Malaya J. Mat. S(1)(2015) 141-152 Jobs scheduling problem in a small scale measure of paper making industry Pervaiz Iqbal a, and P.S. Sehik Uduman b a,b Department of Mathematics and Actuarial Science, B. S. Abdur Rahman University, Chennai, Tamil Nadu, India. Abstract Paper plant is one of the key industrial sectors contributing to the Indian economy. It becomes an essential requirement in our day to day life and is used extensively in our society. The paper product is used not only for writing and publishing purpose, but also used in the form of variety of papers such as tissue papers, cardboards and brown papers etc. This paper developed a framework for proper scheduling of activities (jobs) in paper plant by using job scheduling technique. The problem addressed by using of makespan as a measure of performance where the jobs are sequentially scheduled to the order of priorities to achieve the optimum results. Thus, the result shows that CDS and A1 heuristics are preferred to the traditional method of USO (usual serial method). Accordingly, the CDS heuristic followed by A1 gives the best makespan result. Hence, the operational efficiency of paper plant could lead to the continued sustainability of the Industry and a continued support for economic growth in India. Keywords: Paper plant, mathematical model, job orders, job scheduling, customer s order, performance measure and optimum result. 2010 MSC: 90-XX, 90B25. c 2012 MJM. All rights reserved. 1 Introduction The Indian Paper Industry contributes about 2.6% of the world s total production of paper product. The approximate turnover of the industry is Rs. 50,000 crore and its contribution to the Corresponding author. E-mail address: pervaizmaths@gmail.com (Pervaiz Iqbal).
142 Pervaiz Iqbal et al. / Jobs scheduling problem... exchequer is around Rs. 4500 crore worldwide. The industry provides rich in employment to more than 0.5 million people directly and 1.5 million people indirectly. Most of the paper mills are in existence for a long time and hence present technologies fall in a wide spectrum ranging from oldest to the most modern. The mills use a variety of raw material e.g. wood, bamboo, recycled fibre, bagasse, wheat straw and rice husk etc approximately 31% are based on chemical pulp, 47% on recycled fibre and 22% on agro-residues. The geographical spread of the industry as well as market is mainly responsible for regional balance of production and consumption. The operating capacity of the industry currently stands at 12.75 million tons. The current 2013-14 domestic production is estimated to be 11.38 million tons. As per industry guesstimates, over all paper consumption has now touched 13.10 million tons and per capita consumption is pegged at 10 kg. Demand of paper has been hovering around 8% for some time. So far, the growth in paper industry has mirrored the growth in gross domestic product (GDP). India is the fastest growing market for paper globally and it presents an exciting scenario; paper consumption is poised for a big leap forward in sync with the economic growth and is estimated to touch 13.95 million tons by 2015-16. The futuristic view is that growth in paper consumption would be in multiples of GDP and hence an increase in consumption by one kg per capita would lead to an increase in demand of one million tons. The pulp and paper industry comprises companies that use wood as raw material and produce pulp, paper, board and other cellulose-based products. The pulp and paper industry converts fibrous raw materials into pulp, paper and paperboard. In a first step raw materials are processed into pulp and in a second step paper products are produced out of this pulp. Different plant categories exist depending on whether they only produce pulp (pulp mills) for further processing or only paper out of purchased pulp and/or recycled waste paper (paper mills). The third category is integrated pulp and paper mills, combines the two processes and is most common in the paper industry. There are five basic principal steps in pulp and paper productions, such as: wood preparation, pulping, bleaching, chemical recovery and papermaking. The following step by step description is adapted from the World Energy Council, 1995. 2 Literature Review Oladimeji et al. [1] formulated a multi-criteria maintenance job scheduling model using a weighted multi-criteria integer linear programming maintenance scheduling framework. Three criteria were used and they include criteria for the minimization of equipment idle time, manpower idle time and lateness of job with unit parity. Jarvis [2] developed a heuristic computerized maintenance system for planning and scheduling and the model includes a scheme that allocates
