The present paper deals with a new perfomance measure, the actual mean flow time, defined as a mean of the elapsed time of each job counted from the start time on a schedule to the corresponding due date. For the one machine backward scheduling model with a common due date and independent setup times, LPT schedule is shown as the optimal solution for the proposed measure. An optimal
algorithm is presented for the case with dependent setup times on the basis of the algorithm by Arcelus and Chandra for a n / 1 / F forward scheduling problem. The proposed algorithm is coded in C-language and a computational experience is reported through a 16-bit computer

Miyazaki, Shigeji

Sekoguchi, Kenji

Okayama University Scientific Achievement Repository

Memoirs of the Faculty of Engineering,Okayama University,Vo1.29, No.2, pp.89-94, March 1995Backward Scheduling to Minimize the Actual Mean Flow Timewith Dependent and Independent Setup TimesKenji SEKOGUCHI* and Shigeji MIYAZAKI**(Received January 27 , 1995)The present paper deals with a new perfonnance measure, the actual mean flow time, defmed as amean of the elapsed time of each job counted from the start time on a schedule to the correspondingdue date. For the one machine backward scheduling model with a common due date and independentsetup times, LPT schedule is shown as the optimal solution for the propos.ed measure. An optimalalgorithm is presented for the case with dependent setup times on the basis of the algorithm byArcelus and Chandra for a n / 1 / F forward scheduling problem. The proposed algorithm is codedin C-Ianguage and a computational experience is reported through a 16-bit computer.1. INTRODUCTIONConsider a set of n jobs J i (i = 1,2,··· , n) to be processed on one machine, where the processing time Pi (i = 1,2, ... , n) forjob J i is given and fIxed. The due date of each job is assumed as a common date (d) for all jobs. The ready time Ri (the preparation)of each job is assumed to be able to correspond with the start time Bi on a schedule to be made. Every job should be completed beforeor just on the due date (no tardy jobs allowed), and leave the shop simultaneously with its due date. Other conditions are the same as theordinary scheduling model [1).According to the conditions above, the flow time of each job in the shop inevitably coincides with the elapsed time of the jobcounted from the start time of each job on a schedule to the corresponding due date. In what follows the flow time defIned above iscalled the actual flow time F' which is fonnulated as :~=d-Bi, (i=1,2,"',n), (1)An average of actual flow times over n jobs called actual mean flow time Fa :Fa=(lIn)fF', (2)i-IIn this paper actual mean flow time is used as a perfonnance measure and optimal scheduling algorithms for the setup dependent andindependent problems are proposed to minimize Fa under a constraint of no tardy jobs.The backward scheduling is adopted here for a solution procedure, in which the job is scheduled in reverse order as from the lastposition to the earlier position. The method has a merit of easy scheduling with no tardy jobs. For the one machine model with acommon due date and independent setup times, LPT schedule is shown as the optimum for the proposed measure. A Branch and Boundalgorithm is presented for the case with dependent setup times on the basis of an algorithm for a conventional forward n / 1 / F setupdependent problem with the simultaneously arriving jobs. The proposed algorithm is coded in C-Ianguage and a computationalexperience is reported through a 16-bit micro computer.• The Graduate School of Natural Sience and Technology•• Engineering Mathematics8990 Kenji SEKOGUCHI and Shigeji MIYAZAKI2. SEQUENCE INDEPENDENT SETUP TIMESThe model treated here can be illustrated in Fig. 1, in which the setup time of J i is included in the processing time Pi and thesequence of jobs is being determined along with J 1 through J 3. The completion time of the last job ( J 1) should coincide with thecommon due date (d) according to the constraint of no tardy job and for minimizing Fa .d-F/The orderfor assigningjobs jFa = (1 I 3)2,F~ - min.i=1Fig. 1. Sequence independent case.