Sustainable Production Planning Using the Ranked Positional Weight Technique and Bayesian Estimation

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1 International Journal of Japan Association for Management Systems Vol. 5 No.1 November 2013, pp Sustainable Production Planning Using the Ranked Positional Weight Technique and Bayesian Estimation Takayuki Kataoka a *, Atsushi anafuji a, Masakazu Kanezashi a, Katsumi Morikawa b, and Katsuhiko Takahashi b a Faculty of Engineering, Kinki University, 1, Takaya-Umenobe, igashi-iroshima, Japan b Graduate School of Engineering, iroshima University, 1-4-1, Kagamiyama, igashi-iroshima, , Japan Abstract In recent years, almost every manufacturing site has been supported by a lot of part-time, temporary, or mid-career personnel, given the poor state of the economies all over the world. owever, expert managers of front-line workers have to design more complex methods of production planning, when urgent orders due to uncertain elements like a disaster are taken into consideration. Therefore, this paper proposes a sustainable production planning model using an inference methodology. The method was suggested by Mahadevan et al. in the field of architecture, and some cases of its effectiveness have been shown. In this paper, we try to incorporate that method into our sustainable production planning model. First, a work element that overflows into another process is assumed to be a Dummy Moving Element (DME). Next, DME estimation analysis and improvement using Bayesian estimation are discussed. Finally, the effectiveness of our model is verified by a numerical experiment. Key words: Bayesian estimation, Ranked positional weight technique, Sustainable production planning, Monte Carlo simulation 1. Introduction In recent years, almost every manufacturing site has been supported by a lot of part-time, temporary, or mid-career personnel, given the poor state of the economies all over the world. owever, expert managers of front-line workers must design more complex methods of production planning, to take urgent orders due to uncertain elements like disasters into consideration. Therefore, we feel that it is necessary to discuss a sustainable production planning model using an inference methodology. Focusing on the study of line balancing, Becker et al. conducted a survey of problems and methods in generalized assembly line balancing (ALB) [1,2]. ALB research has traditionally focused on the simple assembly line balancing problem (SALBP) which has some restricting assumptions. Boysen et al. [3] also discuss ALB, namely, which model to use and when. Their paper structures the vast field of ALB according to characteristic, practical settings and highlights the relevant model extensions required to reflect real-world problems. These reports show that the ranked positional weight technique is robust for large-scale problems. owever, they have only focused on ALB and are unable to consider unpredictable urgent orders. On the other hand, Corominas et al. [4] discuss ALB in terms of skilled and unskilled workers. The goal is to minimize the number of temporary workers required, given a cycle time and the team of workers on staff. Otto et al. [5] also discuss incorporating ergonomic risks into ALB. In their article, they show that even though most ergonomic risk estimation methods involve nonlinear functions, they can be integrated into ALB techniques at low additional computational cost. owever, these papers have also only focused on ALB to minimize the number of workers or ergonomic risks. Next, focusing on the study of inference methodologies, Weber et al. [6] discuss complex system reliability modeling with dynamic objectoriented Bayesian networks. owever, we cannot include that in our discussion because special software is necessary to adopt their methodology. owever, Mahadevan et al. [7] propose a methodology to apply a Bayesian network to structure system reliability reassessment. Some cases of its effectiveness have been shown, for example, in the field of architecture. A framed structure with multiple potential locations of plastic hinges and multiple failure sequences are analyzed to illustrate the proposed method. We feel that it is possible to replace Structural Failure in architecture with Process Failure (moving a work *Corresponding author: kataoka@hiro.kindai.ac.jp Received: January 31, 2013 Accepted: September 5,

