Therefore, due to learning and forgetting effects (Shafer, Nembhard, & Uzumeri, 2001; Stratman et al., 2004), changes in unit layout, protocols, and support system in float unit (Dzuiba-Ellis, 2006), cross-trained staff are frequently not as effective as specialists. This raises the question of the impact of productivity on crosstraining configurations: is more cross-training always better when productivity varies? Our paper focuses on the following research questions: what is the impact of productivity on optimal staffing levels (both regular and cross-trained) and the financial benefits for hospitals under different types of decision making? What is the marginal benefit between different cross-training configuration, when productivity of cross-trained nurses varies? We evaluate the impact of productivity of cross-trained nurses on different cross-training configurations in two-unit and multiunit systems. We consider two units that have full flexibility (both units are cross-trained to the other unit), three units that have limited cross-training (partial flexibility – two units are cross-trained to one other unit) and three units that have a minimal complete chain (chaining – each unit is cross-trained to one other unit and no unit receives nurses from two other units) to study the impact of productivity under decentralized and centralized decision making. Hospitals with centralized decision making decide, simultaneously the number of regular nurses and cross-trained nurses; while hospitals with decentralized decision making first determines, the total staffing available for each unit, and later nursing managers for the floor decide on the number of nurses to crosstrain. We formulate and analyze the three cross-training configurations, under two types of decision making, using two-stage stochastic programming, with recourse in the second stage. The first stage decision is the staffing (regular and/or cross-trained nurses) in each unit. We consider general, continuous demand distributions for two and three non-homogenous units, and prove convexity of the objective function with respect to staffing variables for the two-unit system. We derive closed form expressions for the optimal amount of cross-training under the two-units, full-flexibility configuration. When cost of cross-training is high, an increase in productivity leads to an increase in the amount of cross-training. When cost of cross-training is relatively low, however, there is a productivity level beyond which further increases in productivity reduce the amount of cross-training. We derive an analytical expression for the threshold productivity under the two units and numerically evaluate the existence of threshold productivity for other cross-training configurations. Our results show that chaining (complete minimal chain) provides on average 1.20% cost savings compared to partial flexibility (one less cross-training link) even after accounting for unit-level productivity differences. Centralized decision making provides on average 1.13% cost savings compared to decentralized decision making. Our contribution to the literature is twofold: (1) we determine the optimal number of regular and cross-trained staff by accounting for productivity changes; and (2) we evaluate the marginal benefit of different cross-training configurations when productivity varies. This paper is organized as follows. The next section presents the literature review and discusses how our model differs from the staffing and cross-training literature. Section 3 introduces and

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formulates our model of cross-training and scheduling decisions. The model is analyzed in Section 4 as a two-stage stochastic program with recourse. Section 5 shows numerical analysis and highlights its implications for management. Section 6 concludes the paper and discusses possible extensions.

2. Literature review Cross-training in supply chain has widely been used to balance the work load in an assembly system in order to maximize throughput. Hopp et al. (2004) analyze two different cross-training structures, skill chaining and cherry-picking, for a serial production system. They find that when capacity is fairly imbalanced but variability is low, the cherry-picking approach can be used. Jordan, Inman, and Blumenfeld (2004) evaluate the performance of three cross-training configurations in parallel systems using queueing theory and simulation. They conclude that complete chaining gives the maximum benefit and is also robust (Jordan & Graves, 1995). Iravani et al. (2005) find a structural flexibility index that chooses the best pattern among all the alternative patterns of flexibility in parallel systems without having to evaluate all the patterns. Vairaktarakis and Winch (1999) develop heuristics for scheduling work orders through assembly systems so that the cross-training costs are minimized when multi-skilled workers are used. Agnihothri, Mishra, and Simmons (2003) balance the trade-off between customer delay cost and premium for flexibility and models a queueing system to determine the mix of dedicated and crosstrained servers for two job types using simulation, and extend this work to three job types in their 2004 paper Agnihothri and Mishra (2004). Brusco and Johns (1998) present an ILP to evaluate crosstraining configurations for a multi-skilled work force. Brusco, Johns, and Reed (1998) minimize the total number of labor for two skill classes considering productivity. Campbell (2011) develops two stage-stochastic programming for cross-trained nurse allocation and finds that cross-training policies perform better than perfect information under high demand uncertainty. The literature on cross-training policies indicates that total cross-training (each unit has nurses cross-trained to every other unit) can be expensive and time consuming, and benefit obtained may be low due to learning and forgetting effects (Easton, 2011; Hopp et al., 2004; Iravani et al., 2005; Jordan & Graves, 1995). Aksin, Karaesmen, and Ormeci (2007); Batta, Berman, and Wang (2007); Sayin and Karabati (2007); Iravani, Kolfal, and Van Oyen (2007) have discussed in detail the effects of cross-training of agents in call centers and its impact on performance. The existing literature focus mainly on finding the best configuration or flexibility structure (level of cross-training) in different scenarios but do not optimize the amount of cross-training needed for worker. In our paper, we not only determine the optimal staffing but also determine the effects of lowered productivity. Researchers have identified optimal nurse scheduling models considering overtime for regular nurses, use of cross-trained nurses and agency nurses (Bard & Purnomo, 2005; Brusco, 2008; Inman, Blumenfeld, & Ko, 2005; Li & King, 1999; Siferd & Benton, 1992). Bard and Purnomo (2005) formulate an integer programming model considering a 24-hour schedule for regular nurses, float nurses, and outside nurses. The changes in patient acuity and census are updated at the start of each shift and a 24-hour schedule is generated. Li and King (1999) develop a goal programming approach for optimizing the cross-trained staff for subdivided tasks in health care. Inman et al. (2005) use number of shifts where contract nurses are used as their objective to compare different types of cross-training policies for nurses. They simulate cross-training policies considering patient census as poisson arrivals and absenteeism rate of nurses following a binomial

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distribution. All these papers indicate the benefits of using flexible nurses either a priori (Inman et al., 2005; Wright & Bretthauer, 2010) or ad hoc (Bard & Purnomo, 2005). In a recent paper, Wright and Mahar (2013) develop a centralized scheduling system wherein a pool of cross-trained nurses are assigned to different units such that overtime cost and wages are minimized. In our paper, we optimize for the total number of regular and crosstrained nurses required, while Wright and Mahar (2013) optimize the operational assignment of the available pool of nurses. Pinker and Shumsky (2000) analyze a system with specialist and flexible servers when there is a trade off between the efficiency of the specialist and the quality of the flexible servers. McClain, Schultz, and Thomas (2000), show that work-in-process inventory has a significant effect on the productivity of workers when there is work sharing in a serial system. Pinker et al. (2010) and Karuppan (2006) discuss that too much flexibility maybe constraining under different conditions. In our paper, we determine the optimal staffing pattern, which includes regular staff, cross-trained nurses and contract nurses, knowing that patient needs are stochastic in the future and accounting for productivity changes among nurses. We identify the benefits of making a priori staffing and cross-training decisions assuming that demand follows a general, continuous distribution in future time periods. The nursing literature has also evaluated the impact of higher nurse staffing on patient safety outcomes (Czaplinski & Donna, 1998; Hall, 2003; Hall, Doran, & Pink, 2004; Heinz, 2004; Mark, 2004; McCue, Mark, & Harless, 2003; Robertson, Dowd, & Mahmud, 1997; Shullanberger, 2000). Recently, Bae, Mark, and Fried (2010) empirically investigated the effects of using temporary nurses, which includes cross-trained and agency nurses, on quality of patient care. Units that used more than 15% temporary nurses reported higher patient falls and higher back injuries for nurses.

3. Problem definition and model formulation In this section we define and formulate three types of crosstraining configurations (full flexibility for two units; partial flexibility and chaining for three units) depicting two types of decision making (decentralized and centralized) scenarios. We consider non-homogeneous units within a hospital, each specialized yet similar enough to cross-train. Each unit faces a general, continuous demand distribution and is staffed using three types of nurses: regular nurses are assigned to their home unit (regular nurses who are not cross-trained are referred to as dedicated nurses), cross-trained nurses (also referred to as float nurses) can be assigned to other units based on demand and configuration, and contract nurses are hired to meet excess demand. Regular and cross-trained nurses are full-time employees of the hospital and are paid wages regardless of whether they are needed or not. If patient demand is still not met using hospital employees, contract nurses from an outside agency are hired at a higher cost than the wages of regular nurses and cross-trained nurses. We are using the term ‘contract nurses’ to broadly refer to any temporary nurse who can be hired for a short-time period until demand stabilizes. This may include an agency nurse, temporary nurse, travel nurse, or even nurses on call who command higher wages. Depending on the type of configuration and relevant costs, a combination of regular, cross-trained, and contract nurses are used to meet stochastic demand as described here. In the first configuration with two non-homogenous units, nurses in unit i are cross-trained for unit j and vice versa. As shown in Fig. 1: Two-units, Full-flexibility, regular nurses in unit i who are not cross-trained (called dedicated nurses) are allocated to their home unit. To meet additional demand in unit i, cross-trained nurses from unit j not needed in their home unit are allocated

Fig. 1. Cross-training configurations.

