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Sample average approximation method for a new stochastic personnel assignment problem
Accepted Manuscript Sample average approximation method for a new stochastic personnel assignment problem Amir Ghorbani Pour, Zahra Naji-Azimi, Majid Salari PII: DOI: Reference:
S0360-8352(17)30404-7 http://dx.doi.org/10.1016/j.cie.2017.09.006 CAIE 4892
To appear in:
Computers & Industrial Engineering
Received Date: Revised Date: Accepted Date:
21 February 2017 29 August 2017 6 September 2017
Please cite this article as: Pour, A.G., Naji-Azimi, Z., Salari, M., Sample average approximation method for a new stochastic personnel assignment problem, Computers & Industrial Engineering (2017), doi: http://dx.doi.org/ 10.1016/j.cie.2017.09.006
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Sample average approximation method for a new stochastic personnel assignment problem Amir Ghorbani Pour, Zahra Naji-Azimi∗ Department of Management, Ferdowsi University of Mashhad, Mashhad, Iran. {Aghorbani,znajiazimi}@um.ac.ir Majid Salari Department of Industrial Engineering, Ferdowsi University of Mashhad, P.O. Box 91775-1111, Mashhad, Iran. [email protected]
July 8, 2017
∗ Department of Management, Ferdowsi University of Mashhad, Mashhad, Iran. TEL: +98 051 38805352, e-mail: [email protected]
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Sample average approximation method for a new stochastic personnel assignment problem September 7, 2017 Abstract We introduce a practical generalization of the standard personnel assignment problem. In this problem, we have a set of personnel and a set of lost and found centers. Each center receives a stochastic demand. Some constraints, namely, the work skills of different personnel, balancing workload while assigning personnel to the centers and some other practical constraints are taken into account. The objective function is to minimize the assignment cost and the cost occurred by the shortage of personnel in different centers due to the stochastic arrival of the demand over a given time period. We have developed a sample average approximation technique to solve the introduced problem. Computational tests for the studied problem show the effectiveness of the proposed method.
Keywords: Personnel assignment problem, Stochastic optimization, Monte Carlo, Sample average approximation
1
Introduction
The assignment problem (AP) is one of the most studied problems in the combinatorial optimization field. The AP can be considered as assigning n tasks to n agents with the minimum assignment cost in a way that each task is allocated to exactly one agent and each agent is allocated to exactly one task. It can also be considered as the matching between two (or more) sets so that the total cost of the matched pairs is minimized (see, e.g. Burkard, 2002, Pentico, 2007 and Burkard et al., 2012). One of the natural extensions of the AP is called the personnel assignment problem (PAP). In this problem, each task needs some requirements while agents may have various abilities. The problem is to assign a given number of agents to a smaller number of tasks so that the total benefit (score) corresponding to this assignment is maximized. In this paper, we introduce a practical generalization of the standard PAP. In the defined problem, the goal is to assign a set of personnel to a number of work stations over a given time period. The work stations need a different number of skilled workers and the capacity of each station is bounded, so a limited 1
number of personnel can be assigned to each station. In addition, the job rotation has to be taken into account meaning that a person cannot be assigned to the same station within a predetermined number of consecutive time periods. The demands (customers) are arriving to each work station stochastically and for each predetermined number of customers at least a distinct number of personnel has to be available at each station. The objective of the problem is minimizing the assignment cost or maximizing the service quality. The introduced problem has several applications in the real world. Assigning personnel to available branches of different public service centers (e.g., post office, electronic service centers in E-governments policies, and etc.), assigning trainees to different stations for job training (e.g., assigning officer cadet to police centers, interns to different hospital departments or labors to different production lines in a factory) are real examples of this problem. In this paper, we consider the application of the introduced problem by assigning a number of personnel to some lost and found centers.
2
Literature Review
The PAP is NP-complete problem (Toroslu, 2003; Toroslu and Arslanoglu, 2007) and different solution techniques have been developed to solve this problem (see, e.g. Herrera et al., 1999; Karsak, 2000; Toroslu and Arslanoglu, 2007; Lin et al., 2012). In 1974, Trippi et al. considered a large scale military PAP and proposed a mathematical model and an algorithm based on the Ford-Fulkerson network flow approach to solve the problem. Constantopoulos (1989) designed a decision support system to assign a large number of agents to tasks with respect to multiple criteria (Constantopoulos, 1989). In 2003, Toroslu defined the PAP with hierarchical ordering constraints for the organizations in which the agents are categorized into distinct levels such as army power. He showed that this problem is NP-complete and proposed some heuristic and approximate algorithms to solve the problem (Toroslu, 2003). Later in 2007, Toroslu and Arslanoglu extended the PAP with both hierarchical ordering and set restriction constraints and defined an NP-hard multi-objective optimization problem. They proposed a genetic algorithm (GA) to solve different versions of the problem. Cao and Wang (2010) considered the differences among agents and their corresponding effect on the assembly production problem. They proposed a personnel assignment model in which the objectives were maximizing the coefficient of the minimum station fitness and balancing the station fitness for each assembly line. They developed a heuristic algorithm to solve the problem. In 2011, Mayrhofer et al. integrated the sequence planning with the personnel assignment problem in assembly line problem. They presented an approach based on constraint programming with incident discrete simulation to solve the problem. Lin et al. (2012) considered the bi-objective PAP and
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proposed a particle swarm optimization algorithm with random-key encoding scheme. Telhada in 2014, considered a mixed problem of shift scheduling and task assignment problem and developed several MIP formulations to solve the problem. Sir et al. (2015), considered the nurse-patient assignment problem and introduced some models which assign patients to nurses in a balanced manner. To this aim, they used to distribute acuity scores from the patient classification systems and survey-based nurses perceived workload. Sungur and Yavuz (2015) defined an assembly line balancing problem in which tasks need different requirements and qualification levels of workers are ranked hierarchically, while a lower qualified worker can be substituted with a better worker with a higher cost. They formulated the problem as an integer programming model. Recently, Liu et al. (2016) considered worker assignment and the production planning problem simultaneously. In their problem, bottleneck workstation may transfer according to learning and forgetting effects of workers, while the objective is to minimize the back-order and holding cost of inventory. There are some papers dedicated to the PAP in fuzzy environment. A fuzzy multi-criteria decision making (MCDM) algorithm was proposed by Liang and Wang (1994) where for selecting the agents both subjective and objective assessments were taking into account by incorporating a fuzzy MCDM algorithm. Herrera et al. (1999) considered the problem with some verbal information and proposed a GA to solve the problem. A fuzzy multiple-objective programming technique was proposed by Karsak (2000) to select the agents. In the introduced problem, some qualitative factors of agents have been incorporated by fuzzy linguistic variables and the preference between the objectives has been determined by decision makers with linguistic variables. Huang et al. (2009) considered the PAP with fuzzy characteristics and proposed a bi-objective binary integer programming model to formulate the problem. They designed a feedback mechanism to iteratively solve the problem with the consideration of interdependencies among tasks and the differences among the selected agents simultaneously. Some researchers have considered the PAP in a stochastic environment. A stochastic version of the PAP was proposed by king (1965). He believed that the vacant tasks are usually filled with the agents who never have the experience of working on that task. So, he integrated the prediction of an agent’s performance (or job success) during the allocation of tasks to the agents. Punnakitikashem et al. (2013) considered an integrated nurse staffing and assignment problem and presented a stochastic integer programming model with the objective of minimizing the expected excess workload of nurses. They proposed some exact approaches to solve the problem. In this paper, we introduce a practical generalization of the standard PAP. In this problem, we are given a set of lost and found centers and a pre-determined number of personnel to be assigned to the centers. These centers are located in different parts of the city where the demand of each part is a
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stochastic variable. The foundation of a reliable assignment is to allocate the existing personnel to the set of centers in the best possible way in order to meet the demand of customers in different demand scenarios. The capacity of each center for accommodating the personnel is limited and bounded by a given value. In practice, some places are the destination of more tourists than the others and as a consequence the corresponding center has more demand. As a result, to have a fair assignment, the company aims at assigning the personnel to the centers in such a way that personnel work at different centers during a given time period. People arrive to each center stochastically where, according to the standards there should be at least one person in each center to cover each λ units of demand (service quality). Moreover, depending on the location of the center, it may need some skilled workers to be available at that center in order to satisfy the demand of the customers. The objective is to minimize the total assignment cost plus the cost occurred by the shortage of personnel in different centers due to the stochastic arrival of the demand. The new model contains the following differences with respect to those papers available in the literature: 1) in the new introduced model, the arrival demand rate is stochastic, 2) we consider the job rotation for personnel, and 3) the quality of service has not been already considered in the previous papers. The remainder of this paper is organized as follows. Section 3 gives a formal description of the problem. In Section 4, we develop a method based on the combination of the multi criteria decision making tools and sample average approximation (SAA) technique to solve the introduced problem. Computational results for the real case studied problem are presented in Section 5. Finally, Section 6 concludes the paper.
