Computers and Mathematics with Applications

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1 Computes and Mathematics with Applications 58 (009) 9 7 Contents lists available at ScienceDiect Computes and Mathematics with Applications jounal homepage: Bi-citeia single machine scheduling poblem with a leaning effect: Aneja Nai method to obtain the set of optimal sequences V. Mani a, Pei Chann Chang b,, Shih Hsin Chen c a Depatment of Aeospace Engineeing, Indian Institute of Science, Bangaloe, India b Depatment of Infomation Management, Yuan Ze Univesity, 5, Yuan-Dong Road, Tao-Yuan 06, Taiwan ROC c Depatment of Electonic Commece Management, Nanhua Univesity,,Chungkeng, Dalin, Chiayi 68, Taiwan ROC a t i c l e i n f o a b s t a c t Aticle histoy: Received Octobe 007 Received in evised fom Novembe 008 Accepted Januay 009 Keywods: Single machine scheduling Bi-citeia poblem Leaning effect Non-dominated solution In this pape, we conside the bi-citeia single machine scheduling poblem of n jobs with a leaning effect. The two objectives consideed ae the total completion time (TC) and total absolute diffeences in completion times (TADC). The objective is to find a sequence that pefoms well with espect to both the objectives: the total completion time and the total absolute diffeences in completion times. In an ealie study, a method of solving bi-citeia tanspotation poblem is pesented. In this pape, we use the methodology of solving bi-citeia tanspotation poblem, to ou bi-citeia single machine scheduling poblem with a leaning effect, and obtain the set of optimal sequences,. Numeical examples ae pesented fo illustating the applicability and ease of undestanding. 009 Elsevie Ltd. All ights eseved.. Intoduction Duing the past fifty yeas, the single machine scheduling poblem has been studied by many eseaches. A good intoduction to sequencing and scheduling, and also pesenting vaious issues elated to single machine scheduling is []. In single machine scheduling, vaious objectives such as mean flow time, mean tadiness, maximum flow time, maximum tadiness, numbe of tady jobs, weighted mean of ealiness and tadiness ae consideed. The poblem is to find a sequence of jobs that minimizes the consideed objective. In single machine scheduling, the pocessing time of a job is assumed to be a constant. A well-known concept in management science liteatue is leaning effect fist discoveed in []. Because of this leaning effect, the pocessing time of a job is not a constant and depends on its position in the sequence. A suvey of leaning effect is given in []. In the context of single machine scheduling, the leaning effect was fist consideed in []. The objectives consideed in [] ae minimal deviation fom a common due date and minimum flow time, and these objectives ae studied sepaately. An assignment poblem fomulation is pesented fo each of these objectives, and the solution is obtained by solving the assignment poblem. The assignment poblem fomulation given in [] with the objective of due-date assignment poblem, simultaneous minimization of total completion time and vaiation of completion times is pesented in [5]. The scheduling poblem with a leaning effect on paallel identical machines is pesented in [6]. A two machine flowshop with a leaning effect, with the objective of minimizing of total completion time, is studied in [7]. In [7] thee is poposed a banch and bound algoithm to solve this poblem and a heuistic algoithm is also pesented to impove the efficiency of the banch and bound technique. Some impotant studies consideing the leaning effect ae: [8 ]. The bi-citeia single machine scheduling poblem with a leaning effect is consideed in [] and the two objectives consideed ae the total completion Coesponding autho. Tel.: x05, 509; fax: addess: iepchang@satun.yzu.edu.tw (P.C. Chang) /$ see font matte 009 Elsevie Ltd. All ights eseved. doi:0.06/j.camwa

