$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.

Keywords: linear programming, distribution problem, production planning, inventory problem
$offText

Set
   p 'production facilities' / one, two, three /
   d 'distribution centers'  / east, south, west, north /
   c 'customer zones'        / 1*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;

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;

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;

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) < card(m));

display pc, pco, revfac;

Variable
   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 Variable x, y, pn, po, s, h;

Equation
   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;

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;

Model pdi / all /;

solve pdi maximizing profit using lp;