3
Problem Statement
In this section, we develop a mathematical programming model for the introduced problem. We denote by I = {1, · · · , m} and J = {1, · · · , n} the sets of centers and personnel, respectively. In addition, let T = {1, · · · , τ } is the set of time periods and K = {1, · · · , κ} is the set of different work skills. For each i ∈ I and k ∈ K, αik is the minimum number of skilled workers of type k needed in center i. Also, we represent by Sk the set of personnel who are expert in skill k. Since the capacity of each center is limited, the maximum number of personnel that can be accommodated in center i ∈ I is bounded by the threshold βi . In each period t ∈ T , qit shows the number of people (clients) coming to center i to receive services. For each λ clients in a given center, we need at least one worker to cover the demand. Finally,
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cij denotes the cost of assigning worker j to the center i. The variables to define the model are as follows:
xijt
⎧ ⎪ ⎨ 1 = ⎪ ⎩ 0
if worker j is assigned to center i in period t,
∀i ∈ I, j ∈ J, t ∈ T
(1)
otherwise.
The model reads as follows:
Model I)
min Z =
cij xijt
(2)
i∈I j∈J t∈T
subject to:
xijt = 1
∀t ∈ T, ∀j ∈ J,
(3)
∀i ∈ I, ∀t ∈ T, ∀k ∈ K,
(4)
∀i ∈ I, ∀t ∈ T,
(5)
∀i ∈ I, ∀t ∈ T \ {τ }, ∀j ∈ J,
(6)
∀i ∈ I, ∀j ∈ J,
(7)
∀i ∈ I, ∀t ∈ T,
(8)
∀i ∈ I, ∀t ∈ T, ∀j ∈ J.
(9)
i∈I
xijt ≥ αik
j∈Sk
xijt ≤ βi
j∈J
xijt + xijt+1 ≤ 1 xij1 + xijτ ≤ 1 qit xijt ≥ λ j∈J
xijt ∈ {0, 1}
The objective function (2) is to minimize the total assignment cost. Constraint set (3) imposes that in each time period all personnel have to be assigned to different available centers. For each k ∈ K, constraints (4) assure that in each period t ∈ T and each center i ∈ I, at least αik skilled personnel have to be available. Taking into account the capacity of each center i ∈ I, constraint (5) stipulates that at most βi personnel can be assigned to the center i, during different time periods. Constraint sets (6) and (7) impose that nobody is allowed to work in the same center in two consecutive time periods. For each t ∈ T and i ∈ I, constraint set (8) shows that for satisfying the demand of each λ clients, at least one worker has to be available. Finally, the definition of the variables are given in (9). In our studied problem, qit is a stochastic variable with a known distribution function. Suppose Lit is the expected value of personnel’s slackness in center i and period t. Introducing the ηit as the unit cost of assigning not enough personnel to center i in period t, the new model reads as follows:
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min Z =
Model II)
i∈I j∈J t∈T
cij xijt +
ηit Lit
(10)
i∈I t∈T
subject to: (3) to (7), and ⎡
⎤+ q it − Lit = Eqit ⎣ xijt ⎦ λ
∀i ∈ I, ∀t ∈ T,
(11)
∀i ∈ I, ∀t ∈ T, ∀j ∈ J.
(12)
j∈J
xijt ∈ {0, 1}
The first part of the objective function (10) is the minimization of the total assignment cost while the second part minimizes the cost occurred by assigning insufficient personnel to different centers. For each i ∈ I and t ∈ T , constraint (11) represents the expected slackness of the personnel in which X + = max{0, X}
4
Solution Method
In this paper, we integrate the multi criteria decision making tools and the Monte Carlo simulation technique to solve the introduced problem. In particular, to obtain the corresponding values of the assignment cost (cij ) and to rank the personnel according to the job skills, we use the combination of the analytical hierarchy process (AHP) and the simple additive weighting (SAW) methods. To this aim, we extracted the following hierarchy of the required skills (see, Figure 1). In addition, we utilize the Monte Carlo simulation technique to solve the proposed stochastic model (see, Section 3). In the following, we provide some theoretical details concerning the Monte Carlo simulation technique while the detailed information concerning the combination of the AHP and SAW will be provided in Section 5. Insert Figure 1 around here. Several techniques have been introduced to solve the stochastic problems, including Monte Carlo algorithm or Sample Average Approximation Method(see, e.g. Abdullah et al., 2014, Huifu and Zhang, 2014, Koutras et al., 2014 and Wu et al., 2009). Since the exact evaluation of the expected value of (11) is difficult, we use the SAA method in order to approximate the true problem. The corresponding steps of the algorithm are provided in the following.
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• For each center i ∈ I and each period t ∈ T , generate M independent samples of demand, each of Q 1 2 , qit , · · · , qit ). size Q (i.e., qit
• The expected value Lit is estimated as Lit =
Q
s qit s=1 ( λ
+ − n j=1 xijt ) . Q
• The objective function Z is calculated using Model II and substituting Lit from the previous step. It should be mentioned that by increasing the amount of Q, with probability one, Lit converges to the expected value in (11) (see, Kleywegt et al., 2001). • Assuming x∗ to be the optimal solution of Model II, fix the value of variable x and estimate the
true objective value of x∗ by generating Q independent samples where Q is considered to be much larger than Q. ˆ obtained by applying the SAA method. In the following, we Suppose that we have a solution X ˆ to be used as a candidate for solving the real problem. introduce the way of evaluating the quality of X To do so, suppose that we are given M independent samples of size Q. The corresponding steps are as follows: • Let fm represent the objective value of the mth sample obtained by solving Model II. • Compute f¯ and σf2¯ as follows: f¯ = 2 σM,Q =
1 M
M
1 M(M−1)
m=1
fm
M
m=1 (fm
− f¯)2
ˆ (i.e., f˜(X)) ˆ using a large number of scenarios (Q ) for • Calculate the true objective function of X the demand. 2 • Compute σQ as follows: 2 σQ =
1 Q (Q −1)
Q
ˆ
j=1 (F (X, ϑj )
ˆ 2 − f˜(X))
ˆ ϑj ) is the objective function of Model II with a fixed solution X ˆ using the j th In which F (X,
scenario (ϑj ) from the set of Q different scenarios. Now, the upper and lower bounds with 100(1 − α)% confidence are calculated as follows: ˆ + zα σ Upper bound = f˜(X) Q Lower bound = f¯ − tα,v σM,Q In which v = M − 1, tα,v is the α-critical value of t-distribution with v degrees of freedom. In addition, zα is the standard normal critical value. 7
Table 1:
The minimum number of skilled workers required at each center (αik ). Center Foreign language Computer knowledge Social skills
Table 2: Center 1 βi 6 0.1458 ηit
5
1 1 3 1
2 0 1 1
3 0 1 1
4 0 1 0
5 0 1 1
6 2 2 1
7 0 1 1
8 0 1 1
9 0 1 0
10 0 1 1
Some parameters of the model.
2 3 4 3 3 4 0.1250 0.1042 0.1042
5 6 3 5 0.0625 0.1458
7 8 9 3 3 3 0.0625 0.0833 0.0625
10 3 0.1042
Computational Results
In this section, at first we introduce the characteristics of the studied problem. Then we discuss on the results obtained by solving the stochastic model proposed in Section 3. Finally, we propose several tests in order to assess the performance of the developed method.