Pervaiz Iqbal et al. / Jobs scheduling problem... 143 manpower to jobs based on First in First out (FIFO) queue discipline. A parallel machine scheduling model with job processing times controlled by resource allocation was studied by Chen in [3]. He considered both the cases i.e. continuous type and discrete type of processing times. The objective is to minimize the total cost including the cost measured by a scheduling criterion and the cost of allocated resource. He described the problem as follows; there are a set of n jobs, N = {1, 2,..., n} to be processed on m identical parallel machines. For each job j, N has a due date d j and a weight w j which are both externally given. Campbell et al. [4] proposed the Campbell-Dudek-Smith (CDS) heuristic which is a generalization of Johnsons two machine algorithms. It generates a set of m 1 artificial two-machine problems from an original m-machine problem while each of the generated problems is solved using Johnsons algorithm. Pervaiz et al. [5] discussed the mathematical modelling and digestive system of a paper-making plant based on queuing theory. Pervaiz et al. [6] gives a mathematical modelling and performance analysis of stock preparation unit in paper plant industry by using genetic algorithm. Du [7] proposed an artificial intelligence system (AIS) approach for solving the permutation flow shop scheduling problem. While Liaw [8] developed a two-phase heuristic to solve the problem of scheduling two-machine no-wait job shops to minimize the makespan. Pervaiz et al. [9] proposed a genetic algorithm for solving the permutation flowshop scheduling problem to minimize the makespan. Piggott et al. [10] discussed the rice mill firm; rice milling is a crucial step in post-production of rice. The basic objective of a rice milling is to remove the husk and the bran layers to produce an edible, white rice kernel that is sufficiently milled and free of impurities. Depending on the requirements of the customer, the rice should have a minimum of broken kernels. Another class of research of recent interest is the development of a fuzzy logic-based process control system for Gari fermentation plants developed by Odetunji [11]. 3 Paper Making Process Prior to the development of paper, records were documented on clay tablets and papyrus. The modern form of paper used today is believed to have been invented in China in the early 2 nd century AD as an inexpensive option for silk. In recent times paper has become a necessary material which is regularly found in all over the world. Paper has a wide variety of uses including stationary, newspaper, wrapping paper, cardboard boxes, paper bags and envelopes. It is amazing that the fundamental of paper making process has not changed over 2,000 years. As shown in Figure-1, the process involves breaking up raw material to form pulp and then washing, spreading, pressing and drying the pulp to form the paper.
144 Pervaiz Iqbal et al. / Jobs scheduling problem... F igure 1 : F low chart of paper milling process 3.1 Raw Material Feeding The first step of the paper making process is to obtain the raw materials from natural re-sources or recycled sources. Some natural re-sources are softwood trees, hardwood trees, cotton, and other plants. These materials are grown specifically for the paper mill and are harvested like a crop just for this purpose only. The recycled materials include newspaper and mixed paper etc. 3.2 Debarking and Chipping Natural materials need to be neat and clean and cut into pieces before starting the paper making process. Trees are first cut into logs about six feet in length and processed through a debarking drum to strip away the bark. The logs are then processed through chippers, where spinning blades cut the wood into 1 square metre small pieces. 3.3 Kraft Process Natural materials then need to be processed to break down the lignin from the inside walls of the plant. Lignin is the glue or organic polymer that holds the individual plant fibers together. Paper will turn yellow when exposed to light and air if the lignin is not removed from the raw materials. The lignin is broken down through the Kraft process. This involves pressure cooking the material with white liquor and steam in an industrial digester. White liquor is a chemical mixture composed of sodium hydroxide and sodium sulfide. The end products from the Kraft process are pulp fibers and chemicals. The lignin has previously been removed from recycled materials; therefore the recycled
Pervaiz Iqbal et al. / Jobs scheduling problem... 145 materials do not need to go through the Kraft process. It has to go through a repulping process to separate the paper back into individual fibers. 3.4 Washing After the Kraft process, the end products are processed through water washers that separate the pulp fibers from the used chemicals. The pulp fibers move in the opposite direction of the flow of the water within the washers. The used chemicals can be recycled and reused in the Kraft process. 3.5 Fourdrinier Process Next the pulp is processed through a Fourdrinier machine, where the pulp is dropped onto a continuous moving screen made of woven wires to remove large debris and dirt from the pulp. The waste water is drained away through the screen and recycled. The resulting crude paper is squeezed between a series of large rollers to remove more of the remaining water. The pressure from the rollers ensures the paper is smooth and has a uniform thickness. The paper is then processed through heated dryer rollers to remove the remaining water. The Fourdrinier machine is named after the Fourdrinier brothers who devoted their lives to improving paper making machinery. 