Competence-based assignment of tasks to workers in factories with demand-driven manufacturing

The paper is motivated by real problems concerning tasks assignment to workers in medium-sized upholstered furniture plants managed using the Demand-Driven Manufacturing. Although the methodology was developed for furniture plants it can be applied to other types of production plants. We involve competence coefficients, which describe the level of the worker’s skills or capabilities to perform a specific task. The competence coefficients are also used to block the possibility of assigning the given task to a worker that has no skills to do it. Additionally, we involve a dummy worker to the model which guarantees the existence of a solution to the problem. We present and discuss Integer Linear Programming Models for the posted problem that are closely related to the Generalized Assignment Problem. We also discuss the potential use of the presented methodology to solve real-life problems related to production management.


Introduction
We present the problem of assigning tasks to workers in medium-sized upholstered furniture factories managed using Demand-Driven Manufacturing. This problem is motivated by real-life challenges described by managers in upholstered furniture plants in Poland. The main idea in this paper was formulated in 2012 in cooperation with ITM 1 , which provides production management systems in furniture factories. IT implementation of the solution to the presented problem was awarded in 2016 with the 'Debut of the Year' award given by the Rzeczpospolita newspaper. The system based on this solution works, for example, in Primavera Furniture 2 factory.
Demand-Driven Manufacturing (details in Sect. 6), typically used in the factories under consideration, requires a high degree of flexibility in production planning. Some aspects such as unpredictable production disturbances, poor health of the working person, or hidden defects in materials, which can only be noticed while performing the task can be only seen during the execution of the task, require immediate reaction and reallocation. The problem of reallocation was considered, for example, in Jain and Elmaraghy (1997) and Strickera and Lanza 2014. There is a great need to create an effective, flexible and reactive system supporting production management, in particular, human resources management. As far as upholstered furniture factories are concerned, the two stages of production: covers sewing and upholstering most need such a solution.
Besides, the analyzed upholstered furniture factories usually apply the color-coded task priorities (red, yellow, green) based on the Kanban philosophy (see Hammarberg and Joakim 2014). We do not consider any preferences in the stages of production. The proposed methodology applies to every stage independently however, such kinds of preferences are included in the colors of the tasks. Any green task on the given stage of production is assumed to be already produced on the antecedent stages. Every red task has to be finished to use it in the next stages.
We propose different integer linear optimization models for solving the problem of competence-based tasks assignment to workers. Competence coefficients further enhance the model with their 'regular' values from an interval [minC, maxC] that describe the skills or capabilities of each employee to perform each specific task. A subjective appraisal may determine the competence coefficients by the superiors or by some automatic evaluation systems. Alternatively, we can use a mixture of both. It is worth considering the development of an automatic classification model (more on classifiers can be found in Artiemjew 2008;Staruch 2017;Staruch and Staruch 2017). Regardless of the method of evaluation, the values of these factors are to be such that a lower value of the coefficient indicates the greater predisposition of the worker to perform a given task. Thus the competence coefficients play the role of the cost function used in many optimization problems. By setting the values of some competence factors on some high level, we obtain a way of blocking the allocation of tasks to workers with insufficient competence to perform them. On the other hand, the allocation of simple tasks to highly qualified workers can be blocked the same way. The need for blockages is discussed in the domain called task assignment, where binary values of competence coefficients indicating skills or authorization are used (see, e.g., IBM Knowledge Center 2019).
An additional and crucial element of the presented model is a dummy worker (named Dummy for short) with its competence value treated as a penalty for unfinished tasks. Involving Dummy ensures the existence of a solution for the modeled problem.
Further analysis of tasks allocated to Dummy allows deciding if reallocation of tasks should be performed.
The paper is organized as follows. We describe the elements of the problem in Sect. 2. Section 3 is devoted to the description of the role and the form of the competence coefficients. We present optimization ILP models for solving the main problem of this paper in Sect. 4. We discuss their advantages and disadvantages and compare them with the known problems such as of Generalized Assignment Problem and Multi-Knapsack Problem. The proposition of an algorithmic solution is presented in Sect. 5. Section 6 is devoted to discussion of the most important aspects concerning Demand-Driven Manufacturing. We indicate the most important factors influencing the quality of production planning in the form of parameters describing our solution, including modeling of color-coded task priorities. We show how modification of these parameters allows us to solve problems related to production planning effectively and to react to unexpected events.