5.1
Introducing the real problem
To run the model, we use the data provided by a real company. Totally, there are 10 lost and found centers in the city of Mashhad, Iran, which is one of the Iran’s tourism destinations having around 32 million visitors per year. These centers are different in size and the demand rate at each center follows a known distribution function. Totally, 25 personnel with different abilities are available to be assigned to the centers. Each worker has at least one skill from the three distinct skills, namely, foreign language, computer knowledge and social skills, which are required to work in different centers. The number of periods is 47 and to have a reasonable service quality, we set λ = 6. Based on the location of each center, a minimum number of personnel with known skills is needed which are provided in Table 1. Taking into account the capacity of each center, the maximum number of personnel who can work in each center (βi ) is limited and is provided in the second row of Table 2. In addition, the penalty concerning the shortage of personnel at each center i (ηit ), considered as the same value for different time periods, is reported in the last row of Table 2. These values are achieved by normalizing the average penalties proposed by the decision makers for different available center. The distribution functions of the demand in different centers have been estimated using the data over the last two years. Essentially, the data contains the number of people referred to each center to receive services. We have used the EasyFit software in order to obtain the corresponding distribution function at each center. In addition, the Kolmogrov-Smirnov test has been applied to check the fitness of the estimated distribution functions. The assignment cost (cij ) is one of the most important parameters in the introduced model. Since the
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Table 3:
Classifying centers according to the number of demand referring to them. Class Very quiet Quiet Moderate Crowded Center {7,9} {5,8} {3,4,10} {2,6}
Table 4:
Very crowded {1}
The weights and inconsistency rates of the available criteria and sub-criteria.
Criteria
Weight Inconsistency rate
Social Skill
0.64
Computer knowledge 0.27
Foreign language
0.05
0.09
Sub-criteria Old caring Child caring Public relations Problem solving ability Work managing ability Overall knowledge Specific software English Arabic Turkish Ordu
Weight 0.05 0.25 0.22 0.20 0.30 0.25 0.75 0.24 0.36 0.34 0.06
Inconsistency rate
0.06
0.00
0.00
personnel get paid monthly and it is a fixed value for each month, the assignment cost cij is substituted with a parameter which is related to the undesirable assignment of worker j to each center i. In particular, undesirable assignment occurs when assigning a skilled worker to an uncrowded center or assigning an unskilled worker to a crowded center. So, according to the historical data we have classified the centers into five groups from very quiet to very crowded (see, Table 3). In addition, we use the combination of the analytical hierarchy process (AHP) and the simple additive weighting (SAW) in order to rank the personnel according to the job skills. To this aim, we have extracted the hierarchy of the required skills (see, Figure 1). Table 4 represents the weights and inconsistency rates corresponding to the available criteria and sub-criteria. By using the extracted weights and the SAW technique, we obtain a score for each worker and accordingly we classify the workers into five groups from very weak to very capable. Finally, taking into account the group that each center belongs to and the allocated group to each worker, the final values for different cij s are reported in Table 5. The last parameter to be calculated is the set of skilled worker for each k ∈ K (i.e., Sk ). To determine this parameter, we put in Sk all workers whose score in the corresponding skill is greater than 60 percent of the average score for skill k over all personnel.
Table 5:
Allocated cij according to the class of the centers and personnel.
Very capable Capable Moderate Weak Very weak
Very quiet Quiet 1.00 0.75 0.75 0.50 0.50 0.25 0.25 0.00 0.00 0.25
Moderate 0.50 0.25 0.00 0.25 0.50
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Crowded 0.25 0.00 0.25 0.50 0.75
Very crowded 0.00 0.25 0.50 0.75 1.00
Table 6: Q
The objective function and computational time for different values of Q.
Minimum 2 11936.24 12033.93 5 12022.39 7 10 12060.18 12072.57 20 12100.73 30 12109.39 40 12130.71 50 12125.20 60 12137.31 70 12130.79 80 12141.83 90 100 12138.96
Table 7:
Objective value Average Maximum 12105.38 12431.97 12144.62 12268.04 12142.99 12285.13 12142.87 12195.98 12169.76 12259.20 12163.71 12238.75 12166.53 12220.25 12164.34 12217.98 12160.11 12212.30 12161.72 12189.86 12161.65 12194.79 12159.75 12201.77 12163.77 12199.73
Computational Minimum Average 2.14 4.82 2.51 4.92 2.67 8.28 2.73 14.24 14.81 18.69 16.14 22.71 22.07 27.61 26.79 35.85 23.48 31.31 38.17 43.56 46.31 52.09 49.79 58.98 45.28 52.86
time Maximum 10.25 9.37 13.11 26.5 24.51 28.25 31.82 54.6 36.93 50.73 57.6 68.23 63.53
Estimation of the true objective function with Q = 1000. Q 2 5 7 10 20 30 40 50 60 70 80 90 100
5.2
SD 141.41 84.4 72.37 48.69 50.59 38.24 36.36 27.68 29.61 21.29 22.73 18.32 21.97
Minimum 12242.67 12179.23 12173.88 12167.28 12167.36 12166.93 12166.94 12166.79 12166.80 12166.89 12166.97 12166.66 12166.89
Average 12269.91 12195.38 12177.70 12171.33 12168.52 12168.02 12167.96 12167.54 12167.36 12167.35 12167.57 12167.37 12167.48
Maximum 12287.55 12221.09 12182.22 12174.22 12170.00 12169.98 12168.90 12168.85 12168.44 12167.94 12168.39 12168.74 12168.76
SD 12.86 14.68 2.69 2.54 0.76 0.79 0.64 0.66 0.53 0.35 0.52 0.6 0.59
The results
We have used CPLEX 12.1 to solve the stochastic model using the SAA technique. All tests have been performed on a computer with an Intel Core due CPU, 2.66 GHz and 3.5GB of RAM. In our experiments, Q ∈ {2, 5, 7, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100}. In addition, to take randomness into account, for each demand scenario Q, 10 independent samples (M = 10) have been generated. The results are provided in Table 6 in which the first column gives the number of demand scenarios (Q). The next three columns give the minimum, average and maximum value of the objective function over 10 independent samples, respectively. Each entry of the column labeled by “SD” gives the standard deviation of the corresponding row. Finally, the last three columns give, respectively, the minimum, average and maximum computational time (in seconds) over 10 independent samples of each demand scenario. According to the data reported in Table 6, by increasing the number of demand scenarios, the computational time to solve the instances increases while the standard deviation decreases. In order to have an estimation of the true objective function of the solutions obtained by different values of Q, in the next step we calculate a better estimation of the true objective function values by
setting Q = 1000. The results are reported in Table 7. Based on the results reported in Table 7, by increasing the value of Q, the corresponding estimation of
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the true objective function converges. The minimum, average and maximum evaluation of the estimated and true objective function with respect to different number of scenarios are represented in Figures 2 and 3, respectively. As already explained, by increasing the number of demand scenarios (Q), the values of the objective functions are converged.
Insert Figures 2 and 3around here. In the following, we perform a test in order to compare the best solution obtained by the SAA technique and that obtained by solving the deterministic equivalent of the problem. Essentially, by substituting the expected value of the uncertain demands in the model and solving that we will have the expected value solution. By solving the exact model, the objective function of the expected value solution is 12288.4 which is around 1% worse than that of the stochastic solution. Figure 4 shows the corresponding gap between the upper and lower bounds in each demand scenario. According to this figure, the average gap is less than 1% for greater than 70 number of samples.
Insert Figure 4 around here. Since the introduced problem is NP-hard, the large size instances cannot be solved to optimality using CPLEX. As a result, the proposed algorithm should be adopted in order to solve such instances. There are two ways to handle larger size data. The first straight forward way is to consider the solution of a (meta)heuristic approach in the last step of the algorithm instead of the optimal solution (i.e., x∗ ). The second way has been introduced by Juan et al. (2015) as a general framework to solve a stochastic combinatorial optimization problem. In this method the corresponding deterministic problem is achieved by substituting the expected value of available random variable(s). Upon this step, a (meta)heuristic approach is utilized to obtain a set of high quality or promising solutions. In an interactive search, the quality of each promising solution for the stochastic model is calculated using a limited number of iterations of the Monte Carlo simulation to have a rough estimate of the solutions quality. Accordingly, a ranked list of solutions is provided according to the estimated quality of the solutions, while a feedback to metaheuristic can be considered to intensive the search of the (meta)heuristic. After reaching the final time given to (meta)heuristic, the simulation with the larger number of scenarios is performed on the ranked solutions. Finally, the set of elite ranked solutions are updated according to the most accurate estimation and final solutions can be presented to decision makers.
