\$title LogMIP User's Manual Example 1c - Job Scheduling (LOGMIP1C,SEQ=334) \$onText Three jobs (A,B,C) must be executed sequentially in three steps, but not all jobs require all the stages. The objective is to obtain the sequence of tasks which minimizes the completion time. Once a job has started it cannot be interrupted. The objective is to obtain the sequence of task, which minimizes the completion time. In this model we use a precedence formulation. Ref: Raman & Grossmann, Comp. & Chem. Eng., 18, 7, p.563-578, 1994. Aldo Vecchietti, LogMIP User's Manual, 2007, http://www.logmip.ceride.gov.ar/files/pdfs/logmip_manual.pdf Keywords: extended mathematical programming, disjunctive programming, job scheduling, execution sequence \$offText Set j 'jobs' / A, B, C / s 'stages' / 1*3 /; Alias (j,jj), (s,ss); Set less(j,jj) 'upper triangle'; Table p(j,s) 'processing time' 1 2 3 A 5 3 B 3 2 C 2 4 ; Parameter c(j,s) 'stage completion time' w(j,jj) 'pairwise waiting time' pt(j) 'total processing time'; less(j,jj) = ord(j) < ord(jj); c(j,s) = sum(ss\$(ord(ss) <= ord(s)), p(j,ss)); w(j,jj) = smax(s, c(j,s) - c(jj,s-1)); pt(j) = sum(s, p(j,s)); display c, w, pt; Variable t 'completion time' x(j) 'job starting time' pr(j,jj) 'job precedence'; Positive Variable x; Binary Variable pr; Equation comp(j) 'job completion time' seq(j,jj) 'job sequencing j beore jj' dummy 'force names into model'; comp(j).. t =g= x(j) + pt(j); seq(j,jj)\$(ord(j) <> ord(jj)).. x(j) + w(j,jj) =l= x(jj); dummy.. sum(less(j,jj), pr(j,jj)) =g= 0; x.up(j) = 1000; Model m / all /; * by default the convex hull formulation is used File fx /"%lm.info%"/; put fx 'disjunction'; loop(less(j,jj), put / pr(j,jj) seq(j,jj) 'else' seq(jj,j);); putClose; option emp = logmip; solve m using emp minimizing t;