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to unit i. Any remaining demand for unit i is met using contract nurses. A three unit system increases the number of possible crosstraining configurations. Iravani et al. (2005) present all the different types of cross-training configurations. The number of links for a 3 unit system can range anywhere from one (when only one unit is cross-trained) to six (when completely cross-trained). With the criteria of minimizing the number of cross-training links, since more links leads to higher cross-training cost and lower productivity, for the purpose of consistency between two units and three units, and to understand the impact of productivity when all units are cross-trained to only one additional unit, we analyze a partial flexibility configuration and a chaining configuration for the three unit scenario. Chaining has been shown to provide most of the benefit of flexibility obtained from a full flexibility model (Easton, 2011). The marginal benefit obtained from complete cross-training is often not worth the extra cost involved in training and productivity losses. In order to evaluate the value of chaining compared to a partial flexibility configuration that has similar structure to chaining but one less cross-trained unit, we consider a partial flexibility configuration system as presented in Fig. 1: Three-units, Partial-flexibility, where nurses are cross-trained to one additional unit except for the last unit. Nurses in unit 1 are cross-trained for unit 2; nurses in unit 2 are cross-trained for unit 3, while nurses in unit 3 are not cross-trained. In this configuration, unit 2 is the only unit that can float-in (from unit 1) and float-out (to unit 3), unit 1 floats-out nurses to unit 2, and unit 3 can float-in nurses from unit 2. Under the chaining configuration (see Fig. 1: Three-units, Chaining), in addition to cross-trained nurses in unit 1 and 2, nurses in unit 3 are cross-trained for unit 1 thus completing the chain. Units are labeled 1, 2 and 3 for the purpose of readability; there are no inherent ordering/dependencies that may affect modeling and analysis. The allocation process to meet stochastic demand under partial flexibility and chaining is explained below. Regular nurses are used to meet their unit demand, followed by available cross-trained nurses from the other unit, and finally contract nurses are used to meet any remaining demand. Similar to the allocation policy in two-units, full-flexibility, when all units face high demand, regular nurses who are not cross-trained in unit i (dedicated nurses) are first allocated to the unit followed by regular nurses in that unit who are cross-trained. When desirable, we chain the process of allocating cross-trained nurses between units to obtain the maximum benefit of flexibility. For example, to satisfy high demand in unit 3, cross-trained nurses in unit 1 are floated to unit 2 so that cross-trained nurses in unit 2 can be floated to unit 3. Dedicated and cross-trained nurses, due to their tenure and familiarity with the unit, have high productivity in their home unit. On the other hand, cross-trained nurses assigned to the other unit are less familiar with the protocols, unit work environment, unit culture, and unit nurses. Therefore, they are not as productive as the nurses dedicated to that unit. A productivity parameter (qji ) is used to capture this effect for nurses floating from unit j to unit i; i – j. Productivity is measured as the ratio of time taken by a dedicated nurse to do a task to the time taken by a cross-trained nurse to do the same task. We assume that the work quality is the same for cross-trained nurses and dedicated nurses in their home unit. The productivity parameter (qji ) varies from 0 to 1; 0 implies that cross-trained (float) nurses cannot perform any duties in the float unit and 1 implies that they are as productive as the home unit (dedicated) nurses. The productivity of contract nurses is assumed to be 1 since they are multi-skilled and experienced in a variety of tasks. If their productivity is less than 1, the model analysis will remain the same provided the contract nurse cost (si ), is inflated by the productivity of the contract nurse. Notation used in our model formulation is explained below.

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Cost parameters wages for staff in unit i per hour hi training cost for cross-trained nurses in unit i ti per hour si wages for contract nurses in unit i per hour Productivity parameter qji productivity of cross-trained nurse from unit j who floats to unit i Staffing decisions zi number of total staff in unit i ui proportion of cross-trained nurses in unit i ni ¼ ð1  ui Þ  zi number of dedicated nurses in unit i ei ¼ ui  zi number of cross-trained nurses in unit i Staffing allocations xii nurses from home unit i allocated to unit i xji productivity-adjusted cross-trained nurses from unit j allocated to unit i Demand parameters Ui demand distribution (of nursing hours required) for each unit i per hour realization of stochastic demand di i; j ¼ 1; 2; 3; i – j (for three unit models the combination is based on the assumed configuration)

We consider two and three non-homogenous hospital units facing stochastic demand (Ui , for i = 1, 2, 3). Each unit has a pool of nurses (zi ) who are divided into dedicated nurses (ð1  ui Þ  zi ) and cross-trained nurses (ui  zi ). Based on the demand, xii nurses are allocated from home unit i to unit i. Additional demand is met by cross-trained nurses in unit j floating to unit i (xji ). Any excess demand not met either by dedicated or cross-trained nurses is met by contract nurses at a cost of si for unit i. In all three configurations (two-units, full-flexibility; three-units, partial-flexibility; and three-units, chaining), dedicated nurses are assigned to their home unit and cross-trained nurses are either assigned to their home unit or floated to the other unit. In the cases of partial flexibility and chaining, we leverage the benefit of chaining by sequentially allocating cross-trained nurses. For example, if demand in unit 3 is very high, cross-trained nurses in unit 1 are allocated to unit 2, which frees up cross-trained nurses in unit 2 to work in unit 3. In a hospital located in the southeastern United States, administrators decided to implement a cross-training program by training nurses in more than one skill set. Cross-training includes shadowing a regular nurse for a few weeks and demonstrating necessary skills on the new unit. The total staff for the hospital was decided when the unit was constructed and the number of crosstrained nurses was decided later when hospital administrators implemented the cross-training programs. Hospital administrators decided on the number of nurses to be cross-trained, constrained only by the available staff in each unit. We classify such a situation as decentralized decision making. Under decentralized decision making (see Fig. 2), hospital administrators determine the optimal staffing (zi ) for each unit based on stochastic demand. Later, given the optimal staffing in each unit (zi ), the unit planners determine the proportion of nurses to be cross-trained (ui ) to meet stochastic demand at minimum cost. Under the second type of decision making, called centralized decision making as shown in Fig. 2, hospital administrators make all staffing decisions (i.e., choosing the number of dedicated nurses and cross-trained nurses) simultaneously. Total staffing (zi ) and proportion of nurses to be cross-trained (ui ) are decided at the same time.

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Our model does not account for layoffs and absenteeism of regular nurses. It does not explicitly consider on-the-job regular training cost, but this can be added to base wages for regular nurses (hi ) and the model analysis will still hold. The cross-training cost is amortized into a per-hour cost ti . Contract nurses are employed at a cost of si per hour for unit i. The cost of hiring a contract nurse for an hour is higher than wages per-hour and per-hour cost of cross-training a nurse (si P hi þ ti ). Our models are formulated using two stage stochastic programming, with recourse Birge and Louveaux (1997) in the second stage. In the first stage, we decide the staffing levels and crosstraining based on costs and demand distributions, and in the second stage nurses are assigned to units based on actual demand realization. Demand in both units follow general, continuous distributions.. Stage 1: The first stage objective function minimizes the sum of wages, cross-training cost (if applicable) and expected contract nurse cost. Decision variables for this model are total staff and number of nurses to be cross-trained (depending on decision making model). The resource constraints are boundary values for the decision variables. Stage 2: In the second stage, once demand is realized, dedicated and cross-trained nurses are assigned to each unit. Any excess demand is met using contract nurses. All second stage formulations minimize contract nurse cost with resource constraints. Second stage decisions for all types of decision making are convex in their objective functions, so optimal allocations are determined using first order conditions. The expected value of the second stage objective function is then substituted into the first stage objective function to obtain optimal staffing levels. In the following sub-sections, we formulate the three configurations (two-units, full-flexibility; three-units, partial-flexibility; three-units, chaining) under decentralized and centralized decision making models and present the optimal staffing allocations for the second stage.

required that qij ¼ qji , for example unit i could be a sophisticated unit compared to unit j and so nurses cross-trained to work in unit j may find it easier to work in unit j compared to nurses working in unit i from unit j. 3.1.1. Decentralized decision making With decentralized decision making, the first stage has two periods. In the first period, optimal total staffing (zi ) for each unit is determined as a newsvendor solution. The optimal total staffing (zi ) represents dedicated nurses hired without the knowledge that nurses are going to be cross-trained in the future. Any excess demand is met using contract nurses at a cost of si . Given optimal total staffing from period 1 (zi ), the proportion of nurses to be cross-trained (ui ) is determined in the second period. Period 1:

Minzi

X X þ ðhi  zi Þ þ EUi ½si ðdi  zi Þ  i

ð1Þ

i

s:t: zi P 0; i ¼ 1; 2 In the second period, given the newsvendor solution for optimal staffing (zi ) from period 1, the proportion of cross-trained nurses are determined based on the formulation given in Eq. (2). Since there are only two units in this configuration we use subscript i to represent a unit and j to represent the other unit where crosstrained nurses are pulled from and vice versa. Period 2:

Minui

X X þ ðt i  ui  zi Þ þ EUi ½si ðdi  xii  xji Þ  i

s:t: 0 6 ui 6 1;

i

ð2Þ

i; j ¼ 1; 2; i – j

where þ þ xii ¼ minðdi ; zi Þ; xji ¼ min½ðdi  zi Þ ; qji  minððzj  dj Þ ; uj  zj Þ, where xii is the optimal allocation of regular nurses (both dedicated and cross-trained) from unit i to unit i and xji is the optimal allocation of productivity-adjusted cross-trained nurses from unit j to unit i. The second stage objective function is linear in its decision variables and has linear constraints and so the second stage objective function is convex with respect to staffing allocation variables xii , xji .