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Table 8:
The normalized objective function for different values of Q. Q
2 5 7 10 20 30 40 50 60 70 80 90 100
Table 9:
Estimation of the true normalized objective function with Q = 1000. Q 2 5 7 10 20 30 40 50 60 70 80 90 100
5.3
Objective value Minimum Average Maximum 14587.86 15142.44 16245.24 15177.46 15533.99 15958.32 15166.32 15589.93 16006.04 15332.50 15635.82 15799.80 15410.29 15753.60 16050.94 15504.54 15739.47 16005.63 15547.52 15753.88 15949.01 14587.86 15246.64 15547.52 15608.27 15733.14 15920.31 15652.06 15739.26 15840.19 15628.73 15739.19 15857.85 15668.26 15732.20 15882.84 15657.96 15746.36 15875.55
Minimum 16882.06 15908.36 15882.87 15841.40 15781.52 15761.00 15759.53 15760.28 15757.63 15757.98 15757.62 15757.13 15757.47
Average 17051.64 16091.03 15949.78 15894.55 15810.74 15785.29 15771.32 15772.28 15765.63 15761.68 15761.34 15761.55 15760.90
Maximum 17370.44 16244.20 16071.33 15973.24 15859.15 15810.18 15794.69 15788.48 15779.12 15771.27 15768.54 15769.17 15769.70
Sensitivity Analysis
In this section, we perform a test in order to analyze the impact of the stochastic part of the objective function on the overall solution’s cost. In particular, we increase the corresponding coefficient of the stochastic part of the objective function (i.e., parameter ηit ). To this aim, the corresponding values of ηit are set, in a way that both parts of the objective function have, on average, equal contribution to the overall cost of the solution. The new results are reported in Tables 8 and 9, while the minimum, average and maximum evaluation of the true objective function with respect to different number of scenarios have been represented in Figure 5. It is worth mentioning that Tables 8 and 9 report the same information as that reported in Tables 6 and 7.
Insert Figure 5 around here. In this case, the best solution of the stochastic model is 15757.47 with 100 scenarios, while the objective of the related exact model is 17776.91. This means that the stochastic model has improved the quality of the solution by 12.82%. On the other hand, by utilizing the stochastic model and the SAA technique, we can provide several solutions having different risk and varying objective function. For the studied problem, the corresponding box plot is given in Figure 6. Having such solutions, one can decide to choose 12
Table 10:
Comparison of the stochastic and exact models over the set of new instances.
Instance Instance Instance Instance Instance Instance Instance Instance Instance Instance
1 2 3 4 5 6 7 8 9 10
Stochastic obj. function Exact obj. function Improvement Percentage (%) 15759.04 17880.43 13.46 15759.07 18023.17 14.37 15764.32 17705.72 12.32 15759.04 17880.43 13.46 15759.07 18023.17 14.37 15763.69 17494.02 10.98 15760.60 17996.43 14.19 15761.85 18170.68 15.28 15762.13 17620.09 11.79 15763.22 18223.27 15.61
the lowest price solution with more variability in objective function or to select the low risk solution with higher objective function.
Insert Figure 6 around here. Finally, to show the efficiency of the proposed method to find high quality solutions, we have created 10 new instances. These instances have the same characteristics as that already explained for the real sample. The only difference is distribution functions of the demand in different centers, which follows a normal distribution function. The corresponding exact and stochastic results for created instances are reported in Table 10. According to the results, the stochastic model can always produce better solutions with respect to the exact model. In particular, the improvement of the objective function is varying from 10.98 % up to 15.61%.
6
Conclusions
This paper deals with a practical generalization of the standard personnel assignment problem under stochastic demand. In particular, we have a set of personnel and a set of lost and found centers. Each center receives a stochastic demand. The objective function is to minimize the assignment cost and the cost occurred by the shortage of personnel in different centers due to the stochastic arrival of the demand over a given time period. We modeled this problem as a stochastic program and proposed a Sample Average Approximation (SAA) approach to solve it. Computational results show that SAA converges in an efficient and fast manner when applied to the real studied problem. The model’s implementation in practice shows that capable workers are assigned to crowded centers and other constraints like job rotation and space limitation are satisfied properly. So, the proposed model not only helps the manager to assign workers in an appropriate way, but also it can increase the service quality. Since the introduced problem is NP-hard, CPLEX cannot solve the large size instances. So, for the future works, it is recommended to propose a hybrid algorithm based on the combination of the SAA approach and a (meta)heuristic
13
technique to solve the problem. Considering the service time as a stochastic parameter can also be investigated in the future works.
References [1] Abdullah, S., Sabar, N. R., Nazri, M. Z. A., & Ayob, M. (2014). An Exponential Monte-Carlo algorithm for feature selection problems. Computers & Industrial Engineering, 67, 160-167. [2] Burkard, R. E.(2002). Selected topics on assignment problems. Discrete Applied Mathematics, 123(1-3), 257-302. [3] Burkard, R., Dell’Amico, M., & Martello, S.(2012). Assignment Problems: revised reprint. Society for Industrial and Applied Mathematics. [4] Cao, L., & Wang, B.(2010). Personnel assignment for assembly production with worker differences. 10th International Conference on Intelligent Systems Design and Applications (ISDA), 1476-1480. [5] Constantopoulos, P.(1989). Decision support for massive personnel assignment. Decision Support Systems, 5, 355-363. [6] Dias, R., Garcia, N.L., & Zambom, A.Z.(2012). Monte Carlo algorithm for trajectory optimization based on Markovian readings. Computational Optimization and Applications, 305-321. [7] Herrera, F., Lopez, E., Mendana, C., & Rodriguez, M. A.(1999). Solving an assignment-selection problem with verbal information and using genetic algorithm. European Journal of Operational Research, 119, 326-337. [8] Huang, D. K., Chiu, H. N., Yeh, R. H., & Chang, J. H.(2009). A fuzzy multi-criteria decision making approach for solving a bi-objective personnel assignment problem. Computers & Industrial Enginering, 56, 1-10. [9] Huifu, X., & Zhang, D.(2013). Stochastic Nash equilibrium problems: sample average approximation and applications. Computational Optimization and Applications, 55(3), 597-645. [10] Juan, A. A., Faulin, J., Grasman, S. E., Rabe, M., & Figueira, G. (2015). A review of simheuristics: Extending metaheuristics to deal with stochastic combinatorial optimization problems, Operations Research Perspectives, 2, 6272. [11] Karsak, E. E.(2000). A fuzzy multiple objective programming approach for personnel selection. IEEE International Conference on System, Man and Cybernetics, 3, 2007-2012. 14
[12] King, W. R.(1965). A stochastic personnel-assignment model. Operations Research, 13(1), 67-81. [13] Kleywegt, A.J., Shapiro, A., & Homen-De-Mello, T.(2001). The sample average approximation method for stochastic discrete optimization. SIAM Journal on Optimization, 12, 479-502. [14] Koutras, V.P., Malefaki, S., & Platis, A.N.(2014). Rejuvenation effects on the grid environment performance with response time delays using Monte Carlo simulation. Simulation Modelling Practice and Theory, 40, 176-191. [15] Liang, G. S., & Wang, M. J.(1994). Personnel selection using fuzzy MCDM algorithm. European Journal of Operational Research, 78, 22-33. [16] Lin, S.Y., Horng, S.J., Kao, T.W., Fahn, C.S., Huang,D.K., Run, R.S., Wang, Y.R., & Kuo, I.H.(2012). Solving the bi-objective personnel assignment problem using particle swarm Optimization. Applied Soft Computing, 12, 2840-2845. [17] Liu, C., Wang, J., & Leung, J. Y. T. (2016). Worker assignment and production planning with learning and forgetting in manufacturing cells by hybrid bacteria foraging algorithm. Computers & Industrial Engineering, 96, 162-179. [18] Mayrhofer, W., Marz, L., & Sihn, W.(2011). Planning assistance for pearl chain forecasts and personnel assignment planning of sequenced assembly lines. CIRP Annals- Manufacturing Technology, 60, 481-484. [19] Pentico, D. W.(2007). Assignment problems: A golden anniversary survey. European Journal of Operational Research, 176(2), 774-793. [20] Punnakitikashem, P., Rosenberber, J. M., & Buckley-Behan, D. F.(2013). A stochastic programming approach for integrated nurse staffing and assignment. IIE Transactions, 45(10), 1059-1076. [21] Sir, M. Y., Dundar, B., Steege, L.M.B., & Pasupathy, K. S.(2015). Nursepatient assignment models considering patient acuity metrics and nurses perceived workload. Journal of Biomedical Informatics, 55, 237-248. [22] Sungur, B., & Yavuz, Y.(2015). Assembly line balancing with hierarchical worker assignment. Journal of Manufacturing Systems, 37(1), 290-298. [23] Telhada, J.(2014). Alternative MIP formulations for an integrated shift scheduling and task assignment problem. Discrete Applied Mathematics, 164, 328-343.