2 Takayuki Kataoka, Atsushi anafuji, Masakazu Kanezashi, Katsumi Morikawa, and Katsuhiko Takahashi element to another production process) in the field of production management. Therefore, this paper proposes a sustainable production planning model using the inference methodology suggested by Mahadevan et al. In this paper, we try to incorporate their method into a sustainable production planning model. First, a work element that overflows into another production process is assumed to be a Dummy Moving Element (DME). Next, DME estimation analysis and improvement using Bayesian estimation are discussed. Finally, the effectiveness of our model is verified with a numerical experiment. 2. Modeling The proposed method of Mahadevan et al. is effective in the field of structural architecture. In the traditional method, the system failure probability is calculated through the union of all the failure sequences. In the Bayesian network analysis, it is seen that partial event analysis is also needed for the conditional probability calculation. Therefore, the Bayesian network approach has more computational effort than the traditional method. owever, they suggest that this extra effort provides a return by enabling the propagation, that is, reliability re-assessment [7]. That means to analyze the propagations of breaking points and improve the breaking point which the conditional probability is the highest. For example, only a few break points may hardly influence the system failure. On the other hand, it is seen that the propagation of breaking will greatly influence it. An example is shown in Fig. 1. The thick frame shows some observation points and the wavy frame shows some force points added to a building. Some force points have random values by Monte Carlo simulation. If a force point becomes larger, a propagation of breaking (6-4-7) will occur, as shown in Fig. 2. Next, an observation point of the highest conditional probability is selected and an improvement in durability is discussed. Fig. 2. Example of structural failure In this paper, we try to incorporate the above method into a sustainable production planning model. Because of the prior job training, it is not easy to move work elements to another production process. That means the smaller the number of DMEs, the better the model. The detailed explanation is shown in Fig. 3. The thick frame shows some work elements and the wavy frame shows some urgent tasks added to a production process. The urgent tasks have random values by Monte Carlo simulation. If an urgent working hour becomes larger, a propagation of DME (4-5-7) will occur, as shown in Fig. 4. Next, a work element of the highest conditional probability is selected and an improvement in robustness is discussed by changing the assigned process. Fig. 3. Initial condition of process failure Fig. 1. Initial condition of structural failure Fig. 4. Example of process failure International Journal of Japan Association for Management Systems

3 Sustainable Production Planning Using the Ranked Positional Weight Technique and Bayesian Estimation 3. Mathematical Formulas Equation (1) shows the objective function to minimize the total number of DMEs. Equation (2) shows the number of DMEs. Equations (3), (4) and (5) show cycle time, total working hours, and the least number of processes, respectively, in order to design production planning. We can then regard a multiproduct assembly line as a single-model assembly line using Eqs. (6) and (7). Equation (8) is necessary to use the ranked positional weight technique proposed by elgeson and Birnie. Equation (9) shows how work elements are assigned to each process. Equation (10) shows the probability that DMEs occur, and variables of a, b,,n show the work elements. Equation (11) shows the conditional probability of a DME for each work element, based on Eq. (10). [Mathematical Symbols] B : The total number of DMEs V x : The number of DMEs with condition x x : Urgent working hours (x=0,,n) G (mg) : set of work elements assigned to the process M j (M j ) C : cycle time A:operating time on schedule Q i : production workload of item i : number of items N min : least number of processes L : total working hours T i : total working hours to make item i w: work element(w=1,,k) T w : total working hours between periods of work element w 1: if w is necessary to make item "i" δ w = { 0: otherwise t w : average working hours of w per unit within planning cycle W w : positional weight of w t w : working hours of w a ij : value of rank "i" column "j" in precedence matrix (..k,j=1..k) M j : process "j" M j : process "j" including urgent work 1: if system conditon is normal e= { 0: otherwise R : set of unassigned w {R=1,,k} T (Ww ) : working hours of w with weight W S (Ww ) : precedence of w with weight W Δ: comparing work