set PROD;      # products
param T > 0;   # number of weeks

param rate {PROD} > 0;          # tons per hour produced
param inv0 {PROD} >= 0;         # initial inventory
param commit {PROD,1..T} >= 0;  # minimum tons sold in week
param market {PROD,1..T} >= 0;  # limit on tons sold in week

param avail_min {1..T} >= 0;                 # unpenalized hours available
param avail_max {t in 1..T} >= avail_min[t]; # total hours available
param time_penalty {1..T} > 0;

param prodcost {PROD} >= 0;     # cost per ton produced
param invcost {PROD} >= 0;      # carrying cost per ton of inventory
param revenue {PROD,1..T} >= 0; # revenue per ton sold

var Make {PROD,1..T} >= 0;                  # tons produced
var Inv {PROD,0..T};                        # tons inventoried
var Sell {p in PROD, t in 1..T}
   >= commit[p,t], <= market[p,t];          # tons sold

var Use {t in 1..T} >= 0, <= avail_max[t];  # hours used

maximize Total_Profit:
   sum {p in PROD, t in 1..T} (revenue[p,t]*Sell[p,t] -
      prodcost[p]*Make[p,t] - invcost[p]*Inv[p,t]) -
   sum {t in 1..T} <<avail_min[t]; 0,time_penalty[t]>> Use[t];

               # Objective: total revenue less costs in all weeks

subject to Time {t in 1..T}:
   sum {p in PROD} (1/rate[p]) * Make[p,t] = Use[t];

               # Total of hours used by all products
               # may not exceed hours available, in each week

subject to Init_Inv {p in PROD}:  Inv[p,0] = inv0[p];

               # Initial inventory must equal given value

subject to Balance {p in PROD, t in 1..T}:
   Make[p,t] + Inv[p,t-1] = Sell[p,t] + Inv[p,t];

               # Tons produced and taken from inventory
               # must equal tons sold and put into inventory