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[24] Toroslu, I. H.(2003). Personnel assignment problem with hierarchical ordering constraints. Computers & Industrial Enginering, 45, 493-510. [25] Toroslu, I. H., & Arslanoglu, Y.(2007). Genetic algorithm for the personnel assignment problem with multiple objectives. Information Sciences, 177, 787-803. [26] Trippi, R. R., Ash, A. W., & Ravenis, J. V.(1974). A mathematical approach to large scale personnel assignment. Computers & Operations Research, 1(1), 111-117. [27] Wu, F., Dantan, J. Y., Etienne, A., Siadat, A., & Martin, P. (2009). Improved algorithm for tolerance allocation based on Monte Carlo simulation and discrete optimization. Computers & Industrial Engineering, 56(4), 1402-1413.
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Figure 1: The proposed hierarchy to rank the personnel.
Figure 2: The evaluation of the estimated objective functions with respect to different demand scenarios.
17
Figure 3: The evaluation of the estimated true objective functions with respect to different demand scenarios and Q = 1000.
Figure 4: Gap between upper and lower bounds for various sizes of samples.
Figure 5: The evaluation of the estimated true normalized objective functions with respect to different demand scenarios and Q = 1000 .
18
Figure 6: Risk analysis of alternative solutions with different number of scenarios.
19
Highlights:
We introduce a new generalization of the standard personnel assignment problem. In the introduced problem, the allocated demand to personnel is stochastic in nature, which has not been considered in the literature. Moreover, some new constraints have been added to the model of the problem. The new introduced problem is applicable in the real world and we have done it in a real case. We develop a sample average approximation technique to solve the problem. Results for the real studied problem show the effectiveness of the method.
S0360-8352(17)30404-7 http://dx.doi.org/10.1016/j.cie.2017.09.006 CAIE 4892
To appear in:
Computers & Industrial Engineering
Received Date: Revised Date: Accepted Date:
21 February 2017 29 August 2017 6 September 2017
Please cite this article as: Pour, A.G., Naji-Azimi, Z., Salari, M., Sample average approximation method for a new stochastic personnel assignment problem, Computers & Industrial Engineering (2017), doi: http://dx.doi.org/ 10.1016/j.cie.2017.09.006
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Sample average approximation method for a new stochastic personnel assignment problem Amir Ghorbani Pour, Zahra Naji-Azimi∗ Department of Management, Ferdowsi University of Mashhad, Mashhad, Iran. {Aghorbani,znajiazimi}@um.ac.ir Majid Salari Department of Industrial Engineering, Ferdowsi University of Mashhad, P.O. Box 91775-1111, Mashhad, Iran. [email protected]
July 8, 2017
∗ Department of Management, Ferdowsi University of Mashhad, Mashhad, Iran. TEL: +98 051 38805352, e-mail: [email protected]
1
Sample average approximation method for a new stochastic personnel assignment problem September 7, 2017 Abstract We introduce a practical generalization of the standard personnel assignment problem. In this problem, we have a set of personnel and a set of lost and found centers. Each center receives a stochastic demand. Some constraints, namely, the work skills of different personnel, balancing workload while assigning personnel to the centers and some other practical constraints are taken into account. The objective function is to minimize the assignment cost and the cost occurred by the shortage of personnel in different centers due to the stochastic arrival of the demand over a given time period. We have developed a sample average approximation technique to solve the introduced problem. Computational tests for the studied problem show the effectiveness of the proposed method.
Keywords: Personnel assignment problem, Stochastic optimization, Monte Carlo, Sample average approximation
1
Introduction
The assignment problem (AP) is one of the most studied problems in the combinatorial optimization field. The AP can be considered as assigning n tasks to n agents with the minimum assignment cost in a way that each task is allocated to exactly one agent and each agent is allocated to exactly one task. It can also be considered as the matching between two (or more) sets so that the total cost of the matched pairs is minimized (see, e.g. Burkard, 2002, Pentico, 2007 and Burkard et al., 2012). One of the natural extensions of the AP is called the personnel assignment problem (PAP). In this problem, each task needs some requirements while agents may have various abilities. The problem is to assign a given number of agents to a smaller number of tasks so that the total benefit (score) corresponding to this assignment is maximized. In this paper, we introduce a practical generalization of the standard PAP. In the defined problem, the goal is to assign a set of personnel to a number of work stations over a given time period. The work stations need a different number of skilled workers and the capacity of each station is bounded, so a limited 1
number of personnel can be assigned to each station. In addition, the job rotation has to be taken into account meaning that a person cannot be assigned to the same station within a predetermined number of consecutive time periods. The demands (customers) are arriving to each work station stochastically and for each predetermined number of customers at least a distinct number of personnel has to be available at each station. The objective of the problem is minimizing the assignment cost or maximizing the service quality. The introduced problem has several applications in the real world. Assigning personnel to available branches of different public service centers (e.g., post office, electronic service centers in E-governments policies, and etc.), assigning trainees to different stations for job training (e.g., assigning officer cadet to police centers, interns to different hospital departments or labors to different production lines in a factory) are real examples of this problem. In this paper, we consider the application of the introduced problem by assigning a number of personnel to some lost and found centers.
