LP-Model for Job Shop Problems
Transcript
LP-Model for Job Shop Problems
LP-Model for Job Shop Problems Günther Füllerer June 23, 2005 LP-Model: Variables Given Data: n . . . number of jobs m . . . number of machines Di . . . due date of job i pij . . . processing time of job i on machine j seqij . . . position in sequence of job i on machine j Decision Variables: Cmax = max{Ci } Ci . . . completion time of job i Ti . . . tardiness of job i ( 1 if Ti > 0 δi = 0 otherwise startij . . . start of job i on machine j ( 1 if job i follows job j on machine k k yij = 0 otherwise LP-Model: Objective n n n i n i n Objective 1: min Cmax · M + ∑ Ti · 1/M + ∑ δi · 1/M + ∑ Ci · 1/M i n Objective 2: min Cmax · 1/M + ∑ Ti · M + ∑ δi · 1/M + ∑ Ci · 1/M i n i n i n Objective 3: min Cmax · 1/M + ∑ Ti · 1/M + ∑ δi · M + ∑ Ci · 1/M i n i n i i i n Objective 4: min Cmax · 1/M + ∑ Ti · 1/M + ∑ δi · 1/M + ∑ Ci · M Cmax ≥ starti,seqi,m + pi,seqi,m ∀i = 1, . . . , n Ci ≥ starti,seqi,m + pi,seqi,m ∀i = 1, . . . , n i LP-Model: Constraints Precedence constraints starti,seqi,j +1 ≥ starti,seqi,j + pi,seqi,j i = 1, . . . , n j = 1, . . . , m − 1 Disjunction Constraints startik + pik ≤ startjk + M · yijk startjk + pjk ≤ startik + M k = 1, . . . , m · (1 − yijk ) i, j = 1, . . . , n ∧ i < j k = 1, . . . , m i, j = 1, . . . , n ∧ i < j Bounds on Tardiness Ti ≥ starti,seqi,m + pi,seqi,m − Di i = 1, . . . , n Ti ≤ δi · M δi ≤ Ti · M not needed Non-negativity Constraints Ti ≥ 0 i = 1, . . . , n Ci ≥ 0 i = 1, . . . , n Ti ≥ 0 i = 1, . . . , n startij ≥ 0 i = 1, . . . , n j = 1, . . . , m M4 M3 M2 M1 0 4 J1 8 J2 12 J3 16 J4 20 J5 24 J6 28 32 36 40 44 48 52 56 44 48 52 56 Figure: Objective Cmax M4 M3 M2 M1 0 4 J1 8 J2 12 J3 16 J4 20 J5 24 J6 28 32 36 40 Figure: Objective Total Tardiness M4 M3 M2 M1 0 5 J1 10 J2 15 J3 20 J4 25 J5 30 J6 35 40 45 50 55 60 65 70 52 56 Figure: Objective Number of Tardy Jobs M4 M3 M2 M1 0 4 J1 8 J2 12 J3 16 J4 20 J5 24 J6 28 32 36 40 44 Figure: Objective Total Flowtime 48