3.1. Two-units, full-flexibility In two-units, full-flexibility configuration as shown in Fig. 1, nurses are cross-trained in both units. Cross-trained nurses in their home unit have a productivity of 1 and when they float to another unit their productivity levels (qij ) are between 0 and 1. It is not

3.1.2. Centralized decision making In centralized decision making, hospital planners simultaneously determine the optimal staffing in each unit (zi ) and the proportion of nurses to cross-train from each unit (ui ) considering wages, training, and expected contract nurse cost.

Fig. 2. Timelines of decision making.

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Minzi ;ui

" # X X   þ ðhi  zi þ t i  ui  zi Þ þ EUi si ðdi  xii  xji Þ ; i

i

s:t: zi ; ui P 0; ui 6 1; i; j ¼ 1; 2; i – j þ

þ

where xii ¼ min½di ; zi ; xji ¼ min½ðdi  zi Þ ; qji  minððzj  dj Þ ; uj  zj Þ, where xii is the optimal allocation of regular nurses (dedicated and cross-trained) from unit i to unit i and xji is the optimal allocation of productivity-adjusted cross-trained nurses from unit j to unit i. The first stage objective function is the sum of wages for all nurses, cross-training costs, and expected contract cost when demand exceeds dedicated nurse and cross-trained nurse allocations. After demand is realized, available nurses are assigned optimally so that the cost of hiring contract nurses is minimized. The second stage objective function is linear and has linear constraints and so the second stage objective function is convex in its allocation variables. 3.2. Three-unit, partial-flexibility In partial flexibility configuration of three units as shown in Fig. 1, nurses from units 1 and 2 are cross-trained for one additional unit. Unit 3 nurses are not cross-trained. Units are labeled 1, 2, and 3 for the purpose of readability; there are no inherent ordering/dependencies that may affect modeling and analysis. 3.2.1. Decentralized decision making In decentralized decision making, decisions regarding total staffing and proportion of cross-trained nurses are made at different time periods and so there are two periods under first stage optimization. The optimal total staffing (zi ) is determined in first period using newsvendor solution similar to two-units, full-flexibility configuration. Period 1:

Minzi

3 3 X X þ ðhi  zi Þ þ EUi ½si  ðdi  minðzi ; di ÞÞ  i¼1

i¼1

Minui

ti  ui 



þ EUi

i¼1

X i;j2fð1;3Þ;ð2;1Þ;ð3;2Þg

s:t: x31 ¼ 0

ð3Þ

xii ¼

3.3.1. Decentralized decision making In decentralized decision making, optimal total staffing (zi ) for all three units are obtained using newsvendor solution in period 1 (see Eq. 5), without any knowledge of future cross-training. Unmet demand is filled by contract nurses at higher cost. Period 1:

Minzi

3 X

ðhi  zi Þ þ EUi

3 X þ ½si ðdi  minðzi ; di ÞÞ 

ð5Þ

i¼1

In period 2, given the total optimal staffing (zi ) for all three units, the optimal proportion of nurses to be cross-trained (ui ) is determined using Eq. (6). Period 2:

Minui

3 X   t i  ui  zi þ EUi

X

h i þ si ðdi  xii  xji Þ

i;j2fð1;3Þ;ð2;1Þ;ð3;2Þg

ð6Þ

s:t: 0 6 ui 6 1; i ¼ 1; 2; 3 ð4Þ

where



In this configuration (as shown in Fig. 1), unit 1 nurses are cross-trained for unit 2, unit 2 nurses are cross-trained for unit 3, and unit 3 nurses are cross-trained for unit 1.

i¼1

 þ ½si di  xii  xji ;

0 6 ui 6 1; i ¼ 1; 2 (

3.3. Three-units, chaining

i¼1

In the second period, given optimal total staffing (zi , i = 1, 2, 3) from period 1, the proportion of nurses to be cross-trained (ui , i = 1, 2) is determined for units 1 and 2 as given in Eq. (4). Period 2:

zi

3.2.2. Centralized decision making In centralized decision making, both total staffing and proportion of nurses to be cross-trained are decided simultaneously (similar to the centralized decision making of two units, full flexibility) considering wages for nurses in all three units, cross-training costs for nurses in unit 1 and 2, and expected contract nurse cost to meet unmet demand in all three units. The allocation of regular, crosstrained and contract nurses in the second stage is based on the partial flexibility configuration similar to the formulation shown in decentralized decision making. The complete formulation for centralized decision making, partial flexibility is presented in Eq. (27) in the Appendix A.

s:t: zi P 0; i ¼ 1; 2; 3

s:t: zi P 0; i ¼ 1; 2; 3

2 X 

in unit 3, the benefit of unit 1 cross-trained nurses can be leveraged by allocating them to unit 2 thereby freeing up cross-trained nurses in unit 2 to work for unit 3 as presented in optimal allocation for x23 . The second stage objective function is linear in its decision variables and has linear constraints and so the second stage objective function is convex with respect to staffing allocation variables xii , xji .

 

min di ;zi ; i ¼ 1;3    þ min di ; ð1ui Þzi þmin½ui zi ; di ð1ui Þzi xji ; i;j 2 f2;1g

where þ

xii ¼ min½di ; ð1  ui Þzi  þ min½ui zi ; ðdi  ð1  ui Þzi  xji Þ  h h ii þ xji ¼ qji  min ðdi  ð1  ui Þzi Þ ; min ðdj  ð1  uj Þzj  xkj Þ; uj zj ; i; j; k 2 fð1; 3; 2Þ; ð2; 1; 3Þ; ð3; 2; 1Þg

8 0; i ¼ 1;j ¼ 3 > >   > þ > <  þ    ; i;j 2 f2;1g q  min ðd  ð1  u Þz Þ ;min z  d ;u z i i j j ji i j j xji ¼ > > h h ii >  > : q  min di  z þ ;min uj z ;ðdj  ð1  uj Þz  xkj Þþ ; i;j; k 2 f3;2;1g ji i j j

where xii is the optimal allocation of regular nurses (dedicated and cross-trained) from unit i to unit i and xji is the optimal allocation of productivity-adjusted cross-trained nurses from unit j to unit i. The second stage objective function is linear in its decision variables and has linear constraints and so the second stage objective function is convex with respect to staffing allocation variables xii , xji .

where xii is the optimal allocation of regular nurses (dedicated and cross-trained) from unit i to unit i and xji is the optimal allocation of productivity-adjusted cross-trained nurses from unit j to unit i. In this model, unit 3 nurses are not cross-trained to unit 1 and so x31 is constrained to be zero. When cross-trained nurses are needed

3.3.2. Centralized decision making In centralized decision making, both optimal staffing (zi ) and optimal proportion of cross-trained nurses (ui ) for all three units. The complete formulation for centralized decision making, chaining is shown in Eq. (28) in the Appendix A.

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4. Analytical analysis and discussion

4.2. Centralized decision making

This section presents the analysis of the first stage objective function for the two unit configuration under decentralized and centralized decision making, and discusses the analytical insights that can be derived. The expected value of the second stage objective function (expected contract nurse cost) is utilized in the first stage objective function. The Hessian of the first stage objective function is strictly diagonally dominant for all types of decision making for two-units, full-flexibility, so the first stage objective function is convex with respect to the decision variables of interest (see Appendix A for proof). We use first order conditions of the first stage objective function to get optimal values for staffing variables.

The Hessian matrix for the first stage objective function (K1 ðzi ; ui Þ) is strictly diagonally dominant and the first stage objective function is convex in all its decision variables (zi and ui ) for two-units, full-flexibility (see Appendix A for analysis of first stage objective function, Hessian, and proof of convexity). First order conditions given in Eq. (10) and in Eq. (11) are used to determine   the optimal    total staffing zi and proportion of cross-trained nurses ui for unit i.