2
Literature Review
The PAP is NP-complete problem (Toroslu, 2003; Toroslu and Arslanoglu, 2007) and different solution techniques have been developed to solve this problem (see, e.g. Herrera et al., 1999; Karsak, 2000; Toroslu and Arslanoglu, 2007; Lin et al., 2012). In 1974, Trippi et al. considered a large scale military PAP and proposed a mathematical model and an algorithm based on the Ford-Fulkerson network flow approach to solve the problem. Constantopoulos (1989) designed a decision support system to assign a large number of agents to tasks with respect to multiple criteria (Constantopoulos, 1989). In 2003, Toroslu defined the PAP with hierarchical ordering constraints for the organizations in which the agents are categorized into distinct levels such as army power. He showed that this problem is NP-complete and proposed some heuristic and approximate algorithms to solve the problem (Toroslu, 2003). Later in 2007, Toroslu and Arslanoglu extended the PAP with both hierarchical ordering and set restriction constraints and defined an NP-hard multi-objective optimization problem. They proposed a genetic algorithm (GA) to solve different versions of the problem. Cao and Wang (2010) considered the differences among agents and their corresponding effect on the assembly production problem. They proposed a personnel assignment model in which the objectives were maximizing the coefficient of the minimum station fitness and balancing the station fitness for each assembly line. They developed a heuristic algorithm to solve the problem. In 2011, Mayrhofer et al. integrated the sequence planning with the personnel assignment problem in assembly line problem. They presented an approach based on constraint programming with incident discrete simulation to solve the problem. Lin et al. (2012) considered the bi-objective PAP and
2
proposed a particle swarm optimization algorithm with random-key encoding scheme. Telhada in 2014, considered a mixed problem of shift scheduling and task assignment problem and developed several MIP formulations to solve the problem. Sir et al. (2015), considered the nurse-patient assignment problem and introduced some models which assign patients to nurses in a balanced manner. To this aim, they used to distribute acuity scores from the patient classification systems and survey-based nurses perceived workload. Sungur and Yavuz (2015) defined an assembly line balancing problem in which tasks need different requirements and qualification levels of workers are ranked hierarchically, while a lower qualified worker can be substituted with a better worker with a higher cost. They formulated the problem as an integer programming model. Recently, Liu et al. (2016) considered worker assignment and the production planning problem simultaneously. In their problem, bottleneck workstation may transfer according to learning and forgetting effects of workers, while the objective is to minimize the back-order and holding cost of inventory. There are some papers dedicated to the PAP in fuzzy environment. A fuzzy multi-criteria decision making (MCDM) algorithm was proposed by Liang and Wang (1994) where for selecting the agents both subjective and objective assessments were taking into account by incorporating a fuzzy MCDM algorithm. Herrera et al. (1999) considered the problem with some verbal information and proposed a GA to solve the problem. A fuzzy multiple-objective programming technique was proposed by Karsak (2000) to select the agents. In the introduced problem, some qualitative factors of agents have been incorporated by fuzzy linguistic variables and the preference between the objectives has been determined by decision makers with linguistic variables. Huang et al. (2009) considered the PAP with fuzzy characteristics and proposed a bi-objective binary integer programming model to formulate the problem. They designed a feedback mechanism to iteratively solve the problem with the consideration of interdependencies among tasks and the differences among the selected agents simultaneously. Some researchers have considered the PAP in a stochastic environment. A stochastic version of the PAP was proposed by king (1965). He believed that the vacant tasks are usually filled with the agents who never have the experience of working on that task. So, he integrated the prediction of an agent’s performance (or job success) during the allocation of tasks to the agents. Punnakitikashem et al. (2013) considered an integrated nurse staffing and assignment problem and presented a stochastic integer programming model with the objective of minimizing the expected excess workload of nurses. They proposed some exact approaches to solve the problem. In this paper, we introduce a practical generalization of the standard PAP. In this problem, we are given a set of lost and found centers and a pre-determined number of personnel to be assigned to the centers. These centers are located in different parts of the city where the demand of each part is a
3
stochastic variable. The foundation of a reliable assignment is to allocate the existing personnel to the set of centers in the best possible way in order to meet the demand of customers in different demand scenarios. The capacity of each center for accommodating the personnel is limited and bounded by a given value. In practice, some places are the destination of more tourists than the others and as a consequence the corresponding center has more demand. As a result, to have a fair assignment, the company aims at assigning the personnel to the centers in such a way that personnel work at different centers during a given time period. People arrive to each center stochastically where, according to the standards there should be at least one person in each center to cover each λ units of demand (service quality). Moreover, depending on the location of the center, it may need some skilled workers to be available at that center in order to satisfy the demand of the customers. The objective is to minimize the total assignment cost plus the cost occurred by the shortage of personnel in different centers due to the stochastic arrival of the demand. The new model contains the following differences with respect to those papers available in the literature: 1) in the new introduced model, the arrival demand rate is stochastic, 2) we consider the job rotation for personnel, and 3) the quality of service has not been already considered in the previous papers. The remainder of this paper is organized as follows. Section 3 gives a formal description of the problem. In Section 4, we develop a method based on the combination of the multi criteria decision making tools and sample average approximation (SAA) technique to solve the introduced problem. Computational results for the real case studied problem are presented in Section 5. Finally, Section 6 concludes the paper.
3
Problem Statement
In this section, we develop a mathematical programming model for the introduced problem. We denote by I = {1, · · · , m} and J = {1, · · · , n} the sets of centers and personnel, respectively. In addition, let T = {1, · · · , τ } is the set of time periods and K = {1, · · · , κ} is the set of different work skills. For each i ∈ I and k ∈ K, αik is the minimum number of skilled workers of type k needed in center i. Also, we represent by Sk the set of personnel who are expert in skill k. Since the capacity of each center is limited, the maximum number of personnel that can be accommodated in center i ∈ I is bounded by the threshold βi . In each period t ∈ T , qit shows the number of people (clients) coming to center i to receive services. For each λ clients in a given center, we need at least one worker to cover the demand. Finally,
4
cij denotes the cost of assigning worker j to the center i. The variables to define the model are as follows:
xijt
⎧ ⎪ ⎨ 1 = ⎪ ⎩ 0
if worker j is assigned to center i in period t,
∀i ∈ I, j ∈ J, t ∈ T
(1)
otherwise.
The model reads as follows:
Model I)
min Z =
cij xijt
(2)
i∈I j∈J t∈T
subject to:
xijt = 1
∀t ∈ T, ∀j ∈ J,
(3)
∀i ∈ I, ∀t ∈ T, ∀k ∈ K,
(4)
∀i ∈ I, ∀t ∈ T,
(5)
∀i ∈ I, ∀t ∈ T \ {τ }, ∀j ∈ J,
(6)
∀i ∈ I, ∀j ∈ J,
(7)
∀i ∈ I, ∀t ∈ T,
(8)
∀i ∈ I, ∀t ∈ T, ∀j ∈ J.
(9)
i∈I
xijt ≥ αik
j∈Sk
xijt ≤ βi
j∈J
xijt + xijt+1 ≤ 1 xij1 + xijτ ≤ 1 qit xijt ≥ λ j∈J
xijt ∈ {0, 1}
The objective function (2) is to minimize the total assignment cost. Constraint set (3) imposes that in each time period all personnel have to be assigned to different available centers. For each k ∈ K, constraints (4) assure that in each period t ∈ T and each center i ∈ I, at least αik skilled personnel have to be available. Taking into account the capacity of each center i ∈ I, constraint (5) stipulates that at most βi personnel can be assigned to the center i, during different time periods. Constraint sets (6) and (7) impose that nobody is allowed to work in the same center in two consecutive time periods. For each t ∈ T and i ∈ I, constraint set (8) shows that for satisfying the demand of each λ clients, at least one worker has to be available. Finally, the definition of the variables are given in (9). In our studied problem, qit is a stochastic variable with a known distribution function. Suppose Lit is the expected value of personnel’s slackness in center i and period t. Introducing the ηit as the unit cost of assigning not enough personnel to center i in period t, the new model reads as follows:
5
min Z =
Model II)
i∈I j∈J t∈T
cij xijt +
ηit Lit
(10)
i∈I t∈T
subject to: (3) to (7), and ⎡
⎤+ q it − Lit = Eqit ⎣ xijt ⎦ λ
∀i ∈ I, ∀t ∈ T,
(11)
∀i ∈ I, ∀t ∈ T, ∀j ∈ J.
(12)
j∈J
xijt ∈ {0, 1}
The first part of the objective function (10) is the minimization of the total assignment cost while the second part minimizes the cost occurred by assigning insufficient personnel to different centers. For each i ∈ I and t ∈ T , constraint (11) represents the expected slackness of the personnel in which X + = max{0, X}
4
Solution Method
In this paper, we integrate the multi criteria decision making tools and the Monte Carlo simulation technique to solve the introduced problem. In particular, to obtain the corresponding values of the assignment cost (cij ) and to rank the personnel according to the job skills, we use the combination of the analytical hierarchy process (AHP) and the simple additive weighting (SAW) methods. To this aim, we extracted the following hierarchy of the required skills (see, Figure 1). In addition, we utilize the Monte Carlo simulation technique to solve the proposed stochastic model (see, Section 3). In the following, we provide some theoretical details concerning the Monte Carlo simulation technique while the detailed information concerning the combination of the AHP and SAW will be provided in Section 5. Insert Figure 1 around here. Several techniques have been introduced to solve the stochastic problems, including Monte Carlo algorithm or Sample Average Approximation Method(see, e.g. Abdullah et al., 2014, Huifu and Zhang, 2014, Koutras et al., 2014 and Wu et al., 2009). Since the exact evaluation of the expected value of (11) is difficult, we use the SAA method in order to approximate the true problem. The corresponding steps of the algorithm are provided in the following.
