STOCHASTIC MANPOWER ALLOCATION and ... - Semantic Scholar

Proceedings of the 41st International Conference on Computers & Industrial Engineering

STOCHASTIC MANPOWER ALLOCATION and CELL LOADING in CELLULAR MANUFACTURING SYSTEMS Gökhan Eğilmez*, Gürsel A. Süer Industrial and Systems Engineering Department Russ College of Engineering and Technology Ohio University Athens, OH, 45701, USA [email protected]*, [email protected] Tel: (740) 593-1542, Fax (740) 593-0778 In this paper, a stochastic manpower allocation and cell loading problem is studied, where operation times and customer demand are probabilistic. A two-phase hierarchical methodology is proposed to find the optimal manpower assignment and cell loads simultaneously with respect to maximum allowable risk levels. Two non-linear stochastic mathematical models are developed to deal with manpower allocation and cell loading phases. In both models, processing times and demand are normally distributed. Firstly, alternative configurations are generated. Secondly, stochastic manpower allocation is performed and products are loaded to cells. Statistical analysis and Monte-Carlo simulation are employed to link the production rates obtained from the first phase to the capacity requirements that are used in the second phase. The methodology is illustrated with an example problem drawn from a real manufacturing company. According to the results, as risk level increases, lower number of manpower is assigned to cells. Expected utilizations of cells are decreased as the variance of demand and processing times increase. The approach allows decision maker to perform manpower allocation with respect to the desired risk level. Key words: Stochastic manpower allocation, cell loading, monte-carlo simulation, non-linear mathematical models, risk. 1. Introduction As an implementation of Group Technology (GT) philosophy to the manufacturing world, cellular manufacturing (CM) has been one of the most widely-used manufacturing approaches all over the world. Similar products are grouped together as families to be produced in a manufacturing cell, which is a small manufacturing unit consists of workers, equipment and machines where the production flow is unit based and unidirectional. There are two types of manufacturing cells used in CM systems, namely: machineintensive and labor-intensive. In a machine intensive cell, the number of machines is the primary parameter used on determining the output and the impact of labor on the output is limited. On the other hand, in labor-intensive cells, most of the operations require light weight and small machine and equipment where operator is continuously involved in the process. Labor-intensive manufacturing cells can be found in several manufacturing systems such as food, jewelry and shoe manufacturing, medical device and apparel industry. In this paper, a stochastic manpower allocation and cell loading problem in a labor intensive jewelry manufacturing system is studied. Two non-linear mathematical models are developed to deal with manpower allocation and cell loading phases. Processing times and demand are assumed normally distributed. Firstly, stochastic manpower allocation is performed and alternative configurations are obtained. Secondly, products are loaded to minimum number of cells and optimal manpower levels for each cell are determined.

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2. Literature Review Significant amount of work has been done on deterministic manpower allocation and cell loading. Only few of them are cited due to page limitations. Dagli and Suer (1986) worked on determining appropriate manpower levels in the assembly line and proposed a two level approach. Russell, Huang and Leu (1991) suggested various labor scheduling policies. Wirth, Mahmoodi and Mosier (1993) addressed group scheduling for the same problem and provided heuristic methods for labor scheduling policies. Lee and Vairaktarakis (1993) proposed a sequencing approach to minimize total manpower requirement. Cesani and Steudel (2005) studied the labor flexibility in cellular manufacturing systems characterized considering intra-cell operator's mobility. Some of the works in the literature also considered cell-loading in addition to manpower allocation: Suer, Saiz, Dagli and Gonzalez (1995) and Suer, Saiz and Gonzalez (1999), Suer (1999). Suer and Dagli (2005) proposed an approach to minimize the intra-cell manpower transfers and load cells with products. Suer, Cosner and Patten (2009) proposed a three-phased methodology to deal with cell loading and product sequencing in labor intensive cells. Suer (1996) proposed a deterministic hierarchical approach to the problem. In this paper, same problem with uncertain demand and processing times is studied. Two non-linear stochastic mathematical models are developed. Statistical analysis and Monte Carlo simulation are used to link the phases. Finally, experimentation is performed with various manpower levels, products and variances in demand and operation times. 3. Problem Statement The problem is observed in a jewelry manufacturing company, where labor intensive cells are in use. There are five operations and ten products in system. Each product has individual probabilistic demand and processing times. The operations and mean processing times are shown in Table 1. The standard deviations of processing times are assumed as the 10 % of the means. Operation Casting Debarring Linking Stone set and enameling Carding and packing

