prodsch.gms : APEX - Production Scheduling Model
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:
- CDC, APEX-III Reference Manual Version 1.2, Control Data Corporation, Minneapolis, 1980. MIP Sample Problem
Small Model of Type: MIP
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$Title A P E X - Production Scheduling model (PRODSCH,SEQ=9)
$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
$Offtext
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(l,q,s) production segments (sos2 type)
ssb(l,q,s) 0-1 needed for ss 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 in quarter (employees)
f(q) firings in quarter (employees)
Positive Variables p, ss, inv, src, h, f ;
Binary Variables bpl, lease, shift, ssb ;
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)
sbse(q,s) sos shift employment balance (employees)
scc(q,s) sos 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
lssb(l,q,s) ss - ssb 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(l,q,s)) ;
sbse(q,s).. se(q,s) =e= sum(l, pr("labor",l)*ss(l,q,s)) ;
scc(q,s).. sum(l, ss(l,q,s)) =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(l,q,s)) =e= 1;
lssb(l,q,s).. ss(l-1,q,s) + ss(l,q,s) =l= ssb(l-2,q,s) + ssb(l-1,q,s)
+ ssb(l,q,s);
p.up("spring") = .8*card(s)*smax(l, pr("motor",l));
Model prod1 initial employment / acost, ddpc, sbp, sbse, scc, disc, invb
dsrc, dwfc, ed, eb1, messb, lssb /
prod2 steady state / acost, ddpc, sbp, sbse, scc, disc, invb
dsrc, dwfc, ed, eb2, messb, lssb /
Solve prod1 minimizing cost using mip;
Display se.l, p.l ;
Parameter rep1 cost summary , rep2 production summary ;
rep1("direct",q) = dpc.l(q); rep1("storage",q) = isc.l(q);
rep1("hire-fire",q) = wfc.l(q);
rep1("*total*",q) = dpc.l(q) + isc.l(q) + wfc.l(q);
rep2("output",q) = p.l(q); rep2("inventory",q) = inv.l(q) ;
rep2("sales",q) = d(q); rep2("employment",q) = e.l(q);
Display rep1, rep2 ;
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