Conventions and assumptions
We use the abbreviation CBA to indicate the title problem called Competence-Based Assignment of Tasks to Workers.
Assume that there are workers in a plant that are planned to perform a set of tasks, e.g., sewing covers or upholstering furniture. Let us use the following notations: 1. m 0 is the number of workers, who are indicated by indices i = 1, . . . , m 0 ; 2. every worker i has his/her available time denoted by L i ; 3. n is the number of tasks planned to be performed, which are indicated by indices j = 1, . . . , n; 4. every item j has its fixed normative execution time t j , named normative time; 5. each worker may perform any number of tasks within her/his available time; 6. every task can be performed by exactly one worker; 7. each worker i has specific competencies to perform the task j. The level of these competencies is 'normally' presented in the form of a competence coefficient C i j ∈ [minC, maxC] so that the higher the competencies, the lower the competence coefficient value.
The solution to the problem of assigning tasks to workers based on competencies is to assign tasks to workers with a fixed available time so that the total cost of completing the tasks is the 'lowest.' Notice that the CBA problem does not cover interrelations between different stages of production. Changeover times are not considered too, however, the proper use of competence factors (see Sect. 3) gives some 'seriality' effects. this reason, a well-thought-out system for determining competence factors, tailored to the needs of the production department in the factory, is needed. The role of the competence coefficient is to quantify the skill level of the worker, resulting in faster work and higher quality of task performance.
Competence coefficients can be interpreted as a cost (the lower value, the better), so we can think of the competence coefficients as a cost function, which for the taskworker pair i, j takes the cost value C i j .
Interviews with production managers in upholstered furniture factories show that it is standard practice to assign tasks by hand. The shift manager decides on the allocation of tasks based on his own experience and knowledge of the skills of his subordinates.
There is no one-size-fits-all principle to determine which values of competence coefficients are the best. One factory opts for the quality of performance even at the expense of time, while for the other, time reduction is the most important. In the latter case, the competence factors can express a percentage saving of time for each task and hence we can assume that a reasonable range of competence coefficients is between minC = 0.9 and maxC = 1.
Competence coefficients can be used to obtain an assignment of tasks that meet 'seriality' requirements. We understand seriality as allocating a series of the same tasks to a worker, if possible. This seriality saves time by reducing the time it takes to adjust the workstation (changeover time), as well as automation and personal optimization as a consequence of repeating the same tasks many times over. Although the proposed solution does not include any changeover time, tests show that more significant variation in the values of competence coefficients increases the 'seriality' effect. We can obtain the variation mentioned above by taking, for every single task, a large number of different values of competence coefficients, e.g., by specifying these values to three or four decimal places.
The attempt to automate the determination of competence coefficients should start with recording the knowledge of production managers in numerical form. Then the factors affecting the quality of work should be identified, and their numerical evaluation should be recorded in the form of an information table during long-term observation. For example, we quickly identify three of these factors: the number of tasks of a given type carried out so far, the reduction in lead time, the number and importance of corrections (loss of time for corrections, material losses). With such an information table, an appropriate classification model can be used to determine competence coefficients. The obtained coefficients should be monitored on an ongoing basis and, if necessary, corrected.
Machine-learning-based classification methods can be used to solve the problem of determining competence coefficients. There are many models of classifiers in the literature, for example (Artiemjew 2008;Staruch 2017;Staruch and Staruch 2017).

Dummy worker
The next important element in solving the CBA problem is a fictitious worker, whom we call Dummy. Dummy is the last worker (indicated by d) with unlimited available time. Dummy guarantees the existence of a solution and it is used to identify tasks that have not been assigned. Minimization of a total normative time of tasks assigned to Dummy is the desired property in the solution of CBA. Dummy's competence coefficient C d is the same for all the tasks and plays the role of penalty for unassigned tasks.