3.6 Paper Finishing The finished paper is then spooled into large reels and will be cut and finished per customer specifications. Paper can undergo several different finishing techniques depending on its final use. For example, clay can smooth the paper surface, a softening agent can make paper softer and coating it in plastic can make paper water resistant. 4 Problem Formulation In order to schedule the processing of customers orders such that maximum profit is obtained, the principles guiding flow shop scheduling are adopted as presented in the mathematical frame work. In this case customers are free to bring their jobs at any time. However, each customer order (i.e. paper product) passes through the machines in the same order. Since different quantities are brought for processing and the paper milling has the same surface area characteristics, each order requires different amounts of processing time in hours as presented in the scheduling frame work. 4.1 Single Machine Sequencing A single machine sequencing is a flow shop in which jobs visit the machines in the same sequence. The shop characteristic of a single machine shop is given as:
146 Pervaiz Iqbal et al. / Jobs scheduling problem... N/m//F// F Where, N is the number of jobs in the shop m is the number of machines in the shop F is the flow shop F is the mean flow time. N/m is referred to as the hardware and F/ F is referred to as the software of the system. 4.2 Johnson s 2-Machine Algorithm Johnson s 2-machine algorithm is a process in which the jobs are scheduled in the machines in such a sequence that gives the minimum makespan. A typical case of Johnsons 2-machine algorithm with n-jobs is presented below in figure-2. F igure 2 : F low chart of the Johnsons 2 machines algorithm The flow time for job J in the k th position is given by Therefore, F (k) = P (1) + P (2) + P (3) +... + P (k) F (k) = k P (i) (4.1) i=1 Where P (i) is the processing time for the job in the i th position in the sequence. The algorithm supposes that if we have (n) jobs to be scheduled on two machines i.e. positions are possible. J 1, J 2,..., J n then n Total flow time F T = n n k F (k) = P (i) (4.2) k=1 k=1 i=1
Pervaiz Iqbal et al. / Jobs scheduling problem... 147 Mean flow time F = Generally, for the n th positions we have; n k P (i) k=1 i=1 n (4.3) n (n i + 1)P (i) = i=1 n k P (i) k=1 i=1 n (4.4) The optimizing sequence can be obtained from the following process: In this case we have n jobs to be scheduled on two machines i.e. J 1, J 2,..., J n. The optimum solution by Johnson algorithm is obtained in following steps: 1. Set k = 1, l = n. 2. Set the list of unscheduled jobs =J 1, J 2,..., J n. 3. Find the smallest processing times on first and second machines for the currently unscheduled jobs. 4. If the smallest processing time is obtained in step-3 for J i is on the first machine then schedule J i in k th position of processing sequence. Then delete the J i job from the list of unscheduled and decrease k by 1. 5. If the smallest processing time is obtained in step-3 for J i is on the second machine then schedule J i in the 1 st position of processing sequence. Then delete the J i job from the current list of unscheduled jobs and decrease l by 1. 6. Repeat step-(3 to5) for the remaining unscheduled jobs until all the J jobs are scheduled. The main objective of this Johnsons algorithm of sequentially scheduling the jobs to the two machines from step (1 to 6) is to achieve the minimum total makespan for optimum job scheduling, which is obtained by summing up the various processing times obtained by summing up the various processing times gives the makespan for the optimum scheduling. T able 1 : Makespan Results for 24 W eeks
148 Pervaiz Iqbal et al. / Jobs scheduling problem... Makespan Results week A1 CDS USO 1 46.38 47.25 50.24 2 45.32 45.32 50.24 3 45.32 45.33 50.45 4 45.12 45.12 52.42 5 46.04 46.06 51.52 6 47.20 47.18 51.46 7 46.52 46.54 51.08 8 45.33 45.54 51.42 9 43.49 43.40 46.42 10 47.42 47.40 50.54 11 45.82 45.80 51.28 12 45.24 45.24 51.54 13 45.70 45.72 52.50 14 46.04 46.00 53.16 15 50.00 48.42 53.26 16 45.72 45.70 53.22 17 46.32 46.30 50.54 18 45.16 45.18 51.27 19 45.18 44.94 51.34 20 45.24 44.26 51.28 21 46.32 46.02 48.42 22 45.82 45.64 49.08 23 48.26 48.12 48.96 24 46.12 47.08 50.02 T able 2 : Gains in scheduling operation
Pervaiz Iqbal et al. / Jobs scheduling problem... 149 Makespan Results week USO-A1 USO-CDS 1 3.86 2.99 2 4.92 4.92 3 5.13 5.12 4 7.30 7.30 5 5.48 5.46 6 4.26 4.28 7 4.56 4.54 8 6.09 5.88 9 2.93 3.02 10 3.12 3.14 11 5.46 5.48 12 6.30 6.30 13 6.80 6.78 14 7.12 7.16 15 3.26 4.84 16 7.50 7.52 17 4.22 4.24 18 6.11 6.09 19 6.16 6.40 20 6.04 7.02 21 2.10 2.40 22 3.26 3.44 23 0.70 0.84 24 3.90 2.94 T able 3 : P rocess mean and standard deviation Method Mean Makespan Standard Deviation A1 46.04 1.26 CDS 45.98 1.16 USO 50.90 1.59
