Table 1.

Sets, indices, parameters, and decision variables.

Sets
T	Set of time periods in the planning horizon
F	Set of feed formulations produced in Stage 2
M	Set of farms in Stage 1
P	Set of production phases in Stage 1 (total 6 phases until the pig reaches market size)
Indices
t, u	Index of the time periods
P	Index of the production phases
F	Index of the feed formulations
M	Index of the farms
Parameters
$h$	Inventory holding cost of each pig per period (in Stage 1)
$\hat{h}$	Inventory holding cost of any feed formulation per unit per period (in Stage 2)
$s_f$	Setup cost for feed formulation f
$A_{mfut}$	A binary to ensure that, for any farm m, the production period t of feed formulation f must be greater than the starting period u of farm m. (For example, if farm 1 starts in period 2, the production of feed formulation 1 must begin only after period 2). If the value of A is 1, the feed formulation f is produced. Otherwise, the value is 0
$L_m$	Production capacity of farm m (farm size)
$D_t$	Demand for pigs in period t (in Stage 1)
$w_f$	Consumption rate of feed formulation f (in Stage 2)
$C$	Production capacity of the feed mill to produce any feed formulation per period (in Stage 2)
$G$	A large positive number
Decision Variables
$X_{mt}$	= 1; farm m starts production in period t
	= 0; otherwise
$Y_{mft}$	= 1; the feed mill produces feed formulation f for farm m in period t
	= 0; otherwise
$Q_t$	Production quantity of pigs in period t (in Stage 1)
$I_t$	Ending inventory of pigs in period t (in Stage 1)
${\hat{I}}_{ft}$	Ending inventory of feed formulation f in period t (in Stage 2)
$S_{ft}$	= 1; if feed formulation f is produced in period t
	= 0; otherwise
${\hat{D}}_{ft}$	Demand for feed formulation f in period t (in Stage 2)
${\hat{Q}}_{ft}$	Production quantity of feed formulation f in period t (in Stage 2)