4.1. Decentralized decision making

@ K1 ðzi ; ui Þ ¼ ðhi  si Þ  si Uj ½zj  @zi  h i h  i  si 1  Ui zi þ qji  zj  uj Uj 1  uj  zj Z z  h i   j  þ si 1  Uj zj Ui zi  s1  1uj zj

In this section we consider two units that have full flexibility. In the first period, there are no cross-trained nurses (only total staff). We obtain a newsvendor solution (see Eq. (7)) to obtain optimal level of staffing (zi ).

Ui ½zi  ¼

si  hi si

    1  Uj ½zj þ qij zi  di dUi ¼ 0

zi ð1ui Þzi

ð10Þ

ð7Þ

In period 2 we determine the optimal proportion of cross-trained nurses for both units. First order conditions are used to select optimal values of (ui ) given optimal total staffing (zi ) from period 1. (See Appendix A for first order conditions, analysis of Hessian, and proof of convexity.) Eq. (8) gives the closed form expression for the optimum amount of cross-training for each unit i when demand follows a general, continuous distribution.

    Ui 1  ui zi 1  Uj ½qij  ui  zi þ zj  ¼

Z  h  i  1  Ui zi þ qji zj  dj dUj  sj  qij

ti ; qij  sj

i; j ¼ 1; 2; i – j ð8Þ

Given staffing levels, demand distributions, productivity level, cross-training cost, and contract nurse cost, we can determine the optimal number of cross-trained nurses in each unit. Substituting the optimal staffing (zi ) Eq. (7) from period 1 into closed form Eq. (8) for period 2, we obtain the following condition. 







    si  h i 1 si  hi 1 sj  hj  Ui 1  ui U1 U q  u  U U 1  þ j ij i i i j si si sj ti ; i; j ¼ 1; 2 ¼ qij  sj

ð9Þ Looking at Eq. (8), we can draw conclusions regarding the impact of parameter values on staffing decisions. We see that when training cost per  period ðt i Þ increases, the optimal proportion of cross-training ui decreases, as expected. An increase in contract nurse cost of unit j (sj ) results   in an increase in optimal proportion of cross-training in unit i ui . Additional observations are:   1. Optimal amount of cross-training in unit i ui does not depend on the contract cost of unit i (si ).   2. Optimal amount of cross-training in unit i ui does not depend on the training cost per period of unit j (t j).  3. Optimal amount of cross-training in unit i ui does not depend on the productivity of nurses floated from unit j to unit i (qji ). Under partial flexibility and chaining when extending to three units, we find that optimal total staff (zi ) in period 1 is determined using newsvendor solution as shown in Eq. (7) and the optimal proportion of cross-trained nurses (ui ) in period 2 are determined using first order conditions since the hessian for first stage objective function is convex in its decision variables.

 h i h  i @ K1 ðzi ; ui Þ ¼ t i  si 1  Ui zi þ qji  zj  uj Uj 1  uj  zj @ui  si ð1  Ui ½zi Þð1  Uj ½zj Þ  sj Ui ½ð1  ui Þ  zi ð1  Uj ½zj Z z i þ qij  zi  ui Þ  si ð1  Ui ½zi þ qji ðzj  dj ÞÞdU2  sj  qij

Z

ð1ui Þzi

zj

ð1uj Þzj

ð1  Uj ½zj þ qij ðzi  di ÞÞdUi ¼ 0

ð11Þ

Under partial flexibility and chaining, complete enumeration of the search space indicate that the first stage objective function is convex in its decision variables but it was hard to prove analytically. 4.3. Analytical result: Threshold productivity Under any type of decision making and all three configurations, ^ij ) beyond which we find that there exists a productivity level (q further increases in productivity reduce the level of cross-training. Above this threshold, the productivity of cross-trained nurses allow the benefits of flexibility to be obtained with fewer crosstrained nurses. We analytically derive the equation for threshold productivity (Eq. 12), assuming uniform demand distribution ½ai ; bi  for two-units, full-flexibility decentralized decision making, while numerically prove the existence of threshold productivity under other models. 2

^ij ¼ q

sj ðbj  zj Þ þ 4  t i ðbi  ai Þðbj  aj Þ 2  sj ðzi  ai Þðbj  zj Þ

^ij P 1; ¼ 1; if q

i; j ¼ 1; 2;

i–j

;

^ij < 1; i; j ¼ 1; 2; i – j; if q ð12Þ

^ij and conditions under which it holds is The proof for deriving q ^ij P 1, it indicates given in the Appendix A. When the calculated q that the proportion of cross-training always increases as productivity increases. Higher cross-training cost (t i ) increases the likelihood ^ij P 1. of q For centralized decision making in two units and both types of decision making in three units we graphically observe the thresh^ij ) when plotting the optimal proportion of old productivity (q cross-trained nurses against productivity. We observe in all our numeric studies that there exists a threshold productivity beyond which a smaller percentage of nurses are cross-trained. Fig. 3 ^ij for decentralized and centralshows the threshold productivity q

A. Gnanlet, W.G. Gilland / European Journal of Operational Research 238 (2014) 254–269

261

Fig. 3. Threshold productivity when cross-training cost changes.

ized decision making for different levels of cross-training costs under all three configurations. Fig. 4 shows the threshold produc^ij for similar scenarios when contract nurse cost increases. tivity q The threshold value increases with an increase in contract nurse cost. Full numerical results are presented in Section 5. 5. Numerical analysis and discussion In this section, we conduct numerical experimentation using costs obtained from a large hospital located in the southeastern region of the United States. The purpose of numerical experimentation is to obtain insights that cannot be proved analytically. Our numerical analysis compares decentralized and centralized decision making, among three models (full flexibility, partial flexibility, and chaining). 5.1. Parameters To compare the performance of centralized and decentralized decision making in two and three hospital units, we conduct a full factorial study with three levels of contract nurse cost and crosstraining cost, and two levels of mean demand. The cost parameters and demand distributions are given in Table 1. To estimate wages, cross-training cost, and contract nurse cost for dedicated and cross-trained nurses, we interviewed nurses and directors in the nurse staffing and planning departments at a large (over 800 licensed beds) academic hospital. Registered nurses (RNs), who are represented as dedicated nurses in our model, receive an hourly wage between $30 and $35 (excluding benefits) at this

hospital. We chose RNs to represent dedicated and cross-trained nurses since LPNs and NAs are not frequently floated. Hourly wages for dedicated nurses is used as a base cost parameter and hence we selected only one value ($30/hour) for hi . Cross-trained nurses incur cross-training cost in addition to hourly wages. Cross-trained nurses are checked to ensure that they have the necessary skill set to float to another unit. Units that have similar patient care typically have a short training schedule, resulting in low cross-training costs. Some units organize shadowing of nurses as part of training and assign preceptors (mentors who train nurses in a new skill set), resulting in higher cross-training cost. To account for variations in training cost between units, we considered three levels of cross-training cost per hour ($1, $3, and $5). When dedicated and cross-trained nurses are not available, contract nurses are hired from a third party agency to work when needed for a short period of time, and contract nurses have the flexibility to work in multiple units. Therefore, contract nurse costs are higher than the wages for regular nurses and the total cost for cross-trained nurses (wages plus cross-training cost). Depending on the type of unit and floor, contract nurse costs vary and so we considered three levels of cost ($45, $60, and $75). The cost parameters are given in Table 1. For the purpose of numerical analysis, we assume uniform demand distribution ½ai ; bi ; i ¼ 1; 2; 3 for stochastic demand in unit i. For two unit, full flexibility configuration we obtained first order conditions using a general, continuous demand distribution and so insights derived from uniform distribution are generalizable. Demand for unit i is measured as full-time-equivalent (FTE) nurses calculated based on patient care requirements. The FTEs needed

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Fig. 4. Threshold productivity when contract cost changes.