elements each, and picking up w that DME occurred s: number of simulations D w : w that a DME occurred ω(w w ): unassigned list where the ranked positional weight is sorted in descending order [Formulas] n Minimize [B = V x ] (1) N min x=0 V x = G(M j ) G(M j) (2) j=1 C = A/ Q i L = Q i T i (3) (4) N min = L/[C Q i ] (5) T w = Q i δ wi t w t w = T w / Q i K W i = t i + t j max(a ij, 0) M j (: M j ) j=1 = {T (Ww ) [S (W w ) R = ] [max(ω(w w ))] } C w R (6) (7) (8) (9) P(e = 0) = B/s (10) P(D w e = 0) = P(D w, e = 0) P(e = 0) (11) 4. Proposed Method There is the potential for a production line designed by the ranked positional weight technique to have DMEs occur, due to added tasks like an urgent order (urgent work). This method is proposed by elson and Bienie[8]. And, it has a strong point in that the calculation time does not increase largely when the number of urgent work elements on a large-scale production line increases. A flow chart of our proposed method is shown in Fig. 5. At first, urgent tasks that may occur in a production line designed by the ranked positional weight technique are estimated. Next, working hours for the Vol. 5 No. 1 (2013)

4 Takayuki Kataoka, Atsushi anafuji, Masakazu Kanezashi, Katsumi Morikawa, and Katsuhiko Takahashi Start Production line designed by the ranked positional weight technique. (the normal) Monte Carlo simulation toward x (x represents working hours for urgent work) Production line design when urgent work occurs Compare the normal production line with a production line including urgent work Pick up DME propagation Count the number of DMEs with regards to each work element from a set of DME propagation Using Bayesian estimation, calculate DME probability of each work element considering each work element in which a DME occurred Pick up a work element w where DME probability is the next highest Work element w1 is not tested in process Mj ( w1 is tested by positional weight in descending order) w1 maintains precedence and cycle time when w1 added to Mj1 ( Mj1 is the process where w is assigned in first step) W is replaced by W1 The number of simulations < 1000 Examine process Mj, where w is most changeable. Consider robustness from the following Maintain precedence when w added Cycle time not exceeded when w added to Mj w added to Mj Improved process (proposed method) End Fig. 5. Flow chart of proposed method urgent work are estimated, and all DMEs that may occur due to urgent work are checked. Then, some work elements that could be changed and moved to another process are estimated by using Monte Carlo simulation and Bayes estimation toward DMEs. Finally, the most frequent moved element will be previously assigned to another process while maintaining the cycle time and precedence relationships among the work elements. As a result, a sustainable production line design in the case of urgent work can be discussed. 5. Numerical Experiment [Precondition] (1) The working hours are 480 time units, the production size is 64, the number of work elements is 11, and the number of urgent tasks is 1. (2) Fig. 6 shows a precedence diagram that regards a multi-product assembly line as a single model assembly line. (3) Urgent work is element 2. Working hours of element 2 change 0.5 by 0.5 within the parameters of 0 to 6 based on a uniform distribution. The cycle time is fixed by increasing the working hours to continue production. Table 1 shows the normal work elements, working hours, positional weight, and precedence. Fig. 6 shows the normal precedence diagram of Table 1. The cycle time is 7.5 time units according to the preconditions. Table 2 shows how to assign work elements to some process by maintaining cycle time based on the ranked positional weight technique. Table 3 shows an example where the working hours of urgent work element 2 are 3 time units. Fig. 7 shows an urgent work precedence diagram of Table 3. Table 1. Work elements Work Working Positional element hours weight Precedence , 4, , , 10 International Journal of Japan Association for Management Systems

