A benchmark dataset for the retail multiskilled personnel planning under uncertain demand

2.1.Background on PSPs in different industries

Table 1 presents a list of research articles that have addressed PSPs in application contexts other than retail, sometimes considering multiskilled workers and sometimes not. In addition, the articles listed in Table 1 are characterized by having an associated data repository with public access, thus providing valuable datasets for researchers or practitioners to conduct experiments and/or benchmarking. The table categorizes the articles according to the following characteristics:

2.2.Background on PSPs involving multiskilled staff in a retail setting

To illustrate the gap addressed by this data article, Table 2 presents an exhaustive classification of research articles with applied case studies in the retail industry, focusing on PSPs considering multiskilled employees. This classification is based on the following characteristics:

Several aspects can be highlighted from Table 2. First, it can be noted that most articles used MIP models or a combination of MIP models and heuristic approaches as their chosen solution method. Notably, articles such as Henao et al. [20], Henao et al. [23], Abello et al. [1], Fontalvo Echavez et al. [16], Mercado and Henao [32], Mercado et al. [33], and Henao et al. [19] also incorporated optimization techniques such as RO, TSSO, or CF to address the challenge of uncertain demand.

Second, among the 17 articles listed in Table 2, twelve of them addressed a workforce training problem (i.e., [1,16,19–23,32,33,41,44,54]). Therefore, in each of these articles, one of the objectives was to establish the optimal training plan for the staff. However, 8 of these 12 articles specifically addressed a MPAP (i.e., [19–21,23,32,33,44,54]), which is identified in the table by the nomenclature A + WT. Remember that a MPAP simultaneously solves an assignment problem along with a workforce training problem. We emphasize that, unlike other workforce training problems that may also involve shift scheduling, days-off scheduling, or tour scheduling, the MPAP specifically focuses on multiskilling decisions rather than on scheduling decisions (such as weekly schedules with work shifts and rest days).

Third, while Mirrazavi and Beringer [34] did not explicitly define whether they used real data, simulated data, or a combination of both, the articles by Henao et al. [22], Mac-Vicar et al. [30], and Hassani et al. [17] used real data only, while the rest of the articles used a combination of real and simulated data. It is noteworthy that a common element in those articles that used simulated data was the inclusion of staff demand for each store department within the simulated datasets, incorporating various levels of variability in these demands.

Fourth, most of the articles in Table 2 did not disclose 100% of the data used in the experimentation and validation stages of their research. This notable limitation poses a challenge for practitioners and researchers seeking to replicate the published results of such articles with the utmost precision. Continuing the discussion, only five research articles reported the complete datasets used in their investigations. These are as follows:

3.1.Real data

The Chilean workforce management company SHIFT SpA [47] provided us with real data from a prominent home improvement retailer. The real dataset consists of information related to the number of store departments, number of single-skilled workers hired for each department, weekly hours that each worker has to work given his/her labor contract, average staff demand per week per department, and staff costs related to a Chilean retail store. For a better understanding of the data, consider that a retail store has a known number of departments, and these store departments usually have hired a set of workers originally single-skilled and, thus, skilled to work in one department. In addition, each department requires possessing certain basic skills and the working hours of workers depend on what is stipulated in their labor contracts.

Table 3 shows a full description of the parameters and sets associated with the real dataset. Also provided with these sets and parameters is a file named ‘real-data.txt’ written in the mathematical programming language AMPL. This file can be accessed from the Zenodo data repository archived at https://zenodo.org/records/10570229 ([25]). In Table 3, the store departments (L), store workers (I), workers under contract in the department (Il), and store department where each worker was originally skilled (mi) are data associated with the case study. Whereas the weekly hours that each worker must work according to his/her labor contract (h) is set at 45 hours per week, since this is what the Chilean labor law stipulates for a full-time contract. A complete explanation of how the average demand for all departments (rl‾) was obtained and how we estimated the cost of training (c), the cost of understaffing (u), and the cost of overstaffing (b), is presented in Section 4.

Now, it is important to note that the MPAPs addressed in Henao et al. [19], Henao et al. [20], and Henao et al. [23] have two notable assumptions: (1) Unscheduled personnel absenteeism is not considered; that is, all employees are available 100% of the time for which they were hired. (2) Employees are assumed to be homogeneous; thus, all employees have maximum productivity in all departments for which they are trained. However, aiming to enrich the versatility and usefulness of this data article, we have included new real data derived from the experience of Chilean retailers. These data, which focus on unscheduled personnel absenteeism and the phenomena of learning and forgetting, are included in the file named ‘real-data.txt’. Notably, these data have already been used in two of our published research articles, Mac-Vicar et al. [30] and Henao et al. [21].