Table 1 Parameters for numerical study. Parameters

Low

Medium

High

Contract nurse cost (si ) Cross-training cost (t i ) Demand distribution (Ui )

$45/h $1/h [0, 20]

$60/h $3/h –

$75/h $5/h [0, 25]

are variable because of changing patient acuity and census, and the FTE required can vary anywhere from zero to a high value constrained by the bed capacity. Compared to units with highly stable demand such as rehab and psychiatry, units that cross-train have higher variance and significant probability of any demand outcome. Therefore, we use a uniform demand distribution (measured by number of FTEs needed per hour) to represent demand. Higher nurse-to-patient ratio inflates the demand distribution in our model. In this study, we follow Abernathy, Baloff, Hershey, and Wandel (1973) manpower planning framework and we evaluate the tactical decision of determining total number of regular and cross-trained staff, and allocate the nurses to different units in the second stage. However, scheduling of nurses to specific shifts is beyond the scope of this paper. We analyze two demand distributions that are normalized, have equal coefficient of variation, but 25% increase in mean demand. When demand is low, we assume a demand distribution [0, 20] where mean is 10 FTEs; when demand is high, we assume a demand distribution of [0, 25] showing a 2.5 FTE (or 25%) increase in mean demand. We use this to assess the impact on staffing decisions when there is a systematic increase in demand over time. We also assume that physical capacity,

measured in terms of available bed space, is more than enough to accommodate all stochastic demand. Productivity parameters q12 and q21 are measured as the ratio of time taken to perform a task by a dedicated nurse relative to the time needed by a cross-trained nurse to perform the same task. We vary the productivity parameter from 0.1 to 1 in steps of 0.1. Decentralized decision making represents a scenario where staffing decisions are made without the knowledge of future cross-training. Using the above parameters and Eq. (7), we calculated optimal total staffing for decentralized decision making. As contract nurse cost and mean demand increase, optimal staffing values increase and are presented in Table 2. Given these optimal staffing values in period 1, the optimal proportion of cross-training is determined using first order conditions. Under centralized decision making, optimal staff and proportion of cross-training is determined simultaneously using global search. 5.2. Numerical insights This section presents the impact of productivity on staffing and cross-training decisions and examines the cost changes between decentralized and centralized decision making when contract cost, staffing, and mean demand changes. 5.2.1. Productivity threshold In all configurations (full flexibility, partial flexibility, chaining) and both types of decision making (centralized and decentralized), we find that there exists a threshold productivity beyond which the number of cross-trained nurses is reduced. When cross-training cost in unit i ^ij ) increases. For high (ti ) increases, threshold productivity (q

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A. Gnanlet, W.G. Gilland / European Journal of Operational Research 238 (2014) 254–269 Table 2 Optimal nurse staffing (zi ) for decentralized decision making in unit i. Mean demand in unit i

Contract nurse cost in unit i $45

Low [0, 20] High [0, 25]

Table 3 Optimal staffing when demand increases in unit 2 – full flexibility – decentralized decision making (at medium training and contract nurse cost).

6.67 8.33

$60 10 12.5

$75 12 47

cross-training costs, increasing productivity always leads to an increase in the amount of cross-trained nurses (since threshold ^ij ¼ 1). For low cross-training costs, the threshold productivity q productivity is below one. As we can observe from the graphs in Fig. 3, threshold productivity is low when cross-training cost is low (long dashed line). Lower cross-training costs always leads to a higher number of cross-trained nurses. As productivity increases, the higher productivity allows the benefits of cross-training to be realized with a smaller percentage of cross-trained nurses. However, at high cross-training cost (short dashed line and straight line), fewer nurses are cross-trained for any productivity level, resulting in a higher (close to 1.0) threshold productivity. In all three configurations and both types of decision making, we observe that when contract cost in unit j (sj ) increases, thresh^ij ) decreases. For example, in partial flexibility old productivity (q configuration (see Fig. 4), as contract nurse cost increases, threshold productivity decreases from 0.7 to 0.4 in decentralized decision making, and from 0.8 to 0.6 for centralized decision making.

5.2.2. Impact of the demand change on the proportion of cross-trained nurses When mean demand increases in a unit the proportion of crosstrained nurses decreases. Consider the two-units full-flexibility configuration as shown in Tables 3 and 4. When average demand increases in unit 2, the proportion of cross-trained nurses (u2 ) is generally reduced. The number of nurses in the unit with higher demand is increased, so a smaller proportion of nurses need to be cross-trained to give the same degree of flexibility. Under partial flexibility and chaining, the proportion of crosstrained nurses decreases when demand increases in that unit and the proportion of cross-trained nurses increases in the unit where nurses float-in from, for both types of decision making. Consider partial flexibility configuration where demand increases in unit 2 (demand changes from low, low, low to low, high, low) as shown in Tables 5 and 6. We find that u2 decreases (e.g., in Table 6 when q23 ¼ 0:9 and q12 ¼ 0:5, u2 decreases from 0.73 to 0.69) while u1 increases (from 0.44 to 0.45 for the same instance). Similarly, under chaining configuration when demand increases in unit 3 as shown in Tables 7 and 8, we find that u3 decreases (e.g., in Table 8 when q31 ¼ 0:9 and q23 ¼ 0:5, u3 decreases from 0.72 to 0.67) and u2 increases (from 0.41 to 0.42 for the same instance). Because of the ability to obtain optimal total staffing and cross-trained nurses simultaneously for centralized decision making, when comparing the two types of decision making under partial flexibility and chaining, we find that the percentage reduction in the proportion of cross-trained nurses (at high productivity) when mean demand increases in a unit and percentage increase in proportion of crosstrained nurses in other units is significantly less under centralization as compared to decentralized decision making. When there is mean demand increases of 2.5 FTEs in all units, the total staffing increases by 2.5 FTEs but the proportion of nurses cross-trained remains the same. This is true for all models under both centralized and decentralized decision making. In two-units, full-flexibility, when demand increases in a single unit by 25%, the proportion of cross-trained nurses in that unit decreases by 7.2% on average and increases by 7.8% on average in

Productivity

Unit 1 and unit 2 demand Low, low

Low, high

q21

q12

z1

z2

u1

u2

z1

z2

u1

u2

0.1 0.1 0.1 0.5 0.5 0.5 0.9 0.9 0.9

0.1 0.5 0.9 0.1 0.5 0.9 0.1 0.5 0.9

10 10 10 10 10 10 10 10 10

10 10 10 10 10 10 10 10 10

0.00 0.48 0.55 0.00 0.48 0.56 0.00 0.48 0.56

0.00 0.00 0.00 0.48 0.48 0.48 0.56 0.56 0.56

10 10 10 10 10 10 10 10 10

12.5 12.5 12.5 12.5 12.5 12.5 12.5 12.5 12.5

0.00 0.50 0.61 0.00 0.50 0.61 0.00 0.50 0.61

0.00 0.00 0.00 0.45 0.45 0.45 0.50 0.50 0.50

Table 4 Optimal staffing when demand increases in unit 2 – full flexibility – centralized decision making (at medium training and contract nurse cost). Productivity

Unit 1 and unit 2 demand Low, low

q21

q12

z1

0.1 0.1 0.1 0.5 0.5 0.5 0.9 0.9 0.9

0.1 0.5 0.9 0.1 0.5 0.9 0.1 0.5 0.9

10.00 11.61 14.42 8.62 10.12 14.42 5.74 5.74 10.05

Low, high

z2

u1

u2

z1

z2

u1

u2

10.00 8.62 5.74 11.61 10.12 5.74 14.42 14.42 10.05

0.00 0.57 0.70 0.00 0.47 0.68 0.00 0.00 0.55

0.00 0.00 0.00 0.57 0.47 0.00 0.70 0.70 0.55

10 11.68 14.80 8.34 9.85 14.80 5.17 5.16 9.44

12.50 11.08 7.90 14.42 12.90 7.90 17.48 17.48 13.15

0.00 0.59 0.74 0.00 0.48 0.74 0.00 0.05 0.58

0.00 0.00 0.00 0.55 0.46 0.00 0.65 0.65 0.52

the other unit under decentralized decision making. However, for centralized decision making, the proportion of cross-trained nurses decreases by 3.5% and increases by 3.6% in the other unit. When demand increases in a single unit by 25%, the total staff increases by 25% in decentralized decision making while the total staff increases on average by 28.4% in centralized decision making. The percentage cost savings of centralized decision making over decentralized decision making for full flexibility under varying demand distributions for unit 1 and unit 2 is shown in Table 9. The percentage cost savings is 1.15% when average demand increases in one unit and cost savings is 1.17% when demand increases in both units. The benefit of centralization is slightly higher when demand increases in both units under full flexibility configuration. Table 10 shows the percentage cost increase of centralized decision making over decentralized decision making for partial flexibility and chaining under different demand distributions for all three units. We find that the highest benefit (1.47%) is obtained when demand in unit 1 and 2 is low ([0, 20]) and demand in unit 3 is high ([0, 25]) for partial flexibility configuration (see Table 10). In this configuration, unit 3 does not cross-train nurses, however, cross-trained nurses from unit 2 float into unit 3. Therefore, when demand increases in that unit, centralized decision making utilizes cross-trained nurses effectively and provides the maximum benefit. On average, the percentage cost savings of centralized over decentralized when demand increases is 1.23% for partial flexibility and 1.01% under chaining.

