Consider a multi$-$product production planning problem with setup cost where the following

data$/$parameters are available:

Parameters:

$$ : The fixed cost of producing any unit of product $$ in period $.$

$$ : Per unit purchase cost of product $$ in period $.$

$$ : Per unit holding cost of product $$ in period $.$

$$ : Per unit shortage cost of product $$ for each period of shortage.

$$

$$ : Per unit lost sale cost for product $$

$$ : The demand of product $$ in period $.$

$$ : Shelf$-$life of product $$

$$ : The maximum number of shortage periods for product $($the number periods that

the customer will wait to have her$\backslash$his demand satisfied$).$

Sets:

$$ : The set of time periods in the planning horizon.

$$ : The set of products