On the one hand, in Mac-Vicar et al. [30], the authors addressed a tour scheduling problem considering multiskilled employees. In order to assess the effectiveness of a set of flexible labor strategies to mitigate the negative effects of uncertain demand and unscheduled absenteeism, they conducted experiments with three probable absenteeism scenarios: 5%, 10%, and 15%. Specifically, the authors assumed that the probability of an employee missing a scheduled shift followed a Bernoulli distribution, with probabilities set at p=5%,10%,and 20%. These values were determined through consensus among various store managers affiliated with a leading home improvement retailer in Chile.

On the other hand, in Henao et al. [21], the authors addressed a MPAP involving a multiskilled and heterogeneous workforce, which was subject to the learning/forgetting phenomena. In other words, they considered a heterogeneous workforce, where the productivity of multiskilled employees may vary depending on the number of departments to which they are assigned. Therefore, to address their MPAP, the authors modeled three parameters related to the learning and forgetting phenomena, as follows:

The values for Lil and Fil were chosen considering the strong correlation observed between the rates of learning and forgetting. In general, fast learning tends to be associated with fast forgetting.

3.2.Simulated data

The simulated datasets consist of information related to the stochastic demand of the store departments. They consider two sample data types related to the uncertain demand: in-sample and out-of-sample. In-sample refers to the data employed to obtain the in-sample solutions of the MPAP with the TSSO approach. Conversely, out-of-sample refers to the data employed to compare the performance of the reported solutions with the three optimization approaches: TSSO, RO, and CF.

To generate the in-sample data, Monte Carlo Simulation (MCS) was used to randomly create 2,000 demand scenarios for the random parameter rl(s),∀l∈L,s∈S, such that |S|=2,000. In each department, this simulation is carried out for 6 coefficients of variation of the demand: CV=5,10,20,30,40,50%. These six levels of demand variability were established to determine how multiskilling requirements in the workforce can increase as demand uncertainty increases. Specifically, a normal probability distribution was used to create the realizations of the stochastic demand in each store department. Then, two distinct datasets were generated to assess and compare the level of conservatism in the TSSO approach solutions. Each dataset involves a distinct truncation type in the probability density functions (pdfs) for store department demand. In the first dataset the pdfs were truncated at the 5th and 95th percentiles, while in the second dataset the pdfs were zero-truncated. Both datasets avoid creating negative demand values, but the first dataset also avoids creating atypical values. Similarly, a MCS was also used to generate random demand scenarios for the out-of-sample data. In this case, 10,000 demand scenarios were created for each department in a single dataset, following a normal distribution truncated at zero.

Summarizing, this subsection provides three datasets: two in-sample and one out-of-sample. Each dataset contains 6 files (one for each CV) as listed in Table 4. Each file presents the stochastic demand realizations for six store departments, such that each row represents a department and each column represents a demand scenario (2,000 if it is in-sample and 10,000 if it is out-of-sample). The name of the files is coded by three characters i-j-k, where i = IS, OS specifies the type of sample (in-sample, out-of-sample); j = PT, ZT specifies the type of truncation method for the normal distribution (percentile-truncated, zero-truncated); and k=05,10,20,30,40,50 specifies the coefficient of variation (CV=5,10,20,30,40,50%). These files can be accessed from the Zenodo data repository archived at https://zenodo.org/records/10570229 ([25]).

Boxplots were created to visualize the simulated data. Figure 1 graphically compares both truncation types for the in-sample data, considering a coefficient of variation of 50% in the 6 store departments (i.e., ‘IS-PT-50.txt’ vs ‘IS-ZT-50.txt’ files). For the same coefficient of variation (50%) the percentile-truncated data range from 24 to 655 hours, whereas the zero-truncated data is broader and has atypical values, ranging from 0 to 937 hours, considering all departments.

Fig. 1.

Percentile-truncated vs zero-truncated, with a coefficient of variation of 50% in the 6 store departments.

Fig. 2.

Percentile-truncated vs zero-truncated, in the second department with 6 coefficients of variation.

