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Line balancing Description A product is produced on an assembly lines with four
workstations. It is assembled in twelve operations between
which there are certain precedence constraints. The duration
of every task is given. We would like to distribute the
tasks among the workstations in order to balance the line to
obtain the shortest possible cycle time, that is, the total
time required for assembling the product. Every task needs
to be assigned to a single workstation that has to process
it without interruption. Every workstation deals with a
single operation at a time. Further explanation of this example:
'Applications of optimization with Xpress-MP',
Section 7.6 'Assembly line balancing' (b6linebal.mos)
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Data Files linebal_graph.mos (!****************************************************** Mosel Example Problems ====================== file linebal.mos ```````````````` TYPE: Line balancing problem DIFFICULTY: 2 FEATURES: MIP problem, encoding of arcs, `range', formulation of sequencing constraints DESCRIPTION: A product is produced on an assembly lines with four workstations. It is assembled in twelve operations between which there are certain precedence constraints. The duration of every task is given. We would like to distribute the tasks among the workstations in order to balance the line to obtain the shortest possible cycle time, that is, the total time required for assembling the product. Every task needs to be assigned to a single workstation that has to process it without interruption. Every workstation deals with a single operation at a time. FURTHER INFO: `Applications of optimization with Xpress-MP', Section 7.6 `Assembly line balancing' (c) 2008 Fair Isaac Corporation author: S. Heipcke, 2002, rev. Sep. 2017 *******************************************************!) model "Assembly line balancing" uses "mmxprs", "mmsvg" declarations MACH=1..4 ! Set of workstations TASKS=1..12 ! Set of tasks DUR: array(TASKS) of integer ! Durations of tasks ARC: array(RA:range, 1..2) of integer ! Precedence relations between tasks process: array(TASKS,MACH) of mpvar ! 1 if task on machine, 0 otherwise cycle: mpvar ! Duration of a production cycle end-declarations initializations from 'linebal.dat' DUR ARC end-initializations ! One workstation per task forall(i in TASKS) OneMachTask(i):= sum(m in MACH) process(i,m) = 1 ! Sequence of tasks forall(a in RA) Prec(a):= sum(m in MACH) m*process(ARC(a,1),m) <= sum(m in MACH) m*process(ARC(a,2),m) ! Cycle time forall(m in MACH) Cycle(m):= sum(i in TASKS) DUR(i)*process(i,m) <= cycle forall(i in TASKS, m in MACH) process(i,m) is_binary ! Minimize the duration of a production cycle minimize(cycle) ! Solution printing writeln("Minimum cycle time: ", getobjval) forall(m in MACH) do write("Workstation ", m, ":") forall(i in TASKS) write(if(getsol(sum(k in MACH) k*process(i,k)) = m, " "+i, "")) writeln(" (duration: ", getsol(sum(i in TASKS) DUR(i)*process(i,m)),")") end-do ! Solution drawing YFAC:=2 svgsetgraphviewbox(0,0,round(getobjval)+5, MACH.size*YFAC+2) svgsetgraphlabels("Time","Workstations") forall(m in MACH) do cum:=0.0 forall(i in TASKS | getsol(sum(k in MACH) k*process(i,k)) = m ) do svgaddgroup("T"+i, "Task "+i) svgaddrectangle(cum,m*YFAC,DUR(i),0.5) cum+=DUR(i) end-do end-do svgsave("linebal.svg") svgrefresh svgwaitclose("Close browser window to terminate model execution.", 1) end-model | |||||||||||

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