Task blockade
By setting the values of some competence coefficients at a sufficiently high level (indicated by B), we obtain a way of blocking the allocation of tasks to workers with insufficient competences to perform them. On the other hand, it is also possible to block the allocation of simple tasks to highly qualified workers. The need for blockages is discussed and analyzed in a field called Task Assignment, where binary values of competence coefficients indicating skills or authorization are used (see, e.g. [26]).
The value of B should be greater than C d . B = 3 · C d is sufficient to block the assignments as it is the known routine in the classic Transportation Problem.

Optimization models
In this section, we discuss various Integer Linear Programming (abbr. ILP) models that can be used to solve the CBA problem.
We use binary decision variables x i j , where x i j = 1 if and only if the jth task is assigned to the ith worker. We have m = m 0 + 1 constraints describing the requirement that each worker can perform any number of tasks within his or her available time L i , where Dummy is the mth worker with L m = n j=1 l j . We also have n constraints saying that each task can be performed by exactly one worker and m · n binary constraints on x i j .
We consider the following two different optimization models M1, M2.

M1
: M2 : The above models differ in the constraints on workers' available time. In model M1, we assume that competence coefficients do not affect the speed of workers' work and therefore we plan to complete each task in its normative time. M2 presents the assumption that the execution time of the tasks decreases proportionally to the values of competence coefficients. Which model to choose to be implemented in production must be decided by the factory management.
At first glance, we can see that the proposed models are very similar to the minimization problem called in the literature the Generalized Assignment Problem (abbr. GAP) that is known to be NP-hard and even APX-hard to approximate it. There are many algorithmic methods in the literature to obtain a sub-optimal solution to this problem based on different types of heuristics (see Baykasoglu et al. 2007;Cattrysse et al. 1994;Chu and Beasley 1997;Diaz and Fernandez 2001;Lourenço and Serra 2002;Nauss 2003;Osman 1995;Randall 2004;Savelsbergh 1997;Yagiura et al. 2006).
The GAP has the two 2-dimensional functions: the cost function Cost and the weight function W .
M1 is equivalent to the GAP when we take

The use of dummy in optimization models
Another approach to optimization of the CBA problem is the minimization of the total assignment to Dummy. Practical applications justify the use of this type of model because the tasks that are assigned to Dummy should be treated as tasks that will not be performed during the current production shift. Therefore, the model mentioned above pursues a simple objective formulated as 'to assign as many tasks as possible to the workers.' As Example 1 shows, the value of C d should be big enough to obtain both: assigning tasks to workers with the best competencies and minimizing Dummy's assignment.
Hence the objective takes the form of minimization of the sum D = n j=1 t j · x m j . This objective function does not take into account the competence factors and the assumed blockages for assigning some tasks to some workers, as well. Hence for each blocked assignment (with indices i j) we set x i j = 0 and add these equations to the set of constraints. Let DM1 and DM2 denote the described above Dummy optimization models with constraints corresponding to M1 and M2, respectively.
It is easy to see that an equivalent formulation of DM1 and DM2 is replacing the objective function from these models by maximization of the D * = m o i=1 n j=1 t j · x i j . The symbols D * M1 and D * M2 denote these two models. Notice that Dummy is not represented in these two models, as we have no effective constraints on the Dummy's available time as well as Dummy is not included in the objective. Thus the optimization objective is to choose the tasks from the set of all tasks that give maximal profit (total normative time of assigned tasks). Hence, D * M1 and D * M2 are instances of the Multi-Knapsack Problem (abbr. MKP) (see Martello and Toth 1990), where the profits are set as p i j = t j and the weights are described as W i j = t j for M1 and W i j = t j · C i j for M2. Thus we can use some of the sub-optimal algorithms for solving MKP from the known methods described in the rich literature on the subject.
To get the reasonable models DM1, DM2, D * M1, and D * M2, we need the assumption that there are enough tasks so that the optimal solution for M1, M2 includes the nonempty set of tasks assigned to Dummy.
Notice that DM1 (and D * M1 as well) does not include any competence factors, therefore it is useless for solving CBA with differentiated values of competence factors. Now, we discuss the issue of determining the value of the Dummy's competence coefficient C d by comparing the DM2 model with the M2 model. First, it should be noted that they are not equivalent.
Obviously, in the case when all the tasks can be assigned to the 'regular' workers (i.e. D = 0 ) it is enough to take C d > maxC to be sure that any optimal solution for M2 will not assign any task to Dummy. On the other hand, in this situation any solution for DM2 with D = 0 is optimal without guaranteeing optimal solutions for M2.