[Theorem 1)For the nilI Fa backward model with sequence independent setup times, LPT schedule (the shorter job comes later position)gives the optimal solution.Proof: nilI Fa backward model can be transformed into the equivalent nil I F forward model where SPT schedule is theoptimal [2]. The reverse of SPT yields the optimal LPT schedule.3. SEQUENCE DEPENDENT SETUP TIMESTransformation into a forward modelIn the case that the setup time for next job J j is dependent upon the job J i that was immediately processed on the machine, thesetup time can be expressed by sij. The Gantt's chart for the one machine backward model with sequence dependent setup times forminimizing Fa (the model is expressed by n/lI Fa, S ij I b) is illustrated in Fig. 2. In the model the completion time of the last jobshould coincide with the due date ( d) like the previous model.The orderfor assigningjobsFig. 2. nilI Fa, S ij I b model.Fa = (11 n)I Fa/i/;=1=(1ln)I(d-Bfij) - min.i=1If J [iJ denotes the job sequence at the ith position counted from the end position on a time scale as shown in Fig. 2, and if Fa/ i /and P/ i/ denote the actual flow time and the processing time of J [i / ' there exists a relation :Backward Scheduling to Minimize the Actual Mean Flow Time with Dependent and Independent Setup Times 91whereP[jJ[j-l] = Prj] + S[jJ[j-l] (j =2,3,···,n),PflJ[oj = P fl]and S[ jll j-I/ is the setup time attached for J [j/ followed by J I j-I/·The actual mean flow time is given by :Fa=(1ln) ±±Pfjllj-lf=(1ln) ±(n-i+l)p[illi_I/i.Ij_1 i-I(3)(4)(5)Arcelus and Chandra [3] have treated a one machine forward scheduling model with sequence dependent setup times forminimizing F (the model is expressed by n I 1 I F ,sij I f). F is defined as :n nF=(1ln) "2.Fi=(lIn) "2.(Ci- O), (6)i-I j _ 1in which all the jobs are ready at time 0 , and Ci denotes the completion time of J i . The model is illustrated in Fig. 3.oFiIjFin-IjThe order for assigning jobsFig. 3. nl 1 I F,sijl f model.FinjIf J [i] denotes the job sequenced at the ith position counted from the beginning on a time scale in this· case, there exists arelation:F = (1 I nJ.f(n- i +1) P[i-Illi/,.-1whereP[i-I][iJ = p[i] + S[i-IJ[ij> (i =2,3,··· ,n)(7)(8)P[OJ[I] = Pfl]The difference between n I 1 I F ,sij I f and nlIIFa, Sij I b is that S[i-l][i] is substituted by S[i][i-l]' The argumentsabove show that nl1 I Fa, Sij I b problem can be solved by the following procedures. First, the data of setup time matrix sij shouldbe transformed into S ji in a new matrix (the data in ith row and jth column in the original matrix is exchanged to jth row and ithcolumn in a revised matrix). Secondly solve the n I 1 I F ,sij I f problem based on the revised matrix, and the obtained sequenceshould completely be reversed for the solution of nll /pa, sij I b problem.Optimal algorithmFor n / 1 / F ,sij / f problem, an optimal algorithm employing a Branch and Bound method was presented by Arcelus andChandra. (AC algorithm) [3]. AC algorithm is adopting the frontier node search procedure which tends to expand explosively thememory spaces. The algorithm must provide several complicated steps for updating the list of nodes to discard the inactive nodes andrenumber the active nodes consecutively. Although these steps are for saving the memory spaces, they cost much computational time.This paper uses the active new node search procedure for saving the memory spaces beside the simple steps of the algorithm. Theanother advantage of the procedure is that it can rapidly reach trial solutions, even if a sufficient computational time for obtaining theoptimal solution could not be provided.92 Kenji SEKOGUCHI and Shigeji MIYAZAKI. The proposed algorithm for n/I / Fa, Sij / b problem can be listed as follows:[Step 1] The given matrix of setup times (Sij) is transformed into Sji. The element of the new matrix Sji is rewritten as sij again.Make a matrix as shown in Tab. 1 in which the first row shows the processing time Pj of J j and the other rows show Pij defined byPij = Pj + sij. The minimum of each row i is denoted by r( i) (i =0, I,··· , n) and the minimum of each column j is denoted bye(j) (j=l,l,···,n). Rearrange rei) (i=l,l,"',n) and e(j) (j=l,l,· .. ,n) in the nondecreasing order and denote them'asr[I),r[2),···,r[n), and e[I),e[2),···,e[n).Compute RBND(k) ,CBND(k) and BND(k) by :kRBND(k) = i.