In addition, for the second department and each CV, Fig. 2 graphically compares both truncation types for the in-sample data. Remember that the second department has an average weekly demand of 225 hours. Here, it is evident that as the coefficient of variation increases, the range of the weekly hours demand values systematically expands. Similar to Fig. 1, the zero-truncated data is broader, ranging from 1 to 565 hours, whereas the percentile-truncated data ranges from 40 to 410 hours, considering all the coefficients of variation.

The boxplots for the out-of-sample data are not shown, but as expected, they have a similar distribution to the in-sample data. However, in this case, the number of demand scenarios for each box plot is 10,000, instead of 2,000.

4.1.Calculating the mean weekly hours demand and staff costs

SHIFT SpA provided us with the real data for this case study. SHIFT SpA is a company that optimizes the shift schedules of thousands of employees across Latin America, which validates the quality and relevance of the real data they provided. Specifically, they used a specialized software to estimate the average weekly person-hours demand values for each department. This software works in two stages: (1) forecasting transactions and expected sales and (2) generating workforce requirements.

First, the software uses multiple linear regression to forecast the expected sales and number of transactions for each department in the store. The regression requires between 24 and 72 months of historical data to ensure greater accuracy. Second, considering typical customer service times, the software converts the forecast of transactions and expected sales into a staff demand quantified in person-hours.

Regarding the staff costs, it was assumed that each worker has a minimal cost of training (c=1US$−week/employee). Henao et al. [19], Henao et al. [20], Henao et al. [23], Vergara et al. [54], and Mercado et al. [33] expressed that the outcomes found under this supposition represent an upper bound on the possible benefits of using multiskilled staff. In relation to the over/understaffing costs, it was assumed that these are the same across all departments. Using historical data from the retail store, the cost of understaffing is calculated as the average cost of the expected lost sales. Such that u=60US$/hour, a value similar to that reported in [1] and [16]. Also, using historical data from the retail store, the cost of overstaffing is calculated as the average wage cost of having idle staff. Such that b=15US$/hour, a value similar to that reported in [1,16], and [32].

4.2.Monte Carlo simulation

The MCS utilized to generate the stochastic demand realizations in Sect. 3.2 (simulated data), was carried out in an Excel workbook available in the Zenodo data repository archived at https://zenodo.org/records/10570229 ([25]). This workbook contains two worksheets that can be used to generate up to 10,000 scenarios of random demand realizations (outputs). The first worksheet is called ‘percentile-truncated’ and generates a normal distribution truncated at the 5th and 95th percentiles. The second worksheet is called ‘zero-truncated’ and can be used to generate a normal distribution truncated at zero.

The weekly hours demand follows a normal probability distribution; therefore, two parameters are needed to create the stochastic demand realizations in the 6 store departments: (i) the average weekly person-hours demand, which was shown in Table 3; and (ii) the standard deviation in weekly hours, which is calculated as the product between the average value and the coefficient of variation (CV) of the demand. Thus, the CV value can be chosen by the store manager to indicate the degree of uncertainty in the demand that best fits the store’s operations. In the Excel worksheets, both parameters are denoted in yellow cells, indicating that they can be changed.

Some statistics were also calculated in the Excel worksheets. In the ‘percentile-truncated’ worksheet, the 5th and 95th percentiles were calculated. Meanwhile, in the ‘zero-truncated’ worksheet, the standard score (z), denoting a weekly person-hours demand value equal to zero, along with its corresponding quantile in percent, were calculated. These results are presented in cells with gray text, indicating that these cells must not be modified because they are being calculated using Excel formulas.

Then, by using random values and using Excel formulas associated with the inverse normal distribution, it becomes possible to calculate the outputs. In the ‘percentile-truncated’ worksheet, the random values vary between 0.05 and 0.95 with a step size of 0.000001, following a normal distribution. Conversely, in the ‘zero-truncated’ worksheet, the random values vary between the quantile associated with the standard score (z) and 1, with the same step size, ensuring that the realizations of the stochastic demand are non-negative. In both worksheets, the stochastic demand realizations are organized into six rows representing the store departments, and up to 10,000 columns representing the demand scenarios. These results are presented in cells with blue text, indicating that these cells are the results calculated by Excel formulas and must not be modified.

Finally, we emphasize that the detailed description provided above regarding the implementation of the MCS method can be easily cross-referenced and verified by carefully examining the Excel worksheets and the set of Excel formulas used. All these details are readily available to the reader in the Excel workbook.