Example 1
Consider the CBA problem with parameters as in the following table: Then we get the optimal solution for DM2 assigning T ask 1 and T ask 2 to Dummy, T ask 3 to the first worker and T ask 4 to the second worker. Then D 0 = 90, F 0 = 117 + 100 + 90 = 307.
The optimal solution for M2 is the assignment of T ask 3 to Dummy, T ask 4 to the first worker and the rest of the tasks to the second worker. In this case D = 100, F = 130 + 90 + 40 + 45 = 305.
Hence, D > D 0 while F < F 0 . Notice, that a greater value of C d can eliminate this undesirable property. In this example, C d ≥ 1, 5 assures that F ≥ F 0 . This example shows that M2 and DM2 are not equivalent and that the value of C d has an impact on solutions and relationships between different kinds of models.
Experiments carried out in plants mentioned in this paper showed that it is enough to take C d = 2maxC.

An exemplary algorithmic solution
The proposed algorithmic solution is useful for finding an approximate solution to both problems M2 and DM2. It means that the algorithm tries to assign tasks to workers with the best competencies while minimizing the Dummy's assignment.
The first step of the algorithm named Initial is to obtain an initial solution. The greedy algorithm takes the tasks according to the decreasing order of normative times and assigns them to the workers with the lowest possible competence coefficient.
We assume that the tasks are sorted by decreasing normative time. We associate to every worker i his/her remaining time R i , where R i = L i at start. Then for every i = 1, ..., m we search the ith row of the competency matrix and for the minimal value, say c i j and check if c i j · t i ≤ R i . If the answer is 'Yes,' we associate the jth task to the ith worker and update her/his remaining time R i := R i − c i j · t j and then go to the next task. If the answer is 'No,' we search for the next minimal value in the ith row and repeat this sequence of activities until all the tasks are assigned to workers, Dummy included. As a result, we get the assignment of tasks to workers, which we call the initial solution.
The second step called Improve is to improve the initial solution. We suggest using a procedure that minimizes the total assignment to the Dummy while optimizing the objective function for M2. However, a slight modification (i.e. taking t i ≤ R i instead of c i j · t i ≤ R i ) results in solving for M2.
Let's assume that the tasks assigned to the Dummy are sorted by decreasing normative times. The two main functions TWO-FOR-ONE and ONE-FOR-ONE are used at this stage of the algorithm. These functions have two tasks or one task currently unassigned and try to assign them the chosen worker in the place of other task. Such a reassignment should improve the solution. In case of TWO-FOR-ONE function let us assume that we have the two tasks j, k with t j ≥ t k which are taken from Dummy. If we assign them to the worker i in place of the task l, then the value of the objective function is changed and the difference will be called the improveValue We try to maximize the improveValue under assumptions that that The ONE-FOR-ONE function takes a one task Thus the Improve stage is as follows: 1. Take the first two tasks from Dummy j, k and search for a worker i with the task l so that the conditions Cond1. and Cond2. are satisfied and the improveValue is maximal. 2. If such a worker exists, we assign the tasks j, k to him/her, take off the l task and update the remaining time R i = R i − c i j · t j − c ik · t k + c il · t l . Now, we loop the ONE-FOR-ONE function with the current task l, possibly substituting it by other current tasks. If no substitution is done for the current task the loop stops and the task goes to Dummy and takes place according to its normative time. 3. If after searching the entire set of workers (using the TWO-FOR-ONE function) no substitution is done, the task j goes to Dummy to the first place, and the task i goes to the new list of unassigned tasks. 4. While there are at least two tasks assigned to Dummy we repeat steps 1. -3. Finally, Dummy has one task only. This task goes to the new list of rejected parts. 5. Now, all the unassigned tasks are assigned to Dummy, and we can repeat the whole procedure until the list of unassigned tasks is empty. Optionally, we can stop the algorithm and assign all the tasks from unassigned tasks and from rejected tasks to Dummy.
6. An additional option is to perform the ONE-FOR-ONE function for every rejected task, at the end of the algorithm.