Table 1. Mean Processing Times (min) Product 1 0.07 0.45 0.37 0.88 0.38

2 0.05 0.29 0.62 0.29 0.38

3 0.06 0.29 1.18 0.86 0.18

4 0.04 0.31 0.55 0.47 0.4

5 0.08 0.41 0.43 1.38 0.43

6 0.07 0.32 1.18 0.55 0.45

7 0.07 0.41 0.68 0.95 0.33

8 0.04 0.63 1.15 0.91 0.38

9 0.09 0.83 0.82 0.71 0.32

10 0.03 0.18 1.16 1.08 0.27

4. Methodology A hierarchical solution methodology is employed to handle the manpower allocation and cell loading tasks. In the first phase, alternative configurations are generated via the first stochastic non-linear mathematical model. In the second phase, loading products to minimum number of cells and finding optimal operator assignments are simultaneously performed via the second stochastic non-linear mathematical model. 4.1. Generating Alternative Configurations with respect to Maximum Allowable Risk The first model is developed to find the optimal production rates and manpower configurations. The objective is to maximize the production rate (Equation 1). The production rate is the output of bottleneck operation which is less than or equal to the outputs of other operations (Equation 2). Equation 3 is the stochastic constraint which guarantees that there is enough number of operators assigned to each operation to achieve the maximum production rate with respect to the risk level. The number of operators can be assigned to an operation is limited by the number of operators available to perform the operation (Equation 4). Equation 5 guarantees that the total number of operators assigned to all operations cannot

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exceed the total crew size. Finally, the number of operators assigned to an operation can only take positive real integer values up to the maximum number of available operators (Equation 6). Notation M: Number of operators in cell N: Number of operations R: Cell production rate Ri: Operation production rate Xi: The number of operators for operation j of product i i: Operation index p: Operator index µ: Mean processing time σ: Standard deviation of processing time α: Risk level β: Production rate coefficient T: Total number of operators available for cell U: Total number of operators allowed for an operation Z: Z value of standard normal distribution Objective Function Max Z = R

1

Subject to:

1,2, … , 2

µ

1

1,2, … , 3

1,2, … , 4

5 1,

1,2, … ,

6

4.2. Monte-Carlo Simulation The production rates for each product and configuration are obtained via the first mathematical model. In deterministic case, the conversion of production rates to capacity requirements is simply calculated by dividing the demand by the production rate for each product and configuration. In this study, a new approach is required to tackle the probabilistic feature of demand. Monte-Carlo simulation is used to find the fitted distribution of capacity requirements. The steps of simulation are; 1) Sample demand datasets are randomly generated for each product with respect to the probability distribution of demand. 2) Capacity requirements are calculated via dividing demand by corresponding production rates. 3) Finally, obtained capacity requirements are analyzed to find the fitted probability distributions. Chi Square and Kolmogorov-Smirnov tests are used in the analysis. The probability distribution of capacity requirement for each product is used as stochastic processing time in cell loading phase.

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4.3. Finding Optimal Operator Assignment and Loading Cells with respect to Maximum Allowable Risk The model proposed in this section forms the basis for the second phase of the hierarchical methodology. The objective of this model is to minimize the number of operator assignments to the open cells (Equation 8). Equation 9 guarantees that each product is assigned to only one cell. Each cell can have only one manpower configuration (Equation 10). Equation 10 is the probabilistic cell loading constraint. It guarantees that the probability of exceeding the capacity of cell cannot be greater than the risk level. The last equation (Equation 11) is the binary constraints for the product and configuration assignments. Notation bjk: The manpower level required for the configuration k and cell j. Yjk: 1 if alternative configuration k is assigned to cell j, 0 otherwise. Xijk: 1 if product i assigned to configuration k, cell j. M: Number cells N: Number of operations T: Total number of operators available for cell U: Total number of operators allowed for an operation µp: Mean capacity requirement σp: Standard deviation of capacity requirement i: Product index j: Cell index k: Configuration index α: Risk level h: Capacity of a cell (hrs) Kj: number of alternative configurations for cell j Objective Function: K

M

Min Z

b

Y 8

Subject to: M

K

1 i

X

1,2,3, … , N 9

K

Y

1 j

1,2,3, … , M 10 ∑N

X

1

j

1,2,3, … , M and k

1,2,3, … , Kj 11

∑N X and Y

0,1 12

5. Experimentation and Results The experimentation is performed in two steps. In the first phase, alternative configurations are generated via the first non-linear mathematical model. There are ten products and 16 manpower levels (10 to 25 workers) and 40 hours of cell capacity considered. Processing times and demand are assumed to be normally distributed. Due to limited space, only the results obtained for the minimum and maximum manpower levels are shown. 10 % risk level is used for the problem studied.

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Product No 1 2 3 4 5 6 7 8 9 10

Table 2. Results of First Phase and Monte-Carlo Simulation Manpower Configuration Manpower Production Demand O1 O2 O3 O4 O5 Level Rate (µ&σ) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

2 5 2 5 1 3 2 4 1 4 1 3 2 4 2 5 3 8 1 2

2 4 3 9 4 11 3 8 2 4 4 11 3 7 3 9 3 7 4 10

3 10 2 4 3 8 2 6 4 12 2 5 3 9 3 7 2 6 3 9

2 5 2 6 1 2 2 6 2 4 2 5 1 4 1 3 1 3 1 3

10 25 10 25 10 25 10 25 10 25 10 25 10 25 10 25 10 25 10 25

3.33 10.73 4.76 13.71 3.31 9.22 4.17 12.68 2.36 8.61 3.04 9.01 2.95 9.39 2.53 7.61 2.74 8.37 2.7 8.25