5 Sustainable Production Planning Using the Ranked Positional Weight Technique and Bayesian Estimation As a result, when the working hours of urgent work element 2 are 3 time units, work elements 3, 8, 6, 9 and 11 move to another process. With the changing working hours of urgent work element 2, the same procedure is repeated. Fig. 6. Normal precedence diagram Table 2. Traditional method Process Work element Total working hours 1 1, 2, , 3, , 6, , Table 3. Cases when urgent work is 3 time units Work Working Positional element hours weight Precedence , 4, , , Fig. 7. Urgent work precedence diagram Table 4 shows how to assign work elements to some process by maintaining the cycle time based on Table 3. In this experiment, we compared Table 2 with Table 4 and focused on DMEs; work elements 3 and 8 in process 2, work elements 6 and 9 in process 3, and work element 11 in process 4 move to another process Table 4. Traditional method when urgent work is 3 time units Process Work element Total working hours 1 1,2, ,4, ,7, , Next, Monte Carlo simulation is used to find the probability that DMEs will occur on the production line, and each work element will move to another process. In Table 5, the results of 1000 simulations of this experiment are shown. The number of DME propagations is 180. This shows that the total number of work elements 8, 9 and 10 moving to another process is 180 out of 1000 in the case of urgent work element 2. Also, we can calculate the probability that DMEs will occur with each work element by using Bayes estimation. The highest probability of DMEs is considered the most changeable work element. Additionally, it must also maintain precedence. In this experiment, because work element 8 is the most changeable of all elements keeping precedence from Table 6, it is not assigned to process 2 but to process 3 in the first step, compared with the normal ranked potential weight technique (traditional method). Table 7 shows the results of the proposed method. Not only work element 8 but also work element 6 is changed to maintain the cycle time. In this experiment, the case of all elements (not only element 2 ) is discussed. The results of comprehensive solutions are shown in Tables 8 and 9. As a result, the proposed method was able to decrease the total number of DMEs while maintaining the efficiency of line balancing compared with the traditional method, as shown in Tables 8 and 9. Table 5. Results of simulation List of DMEs The number of occurrences Total 808 Vol. 5 No. 1 (2013)

6 Takayuki Kataoka, Atsushi anafuji, Masakazu Kanezashi, Katsumi Morikawa, and Katsuhiko Takahashi Table 6. Conditional probabilities of DMEs Probability of Work element DMEs Table 7. Proposed method Process Work element Total working hours 1 1, 2, , 3, , 8, , Table 8. Number of DME occurrences Urgent element Traditional method Proposed method N. B * * * * * * Total Table 9. Result of comparative experiment Line status Line-balancing efficiency Normal *Expected value: ( )/10= owever, some results of the proposed method are worse than those of the traditional method, as shown in N.B. A new process which consists of only an urgent element is frequently added to maintain precedence when an urgent element is added into an upper process. That is why DMEs occur due to changing elements based on proposed method instead of the traditional method, in which no DMEs occur. It is necessary for us to develop a way to evaluate such a case. 6. Conclusions In this study, our proposed method showed potential for decreasing the number of DMEs while maintaining the line balancing efficiency when compared with the traditional method. In the future, it will be necessary for us to discuss some cases in which the location and number of urgent work elements on a large-scale production line change. ACKNOWLEDGMENT This work was supported by JSPS KAKENI Grant Number References [1] C. Becker and A. Scholl: A survey on problems and methods in generalized assembly line balancing, European Journal of Operational Research, 168 (2006), [2] C. Becker and A. Scholl: Balancing assembly lines with variable parallel workplaces: Problem definition and effective solution procedure, European Journal of Operational Research, 199 (2009), [3] N. Boysen, M. Fliedner, and A. Scholl: Assembly line balancing: Which model to use when?, International Journal of Production Economics, 111 (2008), [4] A. Corominas, R. Pastor, and J. Plans: Balancing assembly line with skilled and unskilled workers, The International Journal of Management Science, 36 (2008), [5] A. Otto, and A. Scholl: Incorporating ergonomic risks into assembly line balancing, European Journal of Operational Research, 212 (2011), [6] P. Weber and L. Jouffe: Complex system reliability modelling with Dynamic Object Oriented Bayesian Networks. (France, 2003, ) [7] S. Mahadevan, R. Zhang, and N. Smith: Bayesian networks for system reliability reassessment, Vanderbilt University, Structural Safety, 23 (2001), [8] W.B. elgeson, and D.P. Birnie: Assembly line balancing using the ranked positional weight technique, Journal of Industrial Engineering, 12 (1961), International Journal of Japan Association for Management Systems