(k-j+l)r[jJ, (k=l,l,···,n-l), (9)j_In-IRBND(n) = n· reO) + '2.(n- j)r[ jJ,j-IkCBND(k) = '2.(k - j+I)e[j), (k -l,l,···,n),j_I(10)(11)BND(k) = max{RBND(k),CBND(k)}, (k = l,l,···,n), (12)where RBND(k) ,CBND(k) and BND(k) stand for lower bounds based on the rows, the columns and the both of them in thePij matrix at the stage of k remaining (unassigned) jobs. BND(n) is used as the lower bound obtained in this stage. Go to Step 2.Tab. 1. Matrix showing Pij (the sum of proccesingtime P j and setup time S ij ).To jobJI h """ Jj " " . In minJI pI p2 ".. Pj ... pn reO)JI-PI2""" Plj """ pIn r(1)h P21 -- "". P2j " .• p2n r(2)Fromjob : : : : :Ji P~~ Pi2 """ py " "" pin rei): : : : : :I npnl pn2" ." pnj ..• - r(n)min e(1)e(2) "" cO) "" ern)[Step 2] Obtain a feasible solution by "closest unvisited city" algorithm for the traveling salesman problem, and set the performancemeasure of the solution as SOLN. If BND(n) = SOLN ,the feasible solution by "closest unvisited city" is the optimal solution, andterminate the algorithm. Otherwise, set the branching level (the number of already assigned jobs) r=O, the branching node N~ = 0,o "the upper bound UB(N,) = SOLN, the performance measure of a best known solution f = SOLN, and go to Step 3. Note thatn - k = r holds, because k is defmed as the remaining (unassigned) jobs.[Step 3] Perform the branching procedure for N~ and create ( n - r +1) kinds of new nodes. Set r = r + 1 and k = k - 1.[Step ~] For each node newly created, compute the total actual flow time of the partial sequence of already assigned jobs PAAF by :rPAAF = .'2.(n-i +I)Pfi-I/lij.._1 (13)If the sum of PAAF and BND(k) by eq. (12) is greater than or equal to f', delete this node from the active node list and applyStep 4 to another node newly created. If no other nodes are remain,ing in this step, go to Step 8. If PAAF + BND(k) < f', go toStep 5.[Step 5] Compute lower bound LB(N,) as follows: Revise the current Pij matrix shown in Tab. 1 by the following procedure. Deleteall the elements in the first row of the current Pij matrix; delete all the elements in the column corresponding to the job (J~) newlyassigned for creating N,; remove the row for J; to the top of rows and shift the other rows accordingly. Compute BND(k) by eq.(12) based on the revised Pij matrix. Set lower boundas LB(N ,) = PAAF + BND(k ) .•Compute SOLN based on the revised Pij matrix through the closest unvisited city algorithm. If the new SOLN is less than f 'set f' = SOLN. Otherwise, maintain the previous f'. If Step 5 has applied to all the nodes newly created, go to Step 6. Otherwise,repeat Step 4 for the next node remaining at the same stage.should be a new trial solution. Set f * = LB(N~), and go to Step 8. IfBackward Scheduling to Minimize the Actual Mean Flow Time with Dependent and Independent Setup Times[Step 6] Choose the node having the minimum lower bound among newly created nodes as the objective node (N~) for a newo • 0 •branching. If r = n, then go to Step 7. If r < nand LB(N r) < f ' then go to Step 3. If r < nand LB(N r» f ,then go to Step8.o • 0[Step 7] If LB(N r) < f ' then the current N r0*0LB(N r)~ f ,then delete N r and go to Step 8.[Step 8] If any active nodes having the lower bounds smaller than f* exist, then choose the node having the minimum lower boundamong the node nearest to the current node N~ . Set the node as the new objective node N~ for a branching. Reset r at the same levelas the choosen_N~. Delete all the newest nodes, and return to Step 3. If there exists no objective node for branching, then go to Step 9.