set MACHINE;
set PRODUCT;

var P{p in PRODUCT} >= 0;
  # P gives the total amount of product p produced

var X{m in MACHINE, p in PRODUCT} >= 0;
  # X[m,p] gives the proportion of the day that
  #  machine m spends producing product j

var TOTAL>=0;
  # the total production

param table{m in MACHINE, p in PRODUCT} >= 0;
  # the table in the book.

param proportion{p in PRODUCT}>=0;
  # the proportion of the total production which is product p

maximize production: TOTAL;

subject to findtot: TOTAL = sum{p in PRODUCT} P[p];

subject to balance {p in PRODUCT}: P[p] = proportion[p]*TOTAL;

subject to definitions {p in PRODUCT}:
     P[p]=sum{j in MACHINE} table[j,p]*X[j,p];
  # amount produced of product p depends on the time spent by each machine
  # making product p

subject to conservation {m in MACHINE}:
     sum{p in PRODUCT} X[m,p] <= 1;
  # X represents a fraction of a day, so the total time spent
  # by machine m must add up to no more than 1.