5000 5000 8000 8000 6000 6000 4000 4000 3000 3000 5000 5000 4000 4000 3000 3000 4500 4500 6000 6000

Capacity Requirement (µ&σ)

685 685 1584 1584 1788 1788 1240 1240 810 810 700 700 600 600 500 500 900 900 1500 1500

1500 467 1690 586 1830 655 955 317 1280 350 1640 557 1360 426 1180 395 1650 538 2220 729

206 63.2 331 114 540 192 303 96.8 344 93.1 230 76.9 203 64.1 197 65.9 325 109 555 184

The results of first phase and Monte Carlo Simulation are shown in Table 2. According to the Table 2, the manpower level on the bottleneck operation is increased significantly which resulted in an increase in the production rate. The fitted distributions of capacity requirements are obtained via statistical analysis and Monte Carlo simulation. According to the results less expected time is required as manpower level increases. Cell No

Table 3. Results of Second Phase 1 2

Family Formation (Cells vs Products) (1, 3, 4) 19 Manpower Required (Total =62) 32.93 Expected Utilization (hrs) 82.30% % Expected Utilization

(2, 6, 8 ,9) 24 35.42 88.50%

3 (5, 7, 10) 19 33.63 84.10%

The results of 2nd phase (stochastic optimal manpower assignment and cell loading) are shown in Table 3. According to the results, the expected utilizations are over 80 %. Total required manpower is 62 and 3 cells are formed with 3-4 products each with respect to 10 % risk level. 6. Conclusion and Future Work In this paper, a stochastic manpower allocation and cell loading problem from jewelry manufacturing company is studied, where demand and operation times are probabilistic. A two phased hierarchical

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methodology is used to solve the problem. Two non-linear stochastic mathematical models are developed. The first model is used to find optimal production rates for a specific manpower level. The second model is employed to load products to available cells and assign the best configuration of manpower. Montecarlo simulation is used to link the phases of proposed hierarchical approach. Based on 10 % risk level, a case problem with 10 products is solved. The products are formed as 3 families to be produced in three dedicated cells. The expected utilizations of cells ranged from 82 % to 89 %.

Risk

Table 4. The impact of risk on Total Manpower Required 10 % 20 % 50 % 80 % 90 %

Optimal Solution

62

58

54

54

54

The impact of risk on required manpower level is analyzed as well (Table 4). As risk level increases, the required manpower decreased until the 50 % of risk level then stayed the same. Even though the experimentation is kept limited due to the page limitations; product splitting among cells, common cell size determination, different demand and processing time distributions and scenarios, skill based manpower allocation, intra-cell manpower transfer, multi-period environment along with heuristics and metaheuristics are other features to be studied as future work. 7. References Cesani, V. & Steudel, H. (2005). A Study of Labor Assignment Flexibility in Cellular Manufacturing Systems. Computers & Industrial Engineering, 48(3), 571-591. Dagli, C. & Suer, G. A. (1986). Scheduling For Flexible Layout. Proceedings of The 17th Midwest Decision Sciences Institute (Pp. 23-25). Nebraska. Lee, C. & Vairaktarakis, G. L. (1993). Job Sequencing In Cellular Layouts to Minimize Total Manpower. Research Report 93-18, University Of Florida, Gainesville, FL. Russell, R. S., Huang, P. Y. & Leu, Y.-Yuh. (1991). A Study of Labor Allocation Strategies in Cellular Manufacturing. Decision Sciences, 22(3), 594-611. Suer, G. A. (1999). Cell Loading In Connected Cells. Proceedings of the 26th International Conference On Computers and Industrial Engineering (Pp. 438-441). New Orleans. Suer, G. A. (1996). Optimal Operator Assignment and Cell Loading in Labor-Intensive Manufacturing Cells. Computers & Industrial Engineering, 31(1), 155-158. Suer, G. A., Saiz, M., Dagli, C. & Gonzalez, W. (1995). Manufacturing Cell Loading Rules and Algorithms for Connected Cells. In A. Kamrani, H. Parsaei, & D. Liles (Eds.), Planning, Design and Analysis of Cellular Manufacturing Systems. Amsterdam: Elsevier. Suer, G., Saiz, Miguel, & Gonzalez, William. (1999). Evaluation of Manufacturing Cell Loading Rules For Independent Cells. International Journal Of Production Research, 37(15), 3445-3468. Süer, G. A., Cosner, J., & Patten, A. (2009). Models for Cell Loading and Product Sequencing In LaborIntensive Cells. Computers & Industrial Engineering, 56(1), 97-105. Elsevier Ltd. Süer, G. A., & Dagli, C. (2005). Intra-Cell Manpower Transfers And Cell Loading In Labor-Intensive Manufacturing Cells. Computers & Industrial Engineering, 48(3), 643-655. Wirth, G. T., Mahmoodi, F., & Mosier, C. T. (1993). An Investigation of Scheduling Policies in A DualConstrained Manufacturing Cell. Decision Sciences, 24(4), 761-788.

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