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BI-CRITERIA IN TWO-MACHINE SCHEDULING UNDER RENTAL SITUATION Laxmi Narain Associate Professor Department of Mathematics, Acharya Narendra Dev College, University of Delhi, Delhi, India [email protected] ABSTRACT This paper studies bi-criteria in 2-machine flow-shop problems under rental policy III. In this paper, Policy III is modified. Here second machine will not be taken on rent at times when the first job is completed on first machine but 2 nd machine will be taken on rent subject to minimum total elapsed time. The objective is: obtain a sequence which minimizes total rental cost subject to minimum total elapsed time. A simple algorithm is developed using Johnson technique. The algorithm is illustrated through a numerical example. Keywords Flow-shop Scheduling, Rental Cost, Rental Time, Elapsed Time, Idle Time 1. INTRODUCTION A study on scheduling literature has revealed the desirability of an optimal schedule being evaluated by more than one performance measures or criteria. Various authors [3-7, 9-19] have studied the flow-shop problems having more than one optimization measures. Gupta and Dudek [7] strongly recommended the use of combination of criteria total flowtime and total elapsed time. Narain and Bagga [12] studied njob, m-machine special flow-shop problems which give minimum possible mean flowtime while minimizing total elapsed time. Narain and Bagga [9] determine the sequence which minimizes the total elapsed time subject to zero total idle time of machines i.e., machines should not remain idle once they start the first job. Narain and Bagga [10] studies njob, m-machines flowshop problems when processing times of jobs on various machines follow certain conditions and the objective is to obtain a sequence which minimizes total elapsed time under no-idle constant. Narain and Bagga [11] studied n-job, 2-machine flowshop problem and provided an algorithm for obtaining a sequence which gives minimum possible mean flowtime under no-idle constraint. In flow-shop problem, situation can occur in practice when one has got the assignment but does not have one's own machines or does not have enough money for the purchase of machines, under these circumstances, may take machines on rent to complete the assignment. Minimization of total rental cost of machines will be the criterion in these types of situations. The following renting policies generally exist: Policy I: All the machines are taken on rent at one time and are returned also at one time. Policy II: All the machines are taken on rent at one time and are returned as and when they are no longer required. Policy III: All the machines are taken on rent as and when they are required and are returned as and when they are no longer required for processing. Bagga [1] studied three-machine problem under Policy I and provide a sequence which minimize the total rental cost of machines. Under Policy II, for three-machine flow-shop problem, Bagga and Ambika [2] provided a Branch-andBound algorithm. This paper studies bi-criteria in 2-machine flow-shop problems under rental Policy III. In this paper, Policy III is modified. Here second machine will not be taken on rent at times when the first job is completed on first machine but 2 nd machine will be taken on rent subject to minimum total elapsed time. Here, the objective is to obtain the sequence which minimize the total elapsed time and provides the total rental cost as minimum as possible. For any sequence S, Total rental cost of machines 2 = n [ p i, j ( S ) Ii , j (S )] cj j 1 i 1 Where pi,j(S) is the processing time of ith job of sequence S on machine Mj, Ii,j(S) is the idle time of machine Mj for ith job of sequence S and Cj is rental cost per unit time of machine M j. Here, the processing times p i,j(S) and rental cost Cj(S) are constant. Therefore, we can only reduce idle times I i,j(S). Idle time on 1st machine is zero. To reduce idle time on 2nd machine, we delay the time of renting of 2 nd machine to process jobs. We have obtained a simple and efficient algorithm, without using Branch-and-Bound technique, which provides optimal sequence having minimum total rental cost without altering the total elapsed time. The algorithm is illustrated through a numerical example. 