Experimental results
This algorithm is not optimal, but it is fast, so in the absence of a better solution it can be successfully used. The computational complexity of the Initial stage is nlogn for sorting and n · m for assigning. Let n 0 denote the number of tasks assigned to Dummy. The Improve stage takes pairs of tasks assigned to Dummy, which is approximated by O(n 2 0 ). Search procedures in steps 1.-3. are approximated by O(n 2 ). The proposed solution is also useful in managing color based priority. First, we run the Initial stage for all the tasks and then perform the Improve stage for the 'red' tasks assigned to Dummy performing steps 1. -5. If there is any 'red' task in rejected tasks we perform ONE-FOR-ONE function for 'red' rejected tasks waiving Cond.1 and taking off 'non-red' tasks, only.
We carried out a series of experiments for data sets that were described by some chosen parameters. All the tests were made on personal notebook (Intel(R) Core(TM) I5-2520 M CPU @ 2.50 GHz 2501 MHz) The number of workers was 10, 30, and 50. The available time for every worker was fixed to 450 minutes. The total normative time of tasks was expressed as the total workers' time multiplied by the given factor (F1 = 1.25 or F2 = 1.5) and next, the set of tasks was generated. We used the following two sequences S1 = (0.2, 0.6, 0.2), S2 = (0, 0.2, 0.8) to describe the percentage share of long, medium and short tasks, where normative times for long tasks were drawn randomly from 60 to 120 minutes, for medium tasks from 30 to 60 minutes, and short tasks from 10 to 30 minutes. The competence factors were randomly drawn from 0.9 to 1 with an accuracy of 0.01. The number of blocked pairs "task-worker," i.e. pairs with the competence factor equal to 3, was on the level of 10 percent of the total time of tasks. The competence factor for Dummy is fixed on 2.
We use notation like 50F1S2 to describe a data set obtained for 50 workers with the F1 factor and the S2 sequence.
We compared our algorithm with the algorithms (cut methods and branch methods) dedicated to solving ILP problems and implemented in the GNU Linear Programming Kit (abbr. GLPK) GLPK 2020. The first one is based on Tomlin-Driebeek heuristic Driebeek 1966 (GLPK parameter -drtom ), and the second one is based on different cut methods (GLPK parameter -cuts). For saving time we set the parameter -tmlim to 900 seconds. The experimental results are presented as an average percentage gap, where the percentage gap was computed as the deviation of the obtained value of the objective function from the lower bound of the instance divided relative to the lower bound. The lower bound for the given instance is the value of the objective function for continuous LP relaxation.
The table below contains the experimental results, where the first two columns describe the used data sets and the number of instances. The next three columns contain results for the algorithm presented in this paper: average runtime of the algorithm, average percentage gap for the Initial stage, and average efficiency after the Improve stage. The last three columns present average percentage gaps for the two ILP methods and also, a share of fails, where the failure means that after 1000 seconds the given algorithm got no solution.
The GLPK methods gave non-optimal results in all the cases and for instances with a rather big number of tasks they often fail. These instances are hard also for our algorithm, where the runtime of the Improve stage is from 900 to 1000 seconds. On the other hand, the Initial stage took less than one second giving quite good results.