5.2.3. Impact of the contract cost on the proportion of cross-trained nurses Consider a unit that allows for both float-in (cross-trained nurses floating in that unit from another unit) and float-out

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Table 5 Optimal staffing when demand increases in unit 2 – partial flexibility – decentralized decision making (at medium training and low contract nurse cost). Productivity

Demand in unit 1, unit 2 and unit 3 Low, low, low

Low, high, low

q23

q12

z1

z2

z3

u1

u2

z1

z2

z3

u1

u2

0.1 0.1 0.1 0.5 0.5 0.5 0.9 0.9 0.9

0.1 0.5 0.9 0.1 0.5 0.9 0.1 0.5 0.9

6.67 6.67 6.67 6.67 6.67 6.67 6.67 6.67 6.67

6.67 6.67 6.67 6.67 6.67 6.67 6.67 6.67 6.67

6.67 6.67 6.67 6.67 6.67 6.67 6.67 6.67 6.67

0.00 0.34 0.56 0.00 0.38 0.59 0.00 0.43 0.62

0.00 0.00 0.00 0.34 0.39 0.45 0.56 0.60 0.66

6.67 6.67 6.67 6.67 6.67 6.67 6.67 6.67 6.67

8.33 8.33 8.33 8.33 8.33 8.33 8.33 8.33 8.33

6.67 6.67 6.67 6.67 6.67 6.67 6.67 6.67 6.67

0.00 0.35 0.58 0.00 0.39 0.67 0.00 0.43 0.64

0.00 0.00 0.00 0.33 0.36 0.41 0.53 0.56 0.60

Table 6 Optimal staffing when demand increases in unit 2 – partial flexibility – centralized decision making (at medium training and low contract nurse cost). Productivity

Demand in unit 1, unit 2 & unit 3 Low, low, low

Low, high, low

q23

q12

z1

z2

z3

u1

u2

z1

z2

z3

u1

u2

0.1 0.1 0.1 0.5 0.5 0.5 0.9 0.9 0.9

0.1 0.5 0.9 0.1 0.5 0.9 0.1 0.5 0.9

6.67 7.76 10.86 6.67 7.76 10.86 6.67 7.67 10.86

6.67 6.15 4.26 7.76 7.25 5.36 10.86 10.35 8.45

6.67 6.67 6.67 6.15 6.15 6.15 4.26 4.26 4.26

0.00 0.43 0.70 0.00 0.44 0.70 0.00 0.44 0.70

0.00 0.00 0.00 0.44 0.47 0.63 0.70 0.73 0.89

6.67 7.81 11.26 6.67 7.81 11.26 6.67 7.81 11.26

8.33 7.80 5.70 9.64 9.12 7.01 13.03 12.50 10.40

6.67 6.67 6.67 6.05 6.05 6.05 3.97 3.97 3.97

0.00 0.45 0.72 0.00 0.45 0.72 0.00 0.45 0.72

0.00 0.00 0.00 0.42 0.44 0.58 0.66 0.69 0.83

Table 7 Optimal staffing when demand increases in unit 3 – chaining – decentralized decision making (at medium training and low contract nurse cost). Productivity q12 ¼ 0:5

Demand in unit 1, unit 2 and unit 3 Low, low, low

Low, low, high

q31

q23

z1

z2

z3

u1

u2

u3

z1

z2

z3

u1

u2

u3

0.1 0.1 0.1 0.5 0.5 0.5 0.9 0.9 0.9

0.1 0.5 0.9 0.1 0.5 0.9 0.1 0.5 0.9

10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00

10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00

10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00

0.48 0.53 0.56 0.55 0.60 0.67 0.61 0.67 0.69

0.00 0.55 0.61 0.00 0.60 0.67 0.00 0.64 0.70

0.00 0.00 0.00 0.53 0.60 0.67 0.61 0.67 0.71

10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00

10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00

12.50 12.50 12.50 12.50 12.50 12.50 12.50 12.50 12.50

0.48 0.54 0.58 0.56 0.62 0.66 0.64 0.69 0.72

0.00 0.58 0.67 0.00 0.63 0.72 0.00 0.65 0.74

0.00 0.00 0.00 0.50 0.56 0.61 0.56 0.59 0.62

Table 8 Optimal staffing when demand increases in unit 3 – chaining – centralized decision making (at medium training and low contract nurse cost). Productivity q12 ¼ 0:5

Demand in unit 1, unit 2 & unit 3 Low, low, low

Low, low, high

q31

q23

z1

z2

z3

u1

u2

u3

z1

z2

z3

u1

u2

u3

0.1 0.1 0.1 0.5 0.5 0.5 0.9 0.9 0.9

0.1 0.5 0.9 0.1 0.5 0.9 0.1 0.5 0.9

7.76 7.76 7.76 7.25 7.22 7.16 5.36 5.25 4.79

6.15 7.25 10.35 6.15 7.22 10.33 6.15 7.16 10.47

6.67 6.15 4.26 7.76 7.22 5.25 10.86 10.33 8.39

0.44 0.44 0.44 0.47 0.45 0.41 0.63 0.55 0.29

0.00 0.47 0.73 0.00 0.45 0.72 0.00 0.41 0.69

0.00 0.00 0.00 0.44 0.45 0.55 0.70 0.72 0.86

7.76 7.76 7.76 7.15 7.12 7.04 5.06 4.95 4.44

6.15 7.29 10.75 6.15 7.26 10.73 6.15 7.20 10.90

8.33 7.80 5.70 9.64 9.09 6.89 13.03 12.48 10.34

0.44 0.44 0.44 0.47 0.45 0.40 0.67 0.58 0.27

0.00 0.48 0.76 0.00 0.46 0.75 0.00 0.42 0.72

0.00 0.00 0.00 0.42 0.43 0.51 0.66 0.67 0.80

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A. Gnanlet, W.G. Gilland / European Journal of Operational Research 238 (2014) 254–269 Table 9 Percentage cost savings of centralized over decentralized decision making over different demand distributions – full flexibility. Unit 1/unit 2

Low – [0, 20] (%)

High – [0, 25] (%)

Low – [0, 20] High – [0, 25]

1.166 1.152

1.152 1.166

(cross-trained nurses floating out to another unit). When contract cost increases in that unit, total staff and proportion of cross-trained nurses increase in that unit while total staff and proportion of crosstrained nurses decreases in the unit where cross-trained nurses float-in from, and total staff decreases and proportion of cross-trained nurses increase in the unit where cross-trained nurses float-out to. For example, consider Table 11 of full flexibility, where cross-trained nurses float-in and float-out from both units 1 and 2. Both z2 and u2 increases when s2 increase from $45 to $75 while z1 and u1 decrease when s2 increases. Similar effects are observed in unit 2 of partial flexibility where cross-trained nurses float-in and floatout (see Table 12 when s2 increases). Also, consider unit 3 of the chaining model (see Table 13), where nurses float-in from unit 2 and float-out to unit 1. Because of increased contract cost (s3 ) in unit 3, both total staff (z3 ) and proportion of cross-trained nurses (u3 ) increases in unit 3, both total staff (z2 ) and proportion of cross-trained nurses (u2 ) decreases in unit 2, and total staff decreases (z1 ) but proportion of cross-trained nurses (u1 ) increases in unit 1. When contract nurse cost increases, the benefit of using centralized compared to decentralized decision making increases under full flexibility (two-units) and chaining (three-units) models. In both these configurations, the highest benefit of centralized over decentralized decision making is obtained when all units have high contract nurse cost, while the lowest average benefit is obtained when all units have low contract nurse cost. Under partial flexibility, because of no cross-training in unit 3, high contract nurse cost in unit 3 and low contract nurse cost in other units provides the least benefit of centralized decision making over decentralized, while low contract nurse cost in unit 3 and high in other units provide the highest benefit of centralized decision making over decentralized decision making. 5.2.4. Impact of the types of decision making on the proportion of cross-trained nurses In this subsection we discuss the benefit of using different cross-training configurations and different types of decision making in a hospital system. Under centralized decision making, the chaining configuration provides a 1.08% cost savings over the partial flexibility configuration. Under decentralized decision making, chaining provides a 1.30% benefit over the partial flexibility configuration. Considering both types of decision making (decentralized and centralized), we find that chaining configuration provides a 1.20% cost savings over partial flexibility configuration. The greatest benefit of centralized over decentralized decision making comes in the partial flexibility configuration. The percent-

Table 11 Average optimal staffing decisions when contract cost changes – full flexibility. z2

u1

u2

8.23 6.01 5.18

8.18 13.29 15.66

0.31 0.19 0.14

0.30 0.46 0.53

$45 $60 $75

13.29 11.36 9.71

6.01 11.32 14.52

0.47 0.35 0.27

0.19 0.35 0.45

$45 $60 $75

15.66 14.52 13.07

5.03 9.71 12.99

0.53 0.45 0.36

0.14 0.27 0.36

s1

s2

$45

$45 $60 $75

$60

$75

z1

Table 12 Average optimal staffing decisions when contract cost changes in unit 2 – partial flexibility. s2

Average over s1

$45 $60 $75

z1

z2

z3

u1

u2

12.96 12.79 12.64

7.21 11.65 14.04

9.75 8.49 7.72

0.48 0.45 0.43

0.49 0.56 0.60

Table 13 Average optimal staffing decisions when contract cost changes in unit 3 – chaining. s3

$45 $60 $75

Average over s1 and s2 z1

z2

z3

u1

u2

u3

12.18 11.00 10.26

11.05 10.83 10.64

6.65 11.23 13.73

0.46 0.49 0.52

0.55 0.50 0.46

0.37 0.49 0.53

age cost savings from using centralized decision making over decentralized is shown in Table 14. We ran statistical significance test on the null hypothesis that the mean cost savings of centralized over decentralized is zero. The null hypothesis was rejected with a ‘p’ value of 0.0001 indicating that the cost savings are highly statistically significant with 99.9999% confidence. Under both types of decision making, chaining provides the greatest benefit over partial flexibility when mean demand is low in unit 2 (where nurses float-in and float-out) and other units have high mean demand. Chaining policy is the least beneficial over partial flexibility when demand is lowest in unit 3 (where nurses only float-in and do not float-out) under decentralized decision making and lowest in unit 2 (where nurses float-in and float-out) under centralized decision making. We find that for full-flexibility (two-units) and chaining (threeunits) models, the percentage savings from using centralized over decentralized decision making is exactly the same when average demand is low for all units and average demand is high for all units as shown in Table 15.