6
• For each center i ∈ I and each period t ∈ T , generate M independent samples of demand, each of Q 1 2 , qit , · · · , qit ). size Q (i.e., qit
• The expected value Lit is estimated as Lit =
Q
s qit s=1 ( λ
+ − n j=1 xijt ) . Q
• The objective function Z is calculated using Model II and substituting Lit from the previous step. It should be mentioned that by increasing the amount of Q, with probability one, Lit converges to the expected value in (11) (see, Kleywegt et al., 2001). • Assuming x∗ to be the optimal solution of Model II, fix the value of variable x and estimate the
true objective value of x∗ by generating Q independent samples where Q is considered to be much larger than Q. ˆ obtained by applying the SAA method. In the following, we Suppose that we have a solution X ˆ to be used as a candidate for solving the real problem. introduce the way of evaluating the quality of X To do so, suppose that we are given M independent samples of size Q. The corresponding steps are as follows: • Let fm represent the objective value of the mth sample obtained by solving Model II. • Compute f¯ and σf2¯ as follows: f¯ = 2 σM,Q =
1 M
M
1 M(M−1)
m=1
fm
M
m=1 (fm
− f¯)2
ˆ (i.e., f˜(X)) ˆ using a large number of scenarios (Q ) for • Calculate the true objective function of X the demand. 2 • Compute σQ as follows: 2 σQ =
1 Q (Q −1)
Q
ˆ
j=1 (F (X, ϑj )
ˆ 2 − f˜(X))
ˆ ϑj ) is the objective function of Model II with a fixed solution X ˆ using the j th In which F (X,
scenario (ϑj ) from the set of Q different scenarios. Now, the upper and lower bounds with 100(1 − α)% confidence are calculated as follows: ˆ + zα σ Upper bound = f˜(X) Q Lower bound = f¯ − tα,v σM,Q In which v = M − 1, tα,v is the α-critical value of t-distribution with v degrees of freedom. In addition, zα is the standard normal critical value. 7
Table 1:
The minimum number of skilled workers required at each center (αik ). Center Foreign language Computer knowledge Social skills
Table 2: Center 1 βi 6 0.1458 ηit
5
1 1 3 1
2 0 1 1
3 0 1 1
4 0 1 0
5 0 1 1
6 2 2 1
7 0 1 1
8 0 1 1
9 0 1 0
10 0 1 1
Some parameters of the model.
2 3 4 3 3 4 0.1250 0.1042 0.1042
5 6 3 5 0.0625 0.1458
7 8 9 3 3 3 0.0625 0.0833 0.0625
10 3 0.1042
Computational Results
In this section, at first we introduce the characteristics of the studied problem. Then we discuss on the results obtained by solving the stochastic model proposed in Section 3. Finally, we propose several tests in order to assess the performance of the developed method.
5.1
Introducing the real problem
To run the model, we use the data provided by a real company. Totally, there are 10 lost and found centers in the city of Mashhad, Iran, which is one of the Iran’s tourism destinations having around 32 million visitors per year. These centers are different in size and the demand rate at each center follows a known distribution function. Totally, 25 personnel with different abilities are available to be assigned to the centers. Each worker has at least one skill from the three distinct skills, namely, foreign language, computer knowledge and social skills, which are required to work in different centers. The number of periods is 47 and to have a reasonable service quality, we set λ = 6. Based on the location of each center, a minimum number of personnel with known skills is needed which are provided in Table 1. Taking into account the capacity of each center, the maximum number of personnel who can work in each center (βi ) is limited and is provided in the second row of Table 2. In addition, the penalty concerning the shortage of personnel at each center i (ηit ), considered as the same value for different time periods, is reported in the last row of Table 2. These values are achieved by normalizing the average penalties proposed by the decision makers for different available center. The distribution functions of the demand in different centers have been estimated using the data over the last two years. Essentially, the data contains the number of people referred to each center to receive services. We have used the EasyFit software in order to obtain the corresponding distribution function at each center. In addition, the Kolmogrov-Smirnov test has been applied to check the fitness of the estimated distribution functions. The assignment cost (cij ) is one of the most important parameters in the introduced model. Since the
8
Table 3:
Classifying centers according to the number of demand referring to them. Class Very quiet Quiet Moderate Crowded Center {7,9} {5,8} {3,4,10} {2,6}
Table 4:
Very crowded {1}
The weights and inconsistency rates of the available criteria and sub-criteria.
Criteria
Weight Inconsistency rate
Social Skill
0.64
Computer knowledge 0.27
Foreign language
0.05
0.09
Sub-criteria Old caring Child caring Public relations Problem solving ability Work managing ability Overall knowledge Specific software English Arabic Turkish Ordu
Weight 0.05 0.25 0.22 0.20 0.30 0.25 0.75 0.24 0.36 0.34 0.06
Inconsistency rate
0.06
0.00
0.00
personnel get paid monthly and it is a fixed value for each month, the assignment cost cij is substituted with a parameter which is related to the undesirable assignment of worker j to each center i. In particular, undesirable assignment occurs when assigning a skilled worker to an uncrowded center or assigning an unskilled worker to a crowded center. So, according to the historical data we have classified the centers into five groups from very quiet to very crowded (see, Table 3). In addition, we use the combination of the analytical hierarchy process (AHP) and the simple additive weighting (SAW) in order to rank the personnel according to the job skills. To this aim, we have extracted the hierarchy of the required skills (see, Figure 1). Table 4 represents the weights and inconsistency rates corresponding to the available criteria and sub-criteria. By using the extracted weights and the SAW technique, we obtain a score for each worker and accordingly we classify the workers into five groups from very weak to very capable. Finally, taking into account the group that each center belongs to and the allocated group to each worker, the final values for different cij s are reported in Table 5. The last parameter to be calculated is the set of skilled worker for each k ∈ K (i.e., Sk ). To determine this parameter, we put in Sk all workers whose score in the corresponding skill is greater than 60 percent of the average score for skill k over all personnel.
Table 5:
Allocated cij according to the class of the centers and personnel.
Very capable Capable Moderate Weak Very weak
Very quiet Quiet 1.00 0.75 0.75 0.50 0.50 0.25 0.25 0.00 0.00 0.25
Moderate 0.50 0.25 0.00 0.25 0.50
9
Crowded 0.25 0.00 0.25 0.50 0.75
Very crowded 0.00 0.25 0.50 0.75 1.00
Table 6: Q
The objective function and computational time for different values of Q.
Minimum 2 11936.24 12033.93 5 12022.39 7 10 12060.18 12072.57 20 12100.73 30 12109.39 40 12130.71 50 12125.20 60 12137.31 70 12130.79 80 12141.83 90 100 12138.96
Table 7:
Objective value Average Maximum 12105.38 12431.97 12144.62 12268.04 12142.99 12285.13 12142.87 12195.98 12169.76 12259.20 12163.71 12238.75 12166.53 12220.25 12164.34 12217.98 12160.11 12212.30 12161.72 12189.86 12161.65 12194.79 12159.75 12201.77 12163.77 12199.73
Computational Minimum Average 2.14 4.82 2.51 4.92 2.67 8.28 2.73 14.24 14.81 18.69 16.14 22.71 22.07 27.61 26.79 35.85 23.48 31.31 38.17 43.56 46.31 52.09 49.79 58.98 45.28 52.86
time Maximum 10.25 9.37 13.11 26.5 24.51 28.25 31.82 54.6 36.93 50.73 57.6 68.23 63.53
Estimation of the true objective function with Q = 1000. Q 2 5 7 10 20 30 40 50 60 70 80 90 100
5.2
SD 141.41 84.4 72.37 48.69 50.59 38.24 36.36 27.68 29.61 21.29 22.73 18.32 21.97
Minimum 12242.67 12179.23 12173.88 12167.28 12167.36 12166.93 12166.94 12166.79 12166.80 12166.89 12166.97 12166.66 12166.89
Average 12269.91 12195.38 12177.70 12171.33 12168.52 12168.02 12167.96 12167.54 12167.36 12167.35 12167.57 12167.37 12167.48
Maximum 12287.55 12221.09 12182.22 12174.22 12170.00 12169.98 12168.90 12168.85 12168.44 12167.94 12168.39 12168.74 12168.76
SD 12.86 14.68 2.69 2.54 0.76 0.79 0.64 0.66 0.53 0.35 0.52 0.6 0.59
The results
We have used CPLEX 12.1 to solve the stochastic model using the SAA technique. All tests have been performed on a computer with an Intel Core due CPU, 2.66 GHz and 3.5GB of RAM. In our experiments, Q ∈ {2, 5, 7, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100}. In addition, to take randomness into account, for each demand scenario Q, 10 independent samples (M = 10) have been generated. The results are provided in Table 6 in which the first column gives the number of demand scenarios (Q). The next three columns give the minimum, average and maximum value of the objective function over 10 independent samples, respectively. Each entry of the column labeled by “SD” gives the standard deviation of the corresponding row. Finally, the last three columns give, respectively, the minimum, average and maximum computational time (in seconds) over 10 independent samples of each demand scenario. According to the data reported in Table 6, by increasing the number of demand scenarios, the computational time to solve the instances increases while the standard deviation decreases. In order to have an estimation of the true objective function of the solutions obtained by different values of Q, in the next step we calculate a better estimation of the true objective function values by
setting Q = 1000. The results are reported in Table 7. Based on the results reported in Table 7, by increasing the value of Q, the corresponding estimation of
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the true objective function converges. The minimum, average and maximum evaluation of the estimated and true objective function with respect to different number of scenarios are represented in Figures 2 and 3, respectively. As already explained, by increasing the number of demand scenarios (Q), the values of the objective functions are converged.