Application of CBA in production planning
This section is devoted to discussion of application of the CBA problem in real-life Demand Driven Institute 2019. Demand-Driven Manufacturing is a modern method of demand and production management that is associated to to a variety of managements methods like Lean (or Just-in-Time) Manufacturing (Ohno 1988;Womack and Jones 2003), Kanban (Anderson 2010;Hammarberg and Joakim 2014;Ladas 2009, Scotland 2018, Product Customization (Hvam et al. 2008), Product Differentiation (Sharp and Dawes 2001).
First, we describe the main characteristics of production management under consideration.
1. Product variety -applied routines like Product Customization or Product Differentiation strengthen this effect 2. Short customer tolerance times -a short time to complete the order, deliver the product on time and meet deadlines at all stages of production 3. On-demand production -all products are manufactured after acceptance of demand, no production for stocks 4. Pressure to less inventory -production planning to save storage space, both for raw materials and finished products 5. Priorities indicating which products should be made first -the most common routine is Kanban based system of the three colors 'Red', 'Yellow,' 'Green.' 6. The need for flexibility in production planning -fast reaction on different kinds of changes and perturbation in production The first aspect called 'product variety' is the main characteristic of manufacturing in plants under consideration. Such a diversity of production affects the size of the CBA problem. However, we claim that CBA is a useful and flexible tool for solving most problems concerning such a kind of production.
The proposed methodology is based on repeated use of the algorithm solving the CBA problem. Therefore, we assume that we have a fast algorithm (abbr.ALG) for solving CBA. There is a variety of sub-optimal algorithmic solutions for GAP or MKP that can be found in the literature. A decision which solution is the most useful in the application of CBA in production planning should be based mainly on the 'speed' criterion. Because of high 'product variety,' a fast algorithm is more desirable even at the expense of accuracy. In the absence of better algorithms, the solution proposed in the previous section can play role of ALG.
CBA problem is described by parameters: number of tasks, normative times of tasks, number of employees, available times of workers, competence matrix. We consider also additional information related to the priority of tasks in the form of color-coding, for example, as in the Kanban method. Red color means that the tasks are urgent and must be completed within the fixed deadline. Yellow indicates tasks that can only be carried out if the red tasks have been assigned in advance. Green indicates a task that can be performed after red and yellow tasks are assigned. You can also consider a preference model that takes into account a more significant number of degrees of preference than the three ones described above.
We are going to show how to use the CBA (supported by ALG) for solving problems caused by the aspects 2.-6. above. The general schema is as follows:

Input Data
We set the values of parameters of CBA basing on the aim of the use of CBA and the use of color-coded priority 2. Run ALG We run ALG for the Input Data 3. Analysis of the Output of ALG We analyze the result of ALG, and we change the Input Data and run ALG again if we need it.
To use CBA to set the date of order completion (aspect 2. 'short customer tolerance times'), we need to set a certain date of completion and flag the previously planned tasks to red, and the new tasks to yellow. If Dummy has no red task in the output of ALG, then the new demand can be considered as being planned on this date. If not, we check the next date of completion.
The aspects 3-5 are reflected in the appropriate prioritization of tasks. Since storing space is minimal, there are few yellow tasks and even fewer green tasks. The typical situation is that production plans are rather tight. Hence the fast reaction on production disturbances (aspect 6.) is necessary. We can run ALG with input data updated after disturbances. If Dummy has any red task assigned to, the manager can react in the following different ways basing on the above general schema:

Summary
We discussed the problem of competence-based tasks assignment to workers in factories that use Demand-Driven Manufacturing. The proposed solution to this problem is based on our interviews with managers in medium-sized upholstered furniture factories in Poland. Furniture production is one of the most important industries in Poland, and there is a need to implement some innovative solutions supporting production management. However, there is no reason preventing CBA to be applied to any type of production plants.
A part of the presented CBA problem was awarded in 2016 with the 'Debut of the Year' award given by the Rzeczpospolita newspaper. We then introduced the coefficients of competence, blockage and virtual Dummy worker. The algorithmic solution based on the continuous relaxation of the M2 model and the repetition of the algorithm in rounds was far from perfect. However, it resulted in such an improvement in the quality of assignment of tasks to the workers that it met with appreciation in the form of verbal comments and the award mentioned above. The solution called VINCI is implemented, for example, in Primavera Furniture 3 factory. This paper is based on a more in-depth formal analysis of the real-life production challenges. We discussed the application of the CBA model in production planning, where the main advantage of our solution is its flexibility. This flexibility has been achieved by highlighting the most critical elements (parameters) influencing the final assignment of tasks to the workers.
The optimization models described in this paper indicate the optimization areas (Generalized Assignment Problem, Multi-Knapsack Problem) which, as they have been known for years, offer many different algorithms to solve them. Thus we can choose the best algorithmic solution that meets the managers' requirements, taking into account the specific way of production management, the practices prevailing in the factory, the specificity of production, directions of optimization.
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