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

Part a$)$ Write a mathematical programming model to minimize the total cost of all products including

purchase cost, holding cost, fixed cost, shortage cost, and lost sale cost. In your formulation, make sure

that you consider the shelf$-$life of products and the maximum number of shortage periods. In your

solution, you need to accurately define variables and new parameters$/$set $($if required$).(15$ points$)$

Part b$)$ Suppose that the volume of each product $$ is $($in cubic meters$)$ and the total available

space in the warehouse is $($in cubic meters$).$ How would you modify the model to consider the limited

space in the warehouse?