2. MATHEMATICAL FORMULATION Notations: S : Sequence of jobs 1, 2,…, n. = max (Zm+1,1, max (Zm+1,1, Zm,2 )) + pm+1,2 Mj : Machine j; j=1, 2, 3. = max (Zm+1,1, Zm,2 ) + pm+1,2 ith pi,j(S) : Processing time of machine Mj. job of sequence S on = Zm+1,2 Ii,j(S) : Idle time of machine Mj for ith job of sequence S. Cj : Rental cost per unit time of machine Mj. Hj(S) : The time when Mj is taken on rent for sequence S. Zi,j(S) : Completion time of ith job of sequence S on machine Mj. Z′i,j(S) : Completion time of ith job of sequence S on machine Mj when Mj starts processing jobs at time Hj(S). T2(S) : Total time for which M2 is required when M2 starts processing jobs at time H2(S). M2 at time H2 > i = 1, 2,…, n and j= 1, 2, 3. increase. Therefore, the result holds for k = m+1 also. Hence, by mathematical induction this theorem holds for all k, where k = 1, 2, ..., n. Under Policy III, machine is taken on rent when it is required and is returned as soon as it completes the last job. By Theorem 2.1; the starting of processing jobs at time H2 on M2 will reduce the idle time of M2 to zero and M2 will be required only for time equivalent to the sum of the processing times of all the jobs on it. Therefore, total rental cost of M 2 will be minimum (least). Total rental cost of M1 will always be minimum (least), since idle time of M1 is always zero. It is obvious from Theorem 2.1, that if we start processing jobs on n Let n jobs require processing over three machines M1, M2 and M3 in the order M1→ M2→ M3. I i,2 , then total elapsed time will i 1 The following algorithm provides the procedure to obtain the sequence which gives minimum possible rental cost while minimizing total elapsed time in 2-machine flow-shop problem under Policy III. Theorem 2.1: If we start processing jobs on M2 at time k H2 = I i,2 , then Zk,2 will remain unaltered. i 1 3. ALGORITHM Algorithm 3.1: Proof: Let Z′i,2 be the completion time of ith job on machine M2 when M2 starts processing jobs at time H2. The proof of the theorem is based on the method of mathematical induction. For k = 1; Step 1: Obtain the sequence having minimum total elapsed time by Johnson algorithm. Let this sequence be S. Step 2: Compute total elapsed time Zn,2(S). Step 3: Compute rental time H2 of M2 for sequence S Z′1,2 = H2 + p1,2 n H2 = Zn,2(S) - 1 = I i,2 p i,2 i 1 + p1,2 i 1 Step 4: Compute total rental cost for sequence S = p1,1 + p1,2 n R(S) = = Z1,2 pi ,1 × C1 + i 1 n p i,2 × C2 i 1 Therefore, the result holds for k = 1. Let the result holds for k = m 4. EXAMPLE For k = m+1; Example 4.1: Z′m+1,2 = max (Zm+1,1, Z′m,2 ) + pm+1,2 m p = max (Zm+1,1, H2 + i,2 ) + pm+1,2 i 1 m 1 = max (Zm+1,1, I i,2 i 1 I i,2 i 1 p i,2 ) + pm+1,2 Jobs i 1 m = max (Zm+1,1, Table 1: Processing Times of Jobs on Machines m + Consider 5-Job, 2-Machine flow-shop problem with processing times (in hours) as given in Table 1 and the rental costs per unit time for machines M1 and M2 are Rs 100 and Rs 200 respectively. m + p i,2 + Im+1,2) + pm+1,2 i 1 = max (Zm+1,1, Zm,2 + max (Zm+1,1- Zm,2 , 0)) + pm+1,2 Machines M1 M2 1 3 5 2 9 2 3 7 4 4 15 11 5 5 6 Operational Research Society, U. K. 43(9) (1992), 871884. Applying Algorithm 3.1; Step 1: Provides sequence S =1-5-4-3-2 Step 2: To obtain minimum total elapsed time, the completion time In-Out of the sequence S is given in table 2. Table 2: Completion Times In-Out. Jobs Machines M1 M2 In-Out In-Out 2 0-3 3 3-8 8-14 4 8-23 23-34 5 23-30 34-38 1 30-39 39-41 [9] Narain, L. and Bagga, P.C. 2003. Minimizing total elapsed time subject to zero idle time of machines in n3 flow-shop problem. Indian Journal of Pure & Applied Mathematics. 