2 0 V. Mani et al. / Computes and Mathematics with Applications 58 (009) 9 7 time and the maximum tadiness. A banch and bound technique is pesented fo the solution and a heuistic algoithm is poposed to seach fo optimal and nea optimal solutions. In this pape, we include the leaning effect as given in []. The pocessing time of a job depends on its position in the sequence and is given as [] p jl = p j l α. In the above equation, p j is the nomal pocessing time of job j, and p jl is the pocessing time of job j if it is in position l of the sequence, and α is the leaning index and α < 0. Fom the above Eq. (), we see that p j > p j > p j... > p jn. Fo example, if p j = and α = 0.55, then p j =, p j =.099, p j =.707, p j =.69, p j5 =.095, and so on. Contibutions of this pape: In this pape, we conside the bi-citeia single machine scheduling poblem of n jobs with a leaning effect. The two objectives consideed ae the total completion time (TC) and total absolute diffeences in completion times (TADC). The objective is to find a sequence that pefoms well with espect to both the objectives: the total completion time and the total absolute diffeences in completion times. A method of solving the bi-citeia tanspotation poblem is pesented in []. In ou study, we use the method of solving bi-citeia tanspotation poblem [], to ou bi-citeia single machine scheduling poblem with a leaning effect, and obtain the set of optimal sequences. This bi-citeia poblem was consideed in an ealie study [5] without the leaning effect. A paametic analysis was pesented to obtain the minimum set of optimal sequences in [5]. In that study, fist the complete set of optimal solutions (CSOS) is obtained and then, fom this CSOS, the minimum set of optimal solutions (MSOS) ae obtained. We also pesent a discussion on the paametic analysis given in [5]. A pseudo-polynomial time dynamic pogamming algoithm is pesented in [6] to solve this biciteia poblem. We also show that, in ou methodology, we can diectly obtain the minimum set of optimal solutions (MSOS). (). Aneja Nai method of solution In this section, we outline Aneja Nai method [], of solving the bi-citeia tanspotation poblem. The algoithm pesented in [], consists of solving the same tanspotation poblem epeatedly but with diffeent objectives. In each iteation, a new efficient exteme point o the change of diection of seach in the objective function is obtained. This algoithm teminates when thee is no moe efficient exteme point. m Minimize z = c ij x ij Minimize z = i= m i= j= d ij x ij j= () subject to the constaints x ij = a i, i =,,..., m whee j= m x ij = b j, i= x ij 0, j =,,..., n foall (i, j), c ij is the cost of tanspoting a unit fom souce i to destination j, d ij is the deteioation of a unit while tanspoting fom i to j, a i is the availability at i, b i is the equiement at j, x ij is the amount tanspoted fom i to j. () The algoithm pesented in [] stats with two points in the objective space. These points ae z () and z (). Points z () and z () ae obtained by minimizing the fist objective and second objective espectively. The value of fist and second objective function at point ae denoted as z (), z(). The value of fist and second objective function at point ae denoted as z(), z (). Using the values of z(), z(), z(), and z(), two new vaiables namely a(,), and a (,) ae detemined. The value of a (,) = z () z (), and a(,) = z () z (). The new tanspotation poblem is fomulated as { } Minimize a (,) c ij + a (,) d ij x ij. () i,j

3 V. Mani et al. / Computes and Mathematics with Applications 58 (009) 9 7 Z () Z, () Z a (,) Z (), Z (), a (,) Z (), Z () Z Fig.. Pictoial explanation of objective space. The solution of the above tanspotation poblem gives a new efficient exteme point o change of diection of seach. If m n thee ae altenate optimal solutions to this poblem, choose an optimal solution fo which i= j= c ijx ij is a minimum. Let the new exteme point obtained be point. The value of fist and second objective function at point ae denoted as z (), z(). This pocess is epeated by consideing points and and also by consideing points and. If a new efficient exteme point is found fom points and, the pocess continues with the new point along with points and. Similaly, if a new efficient exteme point is found fom points and, the pocess continues with the new point along with points and. This algoithm teminates when thee is no moe efficient exteme point. The poof of validity of this algoithm is given in []. Pictoial explanation: In ode to undestand the algoithm given in [], we give below a pictoial epesentation. In that study [], the objective space is consideed instead of the decision space fo finding the exteme points on the nondominated set. The objective space fo the bi-citeia tanspotation poblem can be thought of as an x y plane, in which x-axis epesents the fist objective function value (z ) and y-axis epesents the fist objective function value (z ). Any point in this plane gives the value of fist objective function (z ) and the value of second objective function (z ) fo a given tanspotation poblem. This is shown in Fig.. In this Fig., the points z (), z() and z (), z() ae shown. Also, the value of a (,), a (,) and the thid point z (), z() obtained by solving () ae shown. This pocess is epeated by consideing points and and also by consideing points and.. Poblem fomulation and peliminay analysis We conside the single machine scheduling poblem with a leaning effect. A set of n independent jobs is to be pocessed on a continuously available single machine. The machine can pocess only one job at a time and job peemption and inseting idle times ae not pemitted. Each job has a nomal pocessing time p j, (j =,,..., n) if it is at the fist position in the sequence. The sequence is the ode in which the jobs ae pocessed on the machine. The jobs ae numbeed accoding to shotest nomal pocessing time ule, i.e., p p p n. The two objectives consideed ae the total completion time (TC) and total absolute diffeences in completion times (TADC). TC and TADC fo a given sequence σ ae TC = j= TADC = C j i= C i C j j=i whee C j is the completion time of job j in the given sequence. The single machine scheduling poblem with a leaning effect, is to find the sequence of jobs (σ ) that simultaneously minimizes TC and TADC ae shown in Eqs. (6) and (7), espectively. f TC (σ ) = (n + ) α p [] = = = w,α p [] (6) (5)

4 V. Mani et al. / Computes and Mathematics with Applications 58 (009) 9 7 Table Matching algoithm fo minimization of TC: α = 0.5. Position- w,α Sequence Table Matching algoithm fo minimization of TADC: α = 0.5. Position- w,α Sequence and f TADC (σ ) = ( )(n + ) α p [] = = = w,α p []. (7) In the above equations, p [] is the nomal pocessing time of job in position. The quantities (n + ) α and ( )(n + ) α in the above equations, ae known as positional weights. We can also see if α = 0 Eqs. (6) and (7) educes to the poblem pesented in [5]. The optimal sequence fo any one of the objectives can be obtained by using a matching algoithm [5,7]. Numeical example: We now give a small numeical example to show how the matching algoithm is used to obtain the optimal schedule. Conside the job instance of the poblem given in [5]. The nomal pocessing time of these jobs ae p =, p =, p =, and p =. Let the value of leaning ate α = 0.5. Conside only the fist objective i.e., TC: The poblem is to find the sequence of jobs (σ ) that minimizes f TC (σ ) = (n + ) α p [] = f TC (σ ) = α p [] + α p [] α p [] + α p []. (9) The optimal sequence obtained fo the above poblem using the matching algoithm is { }. The positional weights obtained fom Eq. (6) ae given in Table. The details of obtaining the optimal sequence using the matching algoithm is given in Appendix. The value of TC obtained fo this sequence { } is The value of TADC obtained fo this sequence { } is Conside only the second objective; i.e., TADC: The poblem is to find the sequence of jobs (σ ) that minimizes f TADC (σ ) = ( )(n + ) α p [] = f TADC (σ ) = α p [] + α p [] + α p []. () The optimal sequence obtained fo the above poblem using the matching algoithm is { }. The positional weights obtained fom Eq. (0) ae given in Table. The details of obtaining the optimal sequence using the matching algoithm is given in Appendix. The value of TADC obtained fo this sequence { } is The value of TC obtained fo this sequence { } is 5.5. Ou objective is to find the sequence of jobs that pefoms well with espect to both the objectives TC and TADC. In the next section, we descibe the methodology of obtaining the set of sequences that pefoms well with espect to both the objectives TC and TADC, using the method given in []. (8) (0). Aneja Nai method to bi-citeia single machine scheduling Ou inteest lies in finding a sequence that minimizes both TC and TADC. Now, we will show how the bi-citeia poblem in single machine scheduling with a leaning effect can be solved to obtain the minimum set of optimal schedules using Aneja Nai algoithm []. Fist, we will show the one to one coespondence of vaiables between bi-citeia tanspotation poblem and bi-citeia single machine scheduling poblem with a leaning effect. The one to one coespondence between the two poblems is given in Table. Once this is known, then we can apply the Aneja Nai method to obtain the minimum set of optimal sequences. The algoithm [] fo ou poblem is that of solving the single-machine scheduling poblem (with a leaning effect) epeatedly but with diffeent objectives. In each iteation, we obtain an optimal sequence (efficient point). In this manne, at any iteation, we ae solving the single machine scheduling (with a leaning effect) poblem with a single (but diffeent) objective function, using the matching algoithm given in [7]. We pesent a numeical example fo illustation.