Table 10 Percentage cost savings of centralized over decentralized decision making over different demand distributions – partial flexibility and chaining. Demand distribution

Partial flexibility

Chaining

Unit 3 – [0, 20]

Unit 3 – [0, 25]

Unit 3 – [0, 20]

Unit 3 – [0, 25]

Unit 1/unit 2

[0, 20] (%)

[0, 25] (%)

[0, 20] (%)

[0, 25] (%)

[0, 20] (%)

[0, 25] (%)

[0, 20] (%)

[0, 25] (%)

[0, 20] [0, 25]

1.130 1.097

1.162 1.198

1.472 1.016

1.148 1.131

1.018 1.004

1.024 1.020

1.005 0.997

1.020 1.018

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Table 14 Percentage cost savings of centralized over decentralized decision making under various configurations. % Cost savings

Centralized over decentralized decision making (%)

Two-units, full-flexibility Three-units, partial-flexibility Three-units, chaining

1.160 1.227 1.013

5.2.5. Impact of the total staff on the proportion of cross-trained nurses In most hospitals, total staffing frequently changes due to attrition, technology or policy changes. In order to understand the impact of ad hoc staffing changes on system performance, we analyzed a situation where sub-optimal total staffing levels are set in period 1 and the optimal proportion of nurses are cross-trained in period 2. We found that when total staffing increases in a unit that has the ability to float-in and float-out a cross-trained nurse, productivity has a significant impact on number of nurses to be cross-trained. For example, consider the full flexibility model where nurses in unit 1 can float-out to unit 2 and vice versa. In such units, productivity moderates the impact of total staffing on the proportion of cross-trained nurses. When total staffing increases in that unit (say, z1 of unit 1), the proportion of cross-trained nurses (u1 ) increases at low productivity of cross-trained nurses (q12 ) but decreases at higher productivity of cross-trained nurses (q12 ). It signifies that the benefit of flexibility is obtained at higher productivity and that fewer nurses are cross-trained even when total staffing increases in instances where cross-trained nurses can productively float-in and float-out. This moderation effect of productivity is seen in all units that permit float-in and float-out, such as unit 1 and 2 in full flexibility, unit 2 of partial flexibility, and unit 1, 2, and 3 of chaining model. In all other instances, such as crosstrained nurses only floating-out (unit 1 of partial flexibility) and only floating-in (unit 3 of partial flexibility), the proportion of cross-trained nurses decreases when total staffing increases.

5.3. Managerial insights Our significant contribution is that across all cross-training configurations and types of decision making, there is a threshold productivity beyond which the proportion of cross-trained nurses is reduced. When cost of cross-training is high, the threshold productivity is 1. However, when cost of cross-training is relatively low, there is a productivity level (less than 1) beyond which further increases in productivity reduce the amount of cross-training. Above this threshold, the productivity of cross-trained nurses allow the benefits of flexibility to be obtained with fewer crosstrained nurses. Managers who are aware of the threshold productivity can cross-train fewer nurses to units that will enable crosstrained nurses to be highly productive. The threshold productivity is low when cross-training and contract nurse cost is low. When contract nurse cost increases in a unit that allows crosstrained nurses to float-in and float-out, total staff is increased only in that unit and decreased in other units. The proportion of crosstrained nurses increases in that unit and the unit that nurses floatout to, but decreases in the unit that nurses float-in from. An

Table 15 Percentage cost savings of centralized over decentralized decision making for two demand levels. Average demand

Full flexibility (%)

Chaining (%)

Low in all units High in all units

1.166 1.166

1.018 1.018

increase in contract nurse cost, even in one unit, causes optimal staffing and cross-training decisions to change in all units. When total staff increases for units that allow cross-trained nurses to float-in and float-out, productivity moderates the effect of total staff on optimal cross-training decisions. When total staff increases, managers should not necessarily increase the proportion of cross-trained nurses as well. The proportion of cross-trained nurses increases at higher productivity, but is reduced at lower productivity. In all other cases (when cross-trained nurses only float-out or only float-in), the proportion of cross-trained nurses decreases when total staff is increased. Across all types of decision making, chaining provides on average 1.20% increase in benefits over partial flexibility in a three unit system. When managers plan to increase cross-training breadth, adding a single link that completes a chain will provide on average 1.20% benefit in cost savings and is robust between centralized and decentralized decision making. Centralized decision making, where managers decide both staffing and cross-training levels simultaneously provides on average 1.13% cost savings over decentralized decision making after considering productivity changes. Under both types of decision making, chaining provides the greatest benefit over partial flexibility when mean demand is low in the unit where cross-trained nurses can both float-in and float-out relative to the other units.

6. Conclusion While cross-training of employees has been proven to be beneficial for employees (increased marketability) and service firms (flexible worker assignment, lower cost), we analyzed the impact of productivity changes on cross-training policies and optimal cross-training decisions. Because of procedure, protocol and layout differences, cross-trained nurses not be as productive as regular nurses, consequently impacting the benefit derived from crosstraining. We formulated three different cross-training policies (configurations) in a hospital, and analyzed performance using two stage stochastic programming. We proved convexity and obtained a closed form expression to determine the optimal proportion of cross-trained nurses in a two-units, full-flexibility model. We also found a productivity threshold beyond which fewer cross-trained nurses are needed to derive the benefits of flexibility. We analytically proved the existence of the threshold productivity level in the two unit model and numerically observed it in other configurations. The cross-training and staffing decisions for a two unit, fully flexible unit (where nurses float-in and floatout in both units) are mutually independent when cost and demand changes. A minimal complete chain formation in three units (with three links) provides on average 1.20% cost savings compared to partial flexibility in three units (with two links). Adding a cross-training link to partial flexibility provides statistically significant cost savings for any level of cross-training productivity. Centralized decision making, where total staff and cross training decisions are made simultaneously, provides on average 1.13% cost savings over decentralized decision making, where total staff is determined first and the proportion of cross-trained nurses is decided later. On average, low contract nurse cost in all units provides the least benefit of centralized decision making over decentralized, while high contract cost in all units on average provides the greatest benefit of centralized over decentralized decision making. A limitation to our research is that we have considered average unit-level productivity; while productivity of individual nurses will deviate from average unit productivity based on their experience and education. In addition, with the introduction of the HITECH 2009 Act, all hospitals are progressing towards the imple-

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mentation and use of electronic medical records (EMR) and health care information technology (HIT). The degree of standardization of technological processes is likely to impact the productivity of cross-trained nurses. In this paper, we have not considered the impact of technology on the efficacy of nurses’ work across different units. Further empirical and analytical research is required to identify the consequence of new technology and its impact on both regular nurses and cross-trained nurses.

Centralized decision making – full flexibility: The second stage objective function is convex in the staffing allocation variables. The optimal second stage allocations are given in Section 3.3. Substituting the optimal values for the second stage variables in the first stage objective function we get Eq. (24).

