pdi.gms : ARCNET - Production Distribution and Inventory
This example of a typical two echelon production, distribution and inventory
problem (pdi) is taken from the arcnet user guide.
Reference:
- ARCNET, ARCNET User Guide, Analysis, Research and Computation, Austin, Texas, 1982.
Small Model of Type: LP
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$Title ARCNET - Production Distribution and Inventory (PDI,SEQ=10)
$Ontext
This example of a typical two echelon production, distribution and inventory
problem (pdi) is taken from the arcnet user guide.
ARCNET, ARCNET User Guide, Analysis, Research and Computation,
Austin, Texas, 1982.
$Offtext
Sets p production facilities / one, two, three /
d distribution centers / east, south, west, north /
c customer zones / 1, 2, 3, 4, 5 /
m month / january, february, march, april /
Table pfd(p,*) production facility data (table 12.1)
min-prod max-prod over-prod "prod-cost" over-cost
* (units) (units) (units) ($/unit) ($/unit)
one 5000 1000 35 45
two 1200 3000 500 40 43
three 700 1500 38
Table fdec(p,d) first distribution echelon cost ($ per unit)
east south west north
one 10 12
two 8 4 5
three 6 8
Table sdec(d,c) second distribution echelon cost ($ per unit)
1 2 3 4 5
east 15 19
south 20 22 18
west 16 18 19
north 15 21
Set pd(p,d), dc(d,c); pd(p,d) = yes$fdec(p,d); dc(d,c) = yes$sdec(d,c);
Display pd,dc,sdec;
Table dcd(d,*) distribution center data (table 12.3)
max-invent hold-cost
* (units) ($/unit)
east 3000 2
south 2500 2
west 4000 1
north 2500 3
Table czd(c,*) customer zone data (table 12.5)
min-demand max-demand revenue
* (units) (units) ($/unit)
1 2000 2500 70
2 2500 68
3 2000 3000 65
4 1500 2000 72
5 1500 3000 71
Parameter pc(p,m) production cost normal shift
pco(p,m) production cost overtime
revfac(m) revenue factor / (january,february) = 1, (march,april) = 1.1 /;
pc(p,m) = pfd(p,"prod-cost") + floor(2**(ord(m)-1)/2);
pco(p,m)= pfd(p,"over-cost") + ord(m) - (ord(m) lt card(m));
Display pc, pco, revfac;
Variables x(p,d,m) shipments from production to distribution
y(d,c,m) shipments from distribution centers to markets
pn(p,m) production
po(p,m) production: overtime
s(d,m) storage level
dm(c) demand level
h(d,m) handling
profit, revenue, transport, production, holding
Positive Variables x, y, pn, po, s, h;
Equations ib(d,m) inventory balance
pb(p,m) production balance
hb(d,m) handling balance
db(c,m) demand balance
ar revenue balance
at transport balance
ap production cost balance
ah inventory holding cost definition
apr profit definition ;
ib(d,m).. h(d,m) =e= s(d,m-1) + sum(p$pd(p,d), x(p,d,m));
pb(p,m).. pn(p,m) + po(p,m) =e= sum(d$pd(p,d), x(p,d,m));
hb(d,m).. s(d,m) =e= h(d,m) - sum(c$dc(d,c), y(d,c,m)) ;
db(c,m).. sum(d$dc(d,c), y(d,c,m)) =e= dm(c);
ar.. revenue =e= sum((d,c,m)$dc(d,c), revfac(m)*czd(c,"revenue")*y(d,c,m));
at.. transport =e= sum((d,m), sum(p$pd(p,d), fdec(p,d)*x(p,d,m)) + sum(c$dc(d,c), sdec(d,c)*y(d,c,m)));
ap.. production =e= sum((p,m), pc(p,m)*pn(p,m) + pco(p,m)*po(p,m));
ah.. holding =e= sum((d,m), dcd(d,"hold-cost")*s(d,m));
apr..profit =e= revenue - transport - production - holding + 10 ;
Model pdi / all /;
s.lo(d,"april") = 200; h.up(d,m) = dcd(d,"max-invent");
pn.lo(p,m) = pfd(p,"min-prod"); pn.up(p,m) = pfd(p,"max-prod");
po.up(p,m) = pfd(p,"over-prod"); dm.lo(c) = czd(c,"min-demand"); dm.up(c) = czd(c,"max-demand");
Display h.up, pn.lo;
Solve pdi maximizing profit using lp;
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