5 V. Mani et al. / Computes and Mathematics with Applications 58 (009) 9 7 Table One to one coespondence between the two poblems. Bi-citeia tanspotation poblem z z c ij d ij Exteme point Bi-citeia single machine scheduling poblem TC fist objective TADC second objective (n + ) α = w,α ( )(n + ) α = w,α optimal sequence Table Matching algoithm fo minimization of ():α = 0.5. Position- w,α Sequence Table 5 Matching algoithm fo minimization of ():α = 0.5. Position- w,α Sequence Numeical example: We now use the same job poblem instance given in [5], with a leaning effect (α = 0.5). We show how to apply Aneja and Nai method [] and obtain the minimum set of optimal schedules. The nomal pocessing time of these jobs ae p =, p =, p =, and p =. The value of leaning ate α = 0.5. Following Aneja and Nai [], we fist obtain point in the objective space. This is obtained by minimizing z the fist objective; i.e., the total completion time (TC). The positional weights and the optimal sequence obtained ae shown in Table. The optimal sequence obtained is { }. The value of z fo this sequence is The value of z the second objective; i.e., the total absolute diffeences in completion times (TADC), is Hence, z () = 7.77, and z () = We now obtain point in the objective space. This is obtained by minimizing z the second objective; i.e., the total absolute diffeences in completion times (TADC). The positional weights and the optimal sequence obtained ae shown in Table. The optimal sequence obtained is { }. The values of z and z fo this sequence ({ }) ae 5.5 and espectively. Hence, z () = 5.5, and z () = As mentioned in [], we now use the points and, and obtain the values of a (,) = z () z () = 9.997, and a (,) = z () z () = The new single machine scheduling poblem is fomulated as to find the sequence of jobs (σ ) that minimizes f (σ ) = = a (,) (n + ) α p [] + = a (,) ( )(n + ) α p []. () The positional weights ae w c,α = a (,) (n + ) α p [] + a (,) ( )(n + ) α p []. The positional weights (w c,α ) fo the above combined poblem ae: w c,α = , w c,α = 5.9, w c,α =.9887, and w c,α = The positional weights and the optimal sequence obtained ae shown in Table. The optimal sequence obtained with these weights is { }. The values of z and z fo this sequence ({ }) ae.9 and espectively. We call this point in the objective space and z () =.9, and z () = We now use the points and, and obtain the values of a (,) = z () z () = 6.089, and a(,) = z () z () = The new single machine scheduling poblem is fomulated as to find the sequence of jobs (σ ) that minimizes f (σ ) = = a (,) (n + ) α p [] + = a (,) ( )(n + ) α p []. () The positional weights ae w c,α = a (,) (n + ) α p [] + a (,) ( )(n + ) α p []. The positional weights (w c,α ) fo the above combined poblem ae: w c,α =.96, w c,α = 6.698, w c,α =.509, and w c,α =.695. The positional weights and the optimal sequence obtained ae shown in Table 5. The optimal sequence obtained with these weights is { }. The values of z and z fo this sequence ({ }) ae 8.7 and.576 espectively. We call this point in the objective space and z () = 8.7, and z () =.576. We now use points and, and obtain the values of a (,) = z () z () =.8, and a(,) = z () z () =.975. The new single machine scheduling poblem is fomulated so as to find the sequence of jobs (σ ) that minimizes f (σ ) = = a (,) (n + ) α p [] + = a (,) ( )(n + ) α p []. ()

