$title Simple Warehouse Problem (WHOUSE,SEQ=4)

$onText
A warehouse can store limited units of a commodity. Given an
initial stock, the manager has to decide when to buy or sell in
order to minimize total cost.

Dantzig, G B, Chapter 3.6. In Linear Programming and Extensions.
Princeton University Press, Princeton, New Jersey, 1963.

Keywords: linear programming, warehouse management, inventory
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Set t 'time in quarters' / q-1*q-4 /;

Parameter
   price(t)  'selling price ($ per unit)' / q-1 10, q-2 12, q-3 8, q-4 9 /
   istock(t) 'initial stock      (units)' / q-1 50 /;

Scalar
   storecost 'storage cost  ($ per quarter per unit)' /   1 /
   storecap  'stocking capacity of warehouse (units)' / 100 /;

Variable
   stock(t)  'stock stored at time t (units)'
   sell(t)   'stock sold at time t   (units)'
   buy(t)    'stock bought at time t (units)'
   cost      'total cost                 ($)';

Positive Variable stock, sell, buy;

Equation
   sb(t) 'stock balance at time t (units)'
   at    'accounting: total cost      ($)';

sb(t).. stock(t) =e= stock(t-1) + buy(t) - sell(t)  + istock(t);

at..    cost =e= sum(t, price(t)*(buy(t) - sell(t)) + storecost*stock(t));

stock.up(t) = storecap;

Model swp 'simple warehouse problem' / all /;

solve swp minimizing cost using lp;