[Step 9] Reverse the sequence of the current trial solution t? obtain the optimal solution for the n/l / Fa, Sij / b problem. (END)4. COMPUTATIONAL EXPERIENCEThe proposed algorithm has been coded in C-language and numerical examples were solved through 16-bit micro computer, NECPC-9801 VX, to obtain a computational experience. The average computational time to reach the optimal solution over ten exampleproblems is shown in Fig. 4, of which horizontal axis is the number of jobs included in each problem. The average computational times'through the complete enumeration mothod are estimated as 1.3 years for 14-job problem and 20.1 years for 15-job'problem. Althoughthe computational time through the proposed algorithm increases exponentially, it is less than 2 hrs. for 21-job problem, and less than3.5 hrs. for 22-job problem. This time is rather better than the computational time of other implicit enumeration methods for generalscheduling problems.NEC PC-9801 VX (16-bit micro computer)(Average ever ten example problems for each number ofjobs)93///~ // ~12500~ 10000:{.~ 75001.'"i 5000Iiiu2500o14 15 16 17 18Th. number ofjobs19 20 21 22Fig. 4. Average computational time through the proposed algorithmBy use of the active new node search procedure, the proposed algorithm can yield trial solutions before obtaining the optimalsolution. Each trial solution through the pass to the optimal solution was evaluated by the approximation ratio TJ :OJ-t 'TJ=--xlOO (%) (14)OJ - 0where (0, t and 0 are the performance measures of the worst (maximum), the trial and the optimal (minimum) solutions, respectively.The approximation ratio TJ defined by eq. (14) reaches 100% in the case that the trial solution coincides with the optimal one and 0% inthe case with the worst one. The average of TJ calculated by ten example problems vs. the ordinal number of trial solutions is shown inFig. 5, and the average of TJ vs. the required computational time is in Fig. 6. The average of TJ could attain 95% before obtaining the10th trial solution and within the average computational time of 100 sees. for the example problems including less than 23 jobs.94 Kenji SEKOGUCHI and Shigeji MIYAZAKI100 i--7-------------======::=:¢====----<>-i----<>-- IOjoo. -- 13jobs -I>-- 16joo. --<J-- 19jobs ----<>-- 22jobs93 O-O==';~::::=.-__/__+----=-----'-------=--------=-----=--------'---199 I------,I----------='""""''''--------------==-=-.....=='\'~98I_-----rC---------P<::.----~=--==''===--_o_==-<r-==:::::;~=o-'''::!iJle.~ 97.~ 96 I--,L~---___,L~=____=,....&=:::;',:_~;g::::::&;~::::::::=------_____.ji: 951-----::-:=1,~~~~~c=:::=--------------------__jt 94 I---=~ ~~---------------------_I<92 i?~_~_ _'__~_~_~~___''--'-_~~_~_~~_~_~~_~_~~_--JSOLN 1th 2nd 3,d 4th 5th 6th 7th 8th 9th 10th 11th 12th 13th 14th 15th 16th 17th 18th 19th 20thThe ordinal number of trial solutionsFig. 5.Average approximation ratio vs. the number of trial solutions.100e 99.~ 98c.~ 97E·x 96e"-"-as 95u~~ 94-<9310 100 1000Average computational time (sees.)Fig. 6. Average approximation ratio vs. average computational time.5. CONCLUDING REMARKSThe present paper proposed a new performance neasure, actual mean flow time Fa, which reflects the recent popular productioncircumstance of which typical example can be seen in just-in-time production. The optimal algorithm for n/l / Fa backwardscheduling with sequence dependent setup times was also presented. The algorithm employs a Branch and Bound method for n / 1 / Fforward scheduling problem with a modification into the active new node search procedure.The algorithm was coded in C-language and a computational experience was reported through a NEC PC-9801 VX microcompu"ter. Although the computational time increases exponentially, it succeeded to obtain the optimal solution of 22-job problemwithin one shift duration (eight hours). The trial solutions on the way to the optimal solution can be used as a practical solution, becausethey revealed more than 95% approximationretio (performance measure attained among the range between the worst and the optimalsolution) and the required computational time is within 100 secs. for the problem of which size is less than 23 jobs.REFERENCES[1]. K.R.Baker, Introduction to Sequencing and Scheduling, John Wiley, New York, (1974).[2]. R.W.Conway, W.L.Maxwell and L.W.Miller, Theory of Scheduling, Addison-Wesley, Reading, (1967), pp.27-28.[3]. FJ.Arcelus and R.Chandra, On n / 1 / F setup dependent problems, Engineering Optimization, 7 (1983), pp 58-67.

Backward Scheduling to Minimize the Actual Mean Flow Time with Dependent and Independent Setup Times

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