6 V. Mani et al. / Computes and Mathematics with Applications 58 (009) 9 7 Table 6 Matching algoithm fo minimization of ():α = 0.5. Position- w,α Sequence The positional weights ae w c,α = a (,) (n + ) α p [] + a (,) ( )(n + ) α p []. The positional weights (w c,α ) fo the above combined poblem ae: w c,α =.59, w c,α = 9.7, w c,α = 9.79, and w c,α =.70. The positional weights and the optimal sequence obtained ae shown in Table 6. The optimal sequence is obtained with these weights is { }. The values of z and z fo this sequence ({ }) ae.59 and espectively. We call this point 5 in the objective space and z (5) =.59, and z (5) = When we use the points and, we obtain the same optimal sequence { }. When, we use the points and, we obtain the same optimal sequence obtained is { }. When we use the points and 5, we obtain the same optimal sequence { }. When we use the points and 5, we obtain the same optimal sequence { }. Thee ae no othe optimal sequences and so the algoithm teminates. Based on the above, the minimum set of optimal sequences to this poblem is: { }, { }, { }, { }, and { }. It is shown in [5] that the cadinality of the set MSOS is n, when the leaning effect is not consideed; i.e., α = 0. We can see fom this example, that the cadinality of the set MSOS is not n, when the leaning effect is included; i.e., α Discussions This bi-citeia single machine scheduling poblem without a leaning effect (α = 0) is studied in Bagchi [5]. In that study, a paametic analysis is caied out. In the paametic analysis, the objective is to find the sequence (σ ) that minimizes f (σ ) = δ TC + ( δ) TADC (5) f (σ ) = δ = = (n + )p [] + ( δ) = ( )(n + )p []. (6) = The value of δ is esticted to the open inteval (0,). Fo any given value of δ, the optimal sequence can be obtained by using the matching pocedue. In that study [5], two sets of sequences namely Complete Set of Optimal Schedules (CSOS), and Minimum Set of Optimal Schedules (MSOS), ae defined. The set CSOS has the popety that fo any value of δ fom (0,), all optimal solutions to (5) ae membes of this set. The set MSOS has the popety that fo any value of δ fom (0,), at least one and almost of the optimal solutions to (5) ae membes of this set. So MSOS is a subset of CSOS. It is shown in [5] that thee ae (n ) distinct values of δ given by δ i = n i n i + i =,,..., (n ). A fast algoithm fo obtaining the CSOS is given in [5]. A dynamic pogamming appoach is pesented in [6]. In that study, a condition on δ i is given by consideing thee adjacent exteme points. The minimal set of optimal sequences ae called δ-efficient sequences in [6]. It was shown in Bagchi [5] that the cadinality of the set MSOS is n, and an O(n ) algoithm is pesented to obtain the set MSOS. We pesent a numeical example to show the sets CSOS and MSOS. Numeical example: We now use the same job poblem given in Bagchi [5], without a leaning effect (α = 0). The nomal pocessing time of these jobs ae p =, p =, p =, and p =. Fo this job bi-citeia poblem, thee ae distinct values of δ. They ae: δ = 0.5, δ = /, and δ = We conside the value of δ = 0.5. The objective is to find the sequence (σ ) that minimizes f (σ ) = 0.5 TC TADC (8) f (σ ) = 0.5 = = (n + )p [] = (7) ( )(n + )p []. (9) = The positional weights (w c ) fo the above combined poblem ae: wc =.0000, wc =.0000, wc =.0000, and w c = We see that the positional weights wc = wc, and wc = wc. Because of this, we obtain fou sequences that ae optimal. The fou optimal sequences ae: { }, { },{ }, and { }. Fo all these fou sequences, the value of δ n = TC + ( δ) n = TADC is the same and the value is. We conside the value of δ = /. We conduct a simila analysis as done above. We obtain two sequences that ae optimal. The two optimal sequences ae: { } and { }. Fo the two sequences, the value of δ n = TC + ( δ) TADC is the same and the value is. n =