34 (2003), 219-228. 3-8 [10] Narain, L. and Bagga, P. C. 2004. Flow-shop/no-idle scheduling to minimize total elapsed time. Journal of Global Optimization. 33(2004), 349-367. =39 – 28 = 13 [13] Sen, T. and Gupta, S.K. 1983. A branch and bound procedure to solve a bi- criterion scheduling problem. AIIE Transactions. 15(1983), 84-88. = 9400 (Rupees). Using Algorithm3.1; [14] Sen, T. and Dileepan, P. 1999. A bicriterion scheduling problem involving total flowtime and total tardiness. Journal of Information and Optimization Sciences. 20(2) (1999), 155-170 = 13 x 200 = 2600 (Rupees). 5. CONCLUSION We have proved a theorem to find out the times at which machines should be taken on rent so that total elapsed does not change when we delay the processing of jobs on machines. By using this theorem a simple and efficient Algorithm is developed. From given example it is clear that this Algorithm is very important if the rent of the machines is high. REFERENCES [1] Bagga, P.C. 1969. Sequencing in a rental situation. Jr. of Canadian Operations Research Society. 7(1969), 152153. [2] Bagga, P.C. and Ambika Bhambani. 1996. Minimizing rental costs in three-machine sequencing problem. Jr. of Indian Association for Productivity, Quality and Reliability. 21(1996), 73-77. [3] Bagga, P.C. and Ambika Bhambani. 1996. Bicriteria in flowshop scheduling problems. Journal of Combinatorics, Information and System Sciences. 21(34)(1996). [4] Bagga, P.C. and Ambika Bhambani. 2002. Bicriteria in special flowshop problems. Indian Journal Pure and Applied Mathematics. 33(4) (2002), 435-441 [5] Chandersekharan Rajendran. 1992. production scheduling problem with [11] Narain, L. and Bagga, P. C. 2005. Flow-shop/no-idle scheduling to minimize mean flow-time. ANZIAM. Journal. 47(2005), 265-275. [12] Narain, L. and Bagga, P. C. 2006. Bi-criteria in n × m flow-shop problems. IAPQR Transactions. 31(2006), 5771. = 38 x 100 + 28 x 200 = 3800 +5600 Money saving [7] Gupta, J.N.D. and Dudek, R.A. 1971. Optimality criteria for flowshop schedule. AIIE Transactions. 3(3) (1971), 199-205. [8] Johnson, S.M. 1954. Optimal two and three stage production schedule with set up times included. Naval Research Logistics Quarterly. 1(1954), 61-68. Step 3: Rental time of machine M2 = 41 – (5+2+4+11+6) Step 4: Total Rental cost [6] Dileepan, P. and Sen, T. 1988. Bicriterion state scheduling research for a single machine. Omega. 16 (1988), 53-59. Two-stage bicriteria. [15] Sen, T. and Gupta, S.K. 1983. A branch and bound procedure to solve a bicriterion scheduling problem. AIIE Transactions. 15(1983), 84-88. [16] Sen, T. and Raiszadeh, F. 1997. An algorithm minimize total flowtime and maximum job lateness the two-machine flowshop system. Journal Information and Optimization Sciences. 18(1) (1997), 16. to in of 9- [17] Sen, T., Raiszadeh, F. and Dileepan, P. 1988. A branchand-bound approach to the bicriterion scheduling problem involving total flowtime and range of lateness. Management Science. 34(2) (1988), 254-260. [18] Van Wassenhove, L.N. and Gelders, L.F. 1980. Solving a bicriterion scheduling problem. European Journal of Operational Research. 4(1980), 42-48. [19] Van Wassenhove, L.N. and Baker, K.R. 1982. A bicriterion approach to time/cost trade-off in sequence. European Journal of Operational Research. 11 (1982) 48-54. AUTHOR’S BIOGRAPHY Dr laxmi Narain is currently an Associate Professor in Department of Mathematics, Acharya Narendra Dev College, University of Delhi, Delhi, India. He has specialization in Optimization Technique. He has done Ph.D. in Flow-shop Sequencing Problem from Department of Mathematics, University of Delhi under the Guidance of Dr. P. C. Bagga. He has published research papers in National and International Journals.