150 Pervaiz Iqbal et al. / Jobs scheduling problem... F igure 3 : Graph of processing time for here methods. The data collected here is for a period of 24 different weeks for 24 jobs orders. The processing time, which is the amount of time in hours required to process each customers order on each machine, is considered close to the reality. The scheduling period covers one week which implies that all customers orders for a week are considered and the scheduling activities are prepared on Monday morning before processing of jobs commences. Normally the processing of customers orders (jobs) is on a first-come-first-serve basis. Therefore, the first customer to arrive for service is given a serial order 1; the second customer is given serial order 2, while the third is given serial order 3 etc. However, since it was discovered that the firm processes jobs using this serial order, we referred to this method as usual serial order (USO). The method was included in the program so that it can be evaluated alongside the solution methods. The principle here is to monitor the completion time of the last scheduled customers order. The three methods are adopted and they include: the A1, CDS and USO, which represent two methods developed by Oluleye and Oyetunji (Oluleye et al. 2007) and the traditional method used by the firm. sectionresult and Discussions Table-2 shows the makespan obtained for the three methods (A1, CDS and USO) for the 24-week study period. For all the three methods, the makespan obtained at the ninth period were the minimum, showing 43.49 hrs, 43.40 hrs and 46.42 hrs, respectively for the A1, CDS and USO methods. Similarly, the makespan obtained for the three methods at the eleventh week were the maximum, showing 50.00 hrs, 48.42 hrs and 53.26 hrs respectively. It is seen that from the minimum makespan for instance, A1 and CDS methods performed good with CDS the best, while the traditional approach of USO performed poorly. This implies that if the old approach is continued
Pervaiz Iqbal et al. / Jobs scheduling problem... 151 the jobs for fourteenth week would still stay for an excess of 3.02 hrs in the process before being completed which is equivalent to about an extra half a day wasted in a day of 8 working hours. 5 Conclusion Johnson 2-machine algorithm has successfully applied to job scheduling in a paper making Industry. It has demonstrated that the conventional approach in scheduling of customer orders for paper making in a paper plant based on the plant usual serial order method in which jobs are scheduled as they arrive for processing fails to satisfy the profit maximization objective of the plant. Three methods were used to analyze the data collected from the paper plant. The three methods are A1, CDS and USO which represent two methods (A1 and CDS) developed and the traditional method (USO) used by the plant. We found that CDS performs the best followed by A1, while the worst performance was observed with USO that is clearly seen in Table-2 and Figure-3. Adopting the CDS method will increase the optimum performance of the plant and hence, it is recommended to the plant management to make use of CDS method. References [1] F.O. Oladimeji, S.A. Oke and P.P. Oluwatoyin, A multi-criteria model for maintenance job scheduling, Maejo International Journal of Science and Technology, Vol. 2(2008), 1-12. [2] W.J. Jarvis, Maintenance planning and scheduling in petrochemical manufacturing environment, National Petroleum Refineries Association, Wash. DC, MC, 80(2)(1983), 1-12. [3] Z.L. Chen, Simultaneous job scheduling and resource allocation on parallel machines, Department of Systems Engineering, University of Pennsylvania, Philadelphia, 1999. [4] H.G. Campbell, R.A. Dudek and M.L. Smith, A heuristic algorithm for n-job, m-machine sequencing problem, Journal of Management Science Vol. 16, pp. 630-637, 1970. [5] Pervaiz Iqbal and P.S. Sehik Uduman, Mathematical modeling and behavior of the digestive system of a paper making plant based on queuing theory, International Journal of Pure and Applied Mathematics, Vol. 90(2014), 43-56. [6] Pervaiz Iqbal and Sehik Uduman, Mathematical modeling and performance analysis of stock preparation unit in paper plant industry using genetic algorithm, Mathematical Sciences,, Vol. 34(02)(2014), 1629-1638. International Journal of [7] J. Du, Minimizing mean flow time in two-machine open shops and flow shops, Journal of Algorithms, Vol. 14(1993), 24-44.
152 Pervaiz Iqbal et al. / Jobs scheduling problem... [8] C.F. Liaw, An efficient simple meta-heuristic for minimizing the makespan in two-machine no-wait job shops, Journal of Computers and Operations Research, Vol. 35(10)(2008), 3276-3283. [9] Pervaiz Iqbal and P.S. Sehik Uduman, Genetic algorithm for permutation flowshop scheduling problem to minimize the makespan, International Journal of Computing Algorithm, Vol. 03(02)(2014), 1086-1091. [10] J.R. Piggott, W.R. Morrison and J. Clyne, Changes in Lipids and in Sensory Attributes on Storage of Rice Milled to Different Degrees, International Journal of Food Science and Technology. Vol. 26(6)(2007), 615-628. [11] O.A. Odetunji, O.O. Kehinde, Computer Simulation of Fuzzy Control System for Gari Fermentation Plant, Journal of Food Engineering. Vol. 68(2)(2005), 197-207. Received: April 15, 2015; Accepted: June 13, 2015 UNIVERSITY PRESS Website: http://www.malayajournal.org/