Insert Figures 2 and 3around here. In the following, we perform a test in order to compare the best solution obtained by the SAA technique and that obtained by solving the deterministic equivalent of the problem. Essentially, by substituting the expected value of the uncertain demands in the model and solving that we will have the expected value solution. By solving the exact model, the objective function of the expected value solution is 12288.4 which is around 1% worse than that of the stochastic solution. Figure 4 shows the corresponding gap between the upper and lower bounds in each demand scenario. According to this figure, the average gap is less than 1% for greater than 70 number of samples.
Insert Figure 4 around here. Since the introduced problem is NP-hard, the large size instances cannot be solved to optimality using CPLEX. As a result, the proposed algorithm should be adopted in order to solve such instances. There are two ways to handle larger size data. The first straight forward way is to consider the solution of a (meta)heuristic approach in the last step of the algorithm instead of the optimal solution (i.e., x∗ ). The second way has been introduced by Juan et al. (2015) as a general framework to solve a stochastic combinatorial optimization problem. In this method the corresponding deterministic problem is achieved by substituting the expected value of available random variable(s). Upon this step, a (meta)heuristic approach is utilized to obtain a set of high quality or promising solutions. In an interactive search, the quality of each promising solution for the stochastic model is calculated using a limited number of iterations of the Monte Carlo simulation to have a rough estimate of the solutions quality. Accordingly, a ranked list of solutions is provided according to the estimated quality of the solutions, while a feedback to metaheuristic can be considered to intensive the search of the (meta)heuristic. After reaching the final time given to (meta)heuristic, the simulation with the larger number of scenarios is performed on the ranked solutions. Finally, the set of elite ranked solutions are updated according to the most accurate estimation and final solutions can be presented to decision makers.
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Table 8:
The normalized objective function for different values of Q. Q
2 5 7 10 20 30 40 50 60 70 80 90 100
Table 9:
Estimation of the true normalized objective function with Q = 1000. Q 2 5 7 10 20 30 40 50 60 70 80 90 100
5.3
Objective value Minimum Average Maximum 14587.86 15142.44 16245.24 15177.46 15533.99 15958.32 15166.32 15589.93 16006.04 15332.50 15635.82 15799.80 15410.29 15753.60 16050.94 15504.54 15739.47 16005.63 15547.52 15753.88 15949.01 14587.86 15246.64 15547.52 15608.27 15733.14 15920.31 15652.06 15739.26 15840.19 15628.73 15739.19 15857.85 15668.26 15732.20 15882.84 15657.96 15746.36 15875.55
Minimum 16882.06 15908.36 15882.87 15841.40 15781.52 15761.00 15759.53 15760.28 15757.63 15757.98 15757.62 15757.13 15757.47
Average 17051.64 16091.03 15949.78 15894.55 15810.74 15785.29 15771.32 15772.28 15765.63 15761.68 15761.34 15761.55 15760.90
Maximum 17370.44 16244.20 16071.33 15973.24 15859.15 15810.18 15794.69 15788.48 15779.12 15771.27 15768.54 15769.17 15769.70
Sensitivity Analysis
In this section, we perform a test in order to analyze the impact of the stochastic part of the objective function on the overall solution’s cost. In particular, we increase the corresponding coefficient of the stochastic part of the objective function (i.e., parameter ηit ). To this aim, the corresponding values of ηit are set, in a way that both parts of the objective function have, on average, equal contribution to the overall cost of the solution. The new results are reported in Tables 8 and 9, while the minimum, average and maximum evaluation of the true objective function with respect to different number of scenarios have been represented in Figure 5. It is worth mentioning that Tables 8 and 9 report the same information as that reported in Tables 6 and 7.
Insert Figure 5 around here. In this case, the best solution of the stochastic model is 15757.47 with 100 scenarios, while the objective of the related exact model is 17776.91. This means that the stochastic model has improved the quality of the solution by 12.82%. On the other hand, by utilizing the stochastic model and the SAA technique, we can provide several solutions having different risk and varying objective function. For the studied problem, the corresponding box plot is given in Figure 6. Having such solutions, one can decide to choose 12
Table 10:
Comparison of the stochastic and exact models over the set of new instances.
Instance Instance Instance Instance Instance Instance Instance Instance Instance Instance
1 2 3 4 5 6 7 8 9 10
Stochastic obj. function Exact obj. function Improvement Percentage (%) 15759.04 17880.43 13.46 15759.07 18023.17 14.37 15764.32 17705.72 12.32 15759.04 17880.43 13.46 15759.07 18023.17 14.37 15763.69 17494.02 10.98 15760.60 17996.43 14.19 15761.85 18170.68 15.28 15762.13 17620.09 11.79 15763.22 18223.27 15.61
the lowest price solution with more variability in objective function or to select the low risk solution with higher objective function.
Insert Figure 6 around here. Finally, to show the efficiency of the proposed method to find high quality solutions, we have created 10 new instances. These instances have the same characteristics as that already explained for the real sample. The only difference is distribution functions of the demand in different centers, which follows a normal distribution function. The corresponding exact and stochastic results for created instances are reported in Table 10. According to the results, the stochastic model can always produce better solutions with respect to the exact model. In particular, the improvement of the objective function is varying from 10.98 % up to 15.61%.
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Conclusions
This paper deals with a practical generalization of the standard personnel assignment problem under stochastic demand. In particular, we have a set of personnel and a set of lost and found centers. Each center receives a stochastic demand. The objective function is to minimize the assignment cost and the cost occurred by the shortage of personnel in different centers due to the stochastic arrival of the demand over a given time period. We modeled this problem as a stochastic program and proposed a Sample Average Approximation (SAA) approach to solve it. Computational results show that SAA converges in an efficient and fast manner when applied to the real studied problem. The model’s implementation in practice shows that capable workers are assigned to crowded centers and other constraints like job rotation and space limitation are satisfied properly. So, the proposed model not only helps the manager to assign workers in an appropriate way, but also it can increase the service quality. Since the introduced problem is NP-hard, CPLEX cannot solve the large size instances. So, for the future works, it is recommended to propose a hybrid algorithm based on the combination of the SAA approach and a (meta)heuristic
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technique to solve the problem. Considering the service time as a stochastic parameter can also be investigated in the future works.
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Figure 1: The proposed hierarchy to rank the personnel.
Figure 2: The evaluation of the estimated objective functions with respect to different demand scenarios.
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Figure 3: The evaluation of the estimated true objective functions with respect to different demand scenarios and Q = 1000.
Figure 4: Gap between upper and lower bounds for various sizes of samples.
Figure 5: The evaluation of the estimated true normalized objective functions with respect to different demand scenarios and Q = 1000 .
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Figure 6: Risk analysis of alternative solutions with different number of scenarios.
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Highlights:
We introduce a new generalization of the standard personnel assignment problem. In the introduced problem, the allocated demand to personnel is stochastic in nature, which has not been considered in the literature. Moreover, some new constraints have been added to the model of the problem. The new introduced problem is applicable in the real world and we have done it in a real case. We develop a sample average approximation technique to solve the problem. Results for the real studied problem show the effectiveness of the method.