K1 ðz1 ; z2 ; u1 ; u2 Þ ¼

X ðhi  zi þ t i  zi  ui Þ þ D;

ð24Þ

i

where Appendix A

D ¼ s1

þ s1 ð13Þ

s2

@2K ¼ q21  s1  z22  ðf1  U1 ½q21  u2  z2 þ z1 g  U02 ½ð1  u2 Þz2  @u2 ; u2 ð15Þ þ q21  U2 ½ð1  u2 Þz2   U01 ½q21  u2  z1 þ z2 Þ @2K ¼0 @u1 ; u2

ð16Þ

@2K ¼0 @u2 ; u1

ð17Þ

1

ðd1  z1  q21  z2  u2 ÞdU1 dU2

z1 þðq12 z2 u2 Þ Z 1

z2

z1 þq12 ðz2 d2 Þ

Z

ð1u1 Þz1 0

Z

Z

ðd1  z1  q21 ðz2  d2 ÞÞdU1 dU2 ð25Þ

1

ðd2  z2  q12  z1  u1 ÞdU2 dU1

z2 þðq21 z1 u1 Þ Z 1 z1

þ s2 ð1u Þz Z 1 1 Z1 þ ð1u1 Þz1

z2 þq21 ðz1 d1 Þ 1

ðd2  z2  q12  ðz1  d1 ÞÞdU2 dU1

s1 ðd1  z1 Þ þ s2 ðd2  z2 ÞdU2 dU1

ð26Þ

z2

Second order derivatives for the first stage objective function (Eq. 24) is obtained. Using row reduction techniques on the hessian results in a identity matrix proving that the first stage objective function is convex in it is staffing variables (z1 ; z2 ; u1 ; u2 ). Therefore, first order conditions are used to determine the optimal staffing values and are seen in Eqs. (10) and (11). Centralized decision making – partial flexibility:

K1 ðzi ;ui Þ :

The Hessian matrix with second order conditions with respect to u1 and u2 for the above formulation shows that it is positive definite Blume and Simon (1994) and so the objective function (2) is convex in u1 and u2 . The first order condition for first stage Lagrange function (13) is given below:

l1  u 1 ¼ 0 l2  u 2 ¼ 0 l3  ð1  u1 Þ ¼ 0 l4  ð1  u2 Þ ¼ 0

Z

Z

ð1u2 Þz2

@2K ¼ q12  s2  z21  ðf1  U2 ½q12  u1  z1 þ z2 g  U01 ½ð1  u1 Þz1  @u1 ; u1 ð14Þ þ q12  U1 ½ð1  u1 Þz1   U02 ½q12  u1  z1 þ z2 Þ

@K ¼ l3  l1 þ t 1  z1  q12  s2  U1 ½ð1  u1 Þz1  @u1 f1  U2 ½q12  u1  z1 þ z2 g ¼ 0 @K ¼ l4  l2 þ t 2  z2  q21  s1  U2 ½ð1  u2 Þz2  @u2 f1  U1 ½q21  u2  z2 þ z1 g ¼ 0

ð1u2 Þz2 0

Decentralized decision making – full flexibility: The Lagrangian function for the first stage is given by

Kðu1 ; u2 ; lÞ ¼ t 1  u1  z1 þ t 2  u2  z2 þ EU1 ;U2 Q ðu1 ; u2 ; d1 ; d2 Þ  l1  u1  l2  u2 þ l3  ðu1  1Þ þ l4  ðu2  1Þ

Z

Minzi ;ui

i¼1

ð19Þ ð20Þ ð21Þ ð22Þ ð23Þ

Solving for the Lagrange multipliers we get the following cases: Case 1 (l1 > 0 and l2 > 0): From the constraints (20) and (21), we see that u1 ¼ 0 and u2 ¼ 0, consequently from constraint (22) and (23) we get l3 ¼ 0 and l4 ¼ 0. So constraint (20) and (21) is binding. Case 2 (l1 > 0 and l4 > 0): Implies u1 ¼ 0 and consequently l3 ¼ 0. Since l4 > 0; u2 ¼ 1 from constraint (23) and l2 ¼ 0. Substituting u2 ¼ 1 in constraint (19), we get t 2  z2 ¼ l4 . This is not possible since all the values are positive and so this case is infeasible. Case 3 (l3 > 0 and l2 > 0): Implies u2 ¼ 0 and consequently l4 ¼ 0. But l1 ¼ 0 and so u1 ¼ 1. Substituting in constraint (18) we see that t 1  z1 ¼ l3 . So this case is infeasible. Case 4 (l1 ¼ 0 and l3 ¼ 0; l2 ¼ 0 and l4 ¼ 0): The solution is u1 ¼ ½0; 1Þ and u2 ¼ ½0; 1Þ and satisfies Eqs. (18) and (19).

i¼1

X

þ

½si ðdi  xii  xji Þ 

i;j2fð1;3Þ;ð2;1Þ;ð3;2Þg

s:t: xji ¼ 0; i ¼ 1; j ¼ 3 zi P 0; i ¼ 1; 2; 3 0 6 ui 6 1; i; j ¼ 1; 2; i – j where

xii ¼ ð18Þ

ð27Þ 3 2 X X ðhi  zi Þ þ ðti  ui  zi Þ þ EUi



min½di ; zi ; i ¼ 1; 3 þ min½di ; ð1  ui Þzi  þ min½ðdi  ð1  ui Þzi  xji Þ ; ui zi ; i;j 2 f2;1g

8 > < 0;i ¼ 1;j ¼ 3 þ þ xji ¼ qji  min½ðdi  ð1  ui Þzi Þ ;min½ðzj  dj Þ ;uj zj ; i;j 2 f2;1g > : q  min½ðd  z Þþ ;min½u z ;ðd  ð1  u Þz  x Þþ ; i;j;k 2 f3;2;1g i i j j j j j kj ji where xii is the optimal allocation of regular nurses (dedicated and cross-trained) from unit i to unit i and xji is the optimal allocation of productivity-adjusted cross-trained nurses from other unit j to unit i. Similar to decentralized decision making, while planning for optimal total staffing and proportion of productivity accounted crosstrained nurses, the model considers the use of cross-trained nurses through chaining to meet demand. Cross-trained nurses in unit 1 are allocated to unit 2 when needed so that cross-trained nurses in unit 2 to meet unit 3 demand. In this model, unit 3 nurses are not cross-trained to unit 1 and so x31 is constrained to be zero. The second stage objective function is linear in its decision variables and has linear constraints and so the second stage objective function is convex with respect to staffing allocation variables xii , xji . Centralized decision making – chaining:

Minui

3 X ðhi  zi þ t i  ui  zi Þ þ EUi i¼1

X i;j2fð1;3Þ;ð2;1Þ;ð3;2Þg

s:t: zi P 0; 0 6 ui 6 1; i;j; k ¼ 1; 2; 3; i – j – k

þ

½si ðdi  xii  xj i Þ  ð28Þ

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where þ

xii ¼ min½di ; ð1  ui Þzi  þ min½ui zi ; ðdi  ð1  ui Þzi  xji Þ ; i; j 2 fð1; 3Þ; ð2; 1Þ; ð3; 2Þg þ

þ

xji ¼ qji  min½ðdi  ð1  ui Þzi Þ ; min½uj zj ; ðdj  ð1  uj Þzj  xkj Þ ;; i; j;k 2 fð1; 3; 2Þ; ð2; 1; 3Þ; ð3; 2; 1Þg where xii is the optimal allocation of regular nurses (dedicated and cross-trained) from unit i to unit i and xji is the optimal allocation of productivity-adjusted cross-trained nurses from unit j to unit i. Unit k represents the unit that cross-trains for unit j and unit j represents the unit that cross-trains for unit i. Similar to decentralized decision making, benefit of flexibility is leverage by allocating nurses in sequential units so that demand is met. The second stage objective function is linear in its decision variables and has linear constraints and so the second stage objective function is convex with respect to staffing allocation variables xii , xji . Proof: Threshold Productivity: This section presents the proof for finding the threshold produc^ij ) and corresponding conditions for the threshold productivity (q tivity to hold. Assuming Ui to be uniformly distributed between ½ai ; bi  for all i = 1, 2 we get the following first order condition from Eq. (8). For ui :



qij  ui  zi þ zj ð1  ui Þzi  ai ti ¼  1 bi  ai bj  aj qij  sj

ð29Þ

Combining the terms in Eq. (29), we get a quadratic Eq. (30) in ui .

qij  z2i  u2i  fðbj  zj Þzi  qij  zi ðzi  ai Þg  ui ti þ fðzi  ai Þ  ðbj  zj Þ   ðbj  aj Þ  ðbi  ai Þg ¼ 0 qij  sj

ð30Þ

Solving the quadratic equation in (30), we get an optimal solution for ui as given in Eq. (31).

ui ¼

1 ½K 2  q2ij sj z2i qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  K 2 þ 4  q2ij s2  z2i ðt i ðbj  aj Þðbi  ai Þ  qij  sj  ðbj  zj Þðzi  ai ÞÞ

where

K ¼ qij  sj  zi ðqij ðai  zi Þ  ðbj  zj ÞÞ;

i; j ¼ 1; 2; i – j

ð31Þ

Eq. (31) is re-written as a function of qij . The second derivative of that function is negative and so the function qij is concave under the following condition.

" # pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðbj  zj Þðzi 4  t i  bj  bi  sj  sj ðzi  1Þðbj  zj Þ þ 4  t i  bi  bj qij > þ1 zi sj  zi ðbj  zj Þ ð32Þ This condition holds for all values of the parameters. First order condition of Eq. (31) is used to determine the threshold productivity. The threshold productivity is given in Eq. (33). 2

sj ðbj  zj Þ þ 4  t i ðbi  ai Þðbj  aj Þ ^ij < 1 ; if q 2  sj ðzi  ai Þðbj  zj Þ ^ij P 1; i; j ¼ 1; 2; i – j ¼ 1; if q

^ij ¼ q

ð33Þ ð34Þ

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