7 V. Mani et al. / Computes and Mathematics with Applications 58 (009) We conside the value of δ = /. We conduct a simila analysis as done above. We obtain two sequences that ae optimal. The two optimal sequences ae: { } and { }. Fo the two sequences, the value of δ n = TC + ( δ) n = TADC is the same and the value is.50. The complete set of optimal schedules (CSOS) and the minimum set of optimal schedules (MSOS) ae: CSOS : {{ }, { }, { }, { }, { }, { }}. MSOS : {{ }, { }, { }, { }}. Aneja Nai method: We now show how the Aneja Nai method obtains the MSOS fo the above example, without a leaning effect (α = 0). When α = 0, we use the key idea given in []. The key idea is that when thee is moe than one optimal solution, choose an optimal solution fo which z is minimum. Following Aneja and Nai [], we fist obtain point in the objective space. This is obtained by minimizing z the fist objective; i.e., the total completion time (TC). The positional weights and the optimal sequence obtained is { }. The value of z fo this sequence is The value of z the second objective; i.e., the total absolute diffeences in completion times (TADC), is Hence, z () = , and z () = We now obtain point in the objective space. This is obtained by minimizing z the second objective; i.e., the total absolute diffeences in completion times (TADC). The positional weights ae: w = , w =.0000, w =.0000, and w = We see that the positional weights w = w. Because of this, we obtain two sequences that ae optimal. The two optimal sequences ae: { } and { }. It is given in Aneja and Nai [] that when thee ae moe than one optimal solution, choose an optimal solution fo which z is minimum. The value of z fo the sequence { } is 7, and the value of z fo the sequence { } is 9. Hence, we choose the sequence { }. The values of z and z fo this sequence ({ }) ae and espectively. Hence, z () = , and z () = As mentioned in [], we now use the points and, and obtain the values of a (,) = z () z () =.0000, and a (,) = z () z () = The new single machine scheduling poblem is fomulated so as to find the sequence of jobs (σ ) that minimizes f (σ ) = = a (,) (n + )p [] + = a (,) ( )(n + )p []. (0) The positional weights (w c ) fo the above combined poblem ae: wc =.0000, wc = , wc = , and w c = The optimal sequence obtained with these weights is { }. The values of z and z fo this sequence ({ }) ae.0000 and.0000, espectively. We call this point in the objective space and z () =.0000, and z () = We now use the points and, and obtain the values of a (,) = z () z () = , and a(,) = z () z () = The new single machine scheduling poblem is fomulated so as to find the sequence of jobs (σ ) that minimizes f (σ ) = a (,) (n + )p [] + a (,) ( )(n + )p []. () = = The positional weights (w c ) fo the above combined poblem ae: wc = , wc = , wc = , and w c = The optimal sequence obtained with these weights is { }. The values of z and z fo this sequence ({ }) ae.0000 and espectively. We call this point in the objective space and z () =.0000, and z () = We now use the points and, and obtain the values of a (,) = z () z () =.0000, and a(,) = z () z () = The new single machine scheduling poblem is fomulated so as to find the sequence of jobs (σ ) that minimizes f (σ ) = a (,) (n + )p [] + a (,) ( )(n + )p []. () = = The positional weights (w c ) fo the above combined poblem ae: wc = , wc =.0000, wc =.0000, and w c = We see that the positional weights wc = wc, and wc = wc. Because of this, we obtain fou sequences that ae optimal. The fou optimal sequences ae: { }, { }, { }, and { }. Of these fou sequences, the sequence with a minimum value of z is chosen. The sequence with a minimum z is the sequence { }. In this manne, we obtain the minimum set of optimal schedules (MSOS) and the sequences ae: MSOS : {{ }, { }, { }, { }}. It is shown by [5] that thee ae (n ) distinct values of δ given by the Eq. (7). Using these distinct values, it is shown in [5] that the cadinality of the set MSOS is n, when the leaning effect is not consideed; i.e., α = 0. When the leaning effect is included i.e., α 0, the Eq. (7) is not tue and hence the cadinality of the set MSOS is not n. This fact is shown in this example. The effect of leaning in scheduling poblems has attacted many eseaches and some of the ecent studies ae given in [8 ].

