WegivetheproblemdescriptionbasingonthemodeldevelopedinKatsetal.(2007).The
product(parttype)processingtimeatanymachineisnotfixed,butdefinedbyapairof
minimumandmaximumtimelimits,calledthetimewindowconstraints.Themovements
ofpartsbetweenthemachinesandloading/unloadingstationsareperformedbyarobot,
whichtravelsinanon-negligibletime.Tomoveapart,therobotfirsttravelstothestation
wherethepartislocated,waitifthepartisstillinprocess,unloadthepartandthentravels
tothenextstationspecifiedbyagivensequenceofmaterialhandlingoperationsforthe
robot. The robot is supplied by multiple grippers in order to transport several parts
simultaneouslytoanassemblingmachineorfromandisassemblingmachine.Thereisno
bufferavailablebetweenthemachinesandeachmachinecanprocessonlyoneproductat
time. If different types of products are processed at the same machine, then a non-
negligible setup time between the processing of these products may be required. The
generalproblemistodeterminetheproductsequenceateachmachine,therobotrouteand
the exact processing time of each product at each machine so that the cycle time is
minimized while the time windows, the setup times, and the robot traveling time
constraints are satisfied. Scheduling of the material handling operations of robots to
minimizethecycletime,evenwithasinglepartperMPSandasingleone-gripperrobot,
has been known to be NP-hard in strong sense (Livshits et al. (1974); Lei and Wang
(1989)).
In this chapter, we are interested in a special case of the cyclic scheduling problem
encountered in such a processing network. In particular, we solve the multiple-product
problemofminimizingthecycletimeforaprocessingnetworkwithasinglemulti-gripper
robot, a fixed and known in advance sequence of material handling operations for the
robotto beperformedin eachcycle andtheknown productsequenceat eachmachine.
Throughout the remaining analysis of this chapter, we shall denote this problem as Q.
ProblemQisafurtherextensionoftheschedulingproblemPintroducedandsolvedin
Katsetal.(2007).TheproblemPisthejobshopschedulingproblemwheretechnological
operationsforeachproductarelinkedbysimplechain-likeprecedencerelations(seeFig.
5above).LikeinP,inproblemQthesequenceofrobotmovesisassumedtobefixedand
known.Withthisspecialcase,thesequencingissuefortherobotmovesvanishes,andthe
problemreducestofindingtheexactprocessingtimesfromthegivenintervals.Thiscase
has been shown to be polynomial solvable by several researchers independently via
differentapproaches.RepresentativeworkonthiscanbefoundintheworkbyLivshitset
al. (1974), Matsuo et al. (1991), Lei (1993), Ioachim and Soumis (1995), Chen et al.
(1998), Van de Klundert (1996), Levner et al. (1996, 1997), Levner and Kats (1998),
Crama et al. (2000), Lee (2000), Lei and Liu (2001), Alcaide at al. (2007), Kats et al.
(2007).
In this section, we analyze the properties of Q and show that it can be solved by the
polynomialalgorithm,originatingfromtheparametriccriticalpathmethodbyLevnerand
Kats(1998)forthesingle-productversionoftheproblem.Ourmainobservationisthatthe
technologicalprocessesforproductspresentedbyPERT-typegraphs(seeFig.8)canbe
treatedbythe samemathematicaltoolsasmore primitiveprocessespresentedby linear
chainsconsideredinKatsetal.(2007).
4.2.AformalanalysisofproblemQ
EachgiveninstanceofQhasafixedsequenceofmaterialhandlingoperations,andan
associated MPS with K products and PERT-type precedence relations. The set of