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prodschx.gms : Production Scheduling Model using SOS1 and SOS2

A company specializing in the manufacture of outboard motors faces
highly seasonal demands and wants to minimize production cost. The
three main cost categories are:
  1. direct production cost (nonlinear production relations and shift
     operations are possible)
  2. inventory cost (rent or lease option)
  3. workforce fluctuation cost.
Reference:
Small Model of Type: MIP
$Title Production Scheduling Model using SOS1 and SOS2 (PRODSCHX,SEQ=109) $Ontext A company specializing in the manufacture of outboard motors faces highly seasonal demands and wants to minimize production cost. The three main cost categories are: 1. direct production cost (nonlinear production relations and shift operations are possible) 2. inventory cost (rent or lease option) 3. workforce fluctuation cost. CDC, APEX-III Reference Manual Version 1.2, Control Data Corporation, Minneapolis, 1980. MIP Sample Problem This is a revised version of the model PRODSCH (SEQ=9). The GAMS SOS definition required to change the index positions of the variables ss and SSB. There are three possible ways to use a MIP. $Offtext * Binary variables * SOS1 sets * SOS2 sets * * Some equations are entered twice with a different name to comply * with the strict typing of variables in GAMS. Sets q quarters / summer, fall, winter, spring / s shifts / first, second / l production levels / 1*4 / i(l) production level intervals / 1*3 / Parameters d(q) demand (motors per season) / spring = 24000 / delt(q) discount factor lc(q) leasing cost (dollars per season) / summer = 15000 / ei(q) initial employment / summer = 84 / Scalars mc material cost (dollars per motor) / 100 / sr space rental (dollars per motor) / 2 / invmax upper bound on inventory (motors) hc hiring cost (dollars per employee) / 900 / fc firing cost (dollars per employee) / 150 / ; delt(q) = 1/1.03**(ord(q)-1); invmax = sum(q, d(q)); Table pr(*,l) production relationship 1 2 3 4 labor 20 40 50 60 motor 1000 3000 4500 5800 Table sc(*,s) shift cost ( dollars per shift ) first second fixed 10000 16000 labor 3500 4100 Variables cost total discounted cost per year (1000 $) dpc(q) direct production cost (1000 $ per season) isc(q) inventory storage cost (1000 $ per season) wfc(q) workforce fluctuation cost (1000 $ per season) src(q) space rental cost (1000 $ per season) p(q) production (motors per season) ss(q,s,l) production segments (sos2 type) ssb(q,s,l) 0-1 needed for ss sos2 formulation ss1(q,s,l) SOS1 needed for ss sos2 ss2(q,s,l) SOS2 formulation inv(q) inventory (motors per season) lease lease-rent option e(q) total employment (employees) se(q,s) shift employment (employees per shift) shift(q,s) shift use indicator (binary) h(q) hirings per quarter (employees) f(q) firings per quarter (employees) Positive Variables p, ss, inv, src, h, f ; Binary variables lease, shift, ssb; SOS1 variable ss1; SOS2 variable ss2; Equations acost total cost definition (1000 $) ddpc(q) direct production cost definition (1000 $) disc(q) inventory storage cost definition (1000 $) dwfc(q) workforce fluctuation cost definition (1000 $) sbp(q) sos product balance (motors) sbps2(q) SOS2 product balance (motors) sbse(q,s) sos shift employment balance (employees) sbses2(q,s) SOS2 shift employment balance (employees) scc(q,s) sos shift link sccs2(q,s) SOS2 shift link invb(q) inventory balance (motors) dsrc(q) definition: space rental ed(q) total employment definition (employees) eb1(q) employment balance type 1 (employees) eb2(q) employment balance type 2 (employees) messb(q,s) mutual exclusivity for ssb mess1(q,s) mutual exclusivity for ss1 lssb(q,s,l) ss - ssb linkage lss1(q,s,l) ss - ss1 linkage ; acost.. cost =e= sum(q, delt(q)*( dpc(q) + isc(q) + wfc(q) )); ddpc(q).. dpc(q) =e= (mc*p(q) + sum(s, sc("fixed",s)*shift(q,s) + sc("labor",s)*se(q,s)))/1000; sbp(q).. p(q) =e= sum((s,l), pr("motor",l)*ss (q,s,l)) ; sbps2(q).. p(q) =e= sum((s,l), pr("motor",l)*ss2(q,s,l)) ; sbse(q,s).. se(q,s) =e= sum(l, pr("labor",l)*ss (q,s,l)) ; sbses2(q,s)..se(q,s) =e= sum(l, pr("labor",l)*ss2(q,s,l)) ; scc(q,s).. sum(l, ss (q,s,l)) =e= shift(q,s) ; sccs2(q,s).. sum(l, ss2(q,s,l)) =e= shift(q,s) ; invb(q).. inv(q) =e= inv(q-1) + p(q) - d(q) ; disc(q).. isc(q) =e= (lc(q)*lease + src(q))/1000 ; dsrc(q).. src(q) =g= sr*( inv(q) - invmax*lease ) ; dwfc(q).. wfc(q) =e= (hc*h(q) + fc*f(q))/1000 ; ed(q).. e(q) =e= sum(s, se(q,s)); eb1(q).. e(q) =e= e(q-1) + h(q) - f(q) + ei(q) ; eb2(q).. e(q) =e= e(q--1) + h(q) - f(q) ; messb(q,s).. sum(l, ssb(q,s,l)) =e= 1; mess1(q,s).. sum(l, ss1(q,s,l)) =e= 1; lssb(q,s,l).. ss(q,s,l-1) + ss(q,s,l) =l= ssb(q,s,l-2) + ssb(q,s,l-1) + ssb(q,s,l); lss1(q,s,l).. ss(q,s,l-1) + ss(q,s,l) =l= ss1(q,s,l-2) + ss1(q,s,l-1) + ss1(q,s,l); p.up("spring") = .8*card(s)*smax(l, pr("motor",l)); Model prod1B initial employment / acost, ddpc, sbp, sbse, scc, disc, invb, dsrc, dwfc, ed, eb1, messb, lssb / prod2B steady state / acost, ddpc, sbp, sbse, scc, disc, invb, dsrc, dwfc, ed, eb2, messb, lssb / prod1S1 initial employment / acost, ddpc, sbp, sbse, scc, disc, invb, dsrc, dwfc, ed, eb1, mess1, lss1 / prod2S1 steady state / acost, ddpc, sbp, sbse, scc, disc, invb, dsrc, dwfc, ed, eb2, mess1, lss1 / prod1S2 initial employment / acost, ddpc, sbps2, sbses2, sccs2, disc, invb, dsrc, dwfc, ed, eb1 / prod2S2 steady state / acost, ddpc, sbps2, sbses2, sccs2, disc, invb, dsrc, dwfc, ed, eb2 /; * get global optimum (OPTCR) and force each solve to start from scratch (BRATIO) Option optcr=0,limcol=0,limrow=0,bratio=1; Parameter report summary performance report; Solve prod1b minimizing cost using mip; report('prod1b','objval') = prod1b.objval; report('prod1b','iterusd') = prod1b.iterusd; report('prod1b','nodusd') = prod1b.nodusd; Solve prod1s1 minimizing cost using mip; report('prod1s1','objval') = prod1s1.objval; report('prod1s1','iterusd') = prod1s1.iterusd; report('prod1s1','nodusd') = prod1s1.nodusd; Solve prod1s2 minimizing cost using mip; report('prod1s2','objval') = prod1s2.objval; report('prod1s2','iterusd') = prod1s2.iterusd; report('prod1s2','nodusd') = prod1s2.nodusd; Solve prod2b minimizing cost using mip; report('prod2b','objval') = prod2b.objval; report('prod2b','iterusd') = prod2b.iterusd; report('prod2b','nodusd') = prod2b.nodusd; Solve prod2s1 minimizing cost using mip; report('prod2s1','objval') = prod2s1.objval; report('prod2s1','iterusd') = prod2s1.iterusd; report('prod2s1','nodusd') = prod2s1.nodusd; Solve prod2s2 minimizing cost using mip; report('prod2s2','objval') = prod2s2.objval; report('prod2s2','iterusd') = prod2s2.iterusd; report('prod2s2','nodusd') = prod2s2.nodusd; Display report;