8 6 V. Mani et al. / Computes and Mathematics with Applications 58 (009) Conclusions In a manufactuing system, wokes ae involved in doing the same job o activity epeatedly. Because of epeating the same job o activity, the wokes stat leaning moe about the job o activity. Because of the leaning, the time to complete the job o activity stats deceasing, which is known as leaning effect. Because of this leaning effect, the pocessing time of a job is not a constant and depends on its position in the sequence. In this pape, we consideed the bi-citeia single machine scheduling poblem of n jobs with a leaning effect. The two objectives consideed ae the total completion time (TC) and total absolute diffeences in completion times (TADC). We used Aneja Nai [] method to find a sequence that pefoms well with espect to both the objectives: the total completion time and the total absolute diffeences in completion times. In an ealie study [5] this bi-citeia poblem is consideed without the leaning effect, and a paametic analysis is pesented to obtain the minimum set of optimal sequences. In this study, we used the Aneja Nai method [] of solving biciteia tanspotation poblem, to obtain the set of optimal sequences, to ou bi-citeia single machine scheduling poblem with a leaning effect. A discussion is pesented on the paametic analysis given in [5], and the methodology used in ou study. We have also pesented numeical examples fo ease of undestanding. Acknowledgments The authos would like to thank one of the eviewes fo his/he suggestions which wee useful. The authos would like to thank Edito-in-Chief Pof. Rodin fo his encouagement. Appendix Numeical example: We now give a small numeical example to show how the matching algoithm is used to obtain the optimal sequence. Conside the job poblem given in [5]. The nomal pocessing time of these jobs ae p =, p =, p =, and p =. Let the value of leaning ate α = 0.5. We fist conside only the fist objective i.e.,tc. The poblem is to find the sequence of jobs (σ ) that minimizes f TC (σ ) = (n + ) α p [] = f TC (σ ) = α p [] + α p [] α p [] + α p []. () The positional weights obtained fom Eq. (6) ae w,α =.0000, w,α =.7000, w,α =.69, and w,α = In the matching algoithm, the optimal sequence is obtained by matching the positional weights in descending ode with jobs in ascending ode of thei nomal pocessing times. The positional weights and the optimal sequence obtained ae shown in Table. The optimal sequence obtained is { }, when we conside only the fist (TC) objective. The value of TC fo this sequence { } is obtained as follows: z = = w,α p []. In the above equation p [] is the nomal pocessing time of job in position. We have job in position, job in position, job in position, and job in position. Hence, z is z = w,α p + w,α p + w,α p + w,α p z = = (6) The value of TADC fo this sequence { } is obtained as follows: z = = w,α p []. We have job in position, job in position, job in position, and job in position. Hence, z is z = w,α p + w,α p + w,α p + w,α p z = = (8) We now conside only the second objective; i.e.,tadc. The poblem is to find the sequence of jobs (σ ) that minimizes f TADC (σ ) = ( )(n + ) α p [] = f TADC (σ ) = α p [] + α p [] + α p []. (0) () (5) (7) (9)

9 V. Mani et al. / Computes and Mathematics with Applications 58 (009) The positional weights obtained fom Eq. (7) ae w,α = , w,α =.7000, w,α =.88, and w,α =.00. In the matching algoithm, the optimal sequence is obtained by matching the positional weights in descending ode with jobs in ascending ode of thei nomal pocessing times. The positional weights and the optimal sequence obtained ae shown in Table. The optimal sequence obtained is { }, when we conside only the second (TADC) objective. The value of TC fo this sequence { } is obtained as follows: z = = w,α p []. We have job in position, job in position, job in position, and job in position. Hence, z is z = w,α p + w,α p + w,α p + w,α p z = = 5.5. () The value of TADC fo this sequence { } is obtained as follows: z = = w,α p []. We have job in position, job in position, job in position, and job in position. Hence, z is z = w,α p + w,α p + w,α p + w,α p z = = () () () Refeences [] K.R. Bake, Intoduction to Sequencing and Scheduling, John Wiley & Sons, New Yok, 97. [] T.P. Wight, Factos affecting the cost of aiplanes, Jounal of Aeonautical Science (96) 8. [] L.E. Yelle, The leaning cuve: Histoical eview and compehensive suvey, Decision Science 0 (979) 0 8. [] D. Biskup, Single-machine scheduling with leaning consideations, Euopean Jounal of Opeational Reseach 5 (999) [5] G. Mosheiov, Scheduling poblems with a leaning effect, Euopean Jounal of Opeational Reseach (00) [6] G. Mosheiov, Paallel machine scheduling with a leaning effect, Jounal of the Opeational Reseach Society 5 (00) [7] W.C. Lee, C.C. Wu, Minimizing total completion time in a two-machine flowshop with a leaning effect, Intenational Jounal of Poduction Economics 88 (00) [8] D. Biskup, D. Simons, Common due date scheduling with autonomous and induced leaning, Euopean Jounal of Opeational Reseach 59 (00) [9] W.H. Kuo, D.L. Yang, Minimizing the makespan in a single machine scheduling poblem with a time based leaning effect, Infomation Pocessing Lettes 97 (006) [0] G. Mosheiov, J.B. Sidney, Note on scheduling with geneal leaning cuves to minimize the numbe of tady jobs, Jounal of the Opeational Reseach Society 56 (005) 0. [] A. Bachman, A. Janiak, Scheduling jobs with position-dependent pocessing times, Jounal of the Opeational Reseach Society 55 (00) [] J.B. Wang, Z.Q. Xia, Flowshop scheduling with a leaning effect, Jounal of the Opeational Reseach Society 56 (005) 5 0. [] W.C. Lee, C.C. Wu, H.J. Sung, A bi-citeia single-machine scheduling poblem with leaning consideations, Acta Infomatica 0 (00) 0 5. [] Y.P. Aneja, K.P.K. Nai, Biciteion tanspotation poblem, Management Science 5 (979) [5] U. Bagchi, Simultaneous minimization of mean and vaiation of flow time and waiting time in single machine systems, Opeations Reseach 7 (989) 8 5. [6] P. De, J.B. Ghosh, C.E. Wells, On the minimization of completion time vaiance with a bi-citeia extension, Opeations Reseach 0 (99) [7] S.S. Panwalka, M.L. Smith, A. Seidmann, Common due-date assignment to minimize total penalty fo the one machine scheduling poblem, Opeations Reseach 0 (98) [8] D. Biskup, A state-of-the-at eview on scheduling with leaning effects, Euopean Jounal of Opeational Reseach 88 (008) 5 9. [9] A. Janiak, R. Rudek, The leaning effect: Getting to the coe of the poblem, Infomation Pocessing Lettes 0 (007) [0] A. Janiak, R. Rudek, A new appoach to the leaning effect: Beyond the leaning cuve estictions, Computes & Opeations Reseach 5 (008) [] C. Koulamas, G.J. Kypaisis, Single-machine and two-machine flow-shop scheduling with geneal leaning functions, Euopean Jounal of Opeational Reseach 78 (007) [] J.B. Wang, C.T.D. Ng, T.C.E. Cheng, L.L. Liu, Single-machine scheduling with a time-dependent leaning effect, Intenational Jounal of Poduction Economics (008) [] C.C. Wu, W.C. Lee, Single-machine scheduling poblems with a leaning effect, Applied Mathematical Modelling (008) 9 97.