Problem $1.$

Warmzies are hooked on optimization after successfully utilizing it to procure wool for their products. This time, they would like to optimize their production and inventory planning for their cashmere sweaters, which only uses the ultrafine type of wool.

For the cashmere sweaters, they have to honor the following customer contracts for the coming five months given the table below.

$\backslash$table$[[,$Month $1,$Month $2,$Month $3,$Month $4,$Month $5],[$Demand $($in $1\mathrm{.}000$s$),50,40,70,80,60],[$Production Cost $($$ per sweater$),30,29,32\mathrm{.}5,36,37\mathrm{.}5]]$

Warmzies $\text{'}$s demand and production cost for each month

These customer contracts must be met exactly at each month. Because of fluctuating market prices for wool during this season, the production costs are different for each month, as indicated in the table $($see also problem$\_$set$2\_$S$24$PT$.$xlsx$).$

Warmzies' production plant can produce at most $70,000$ cashmere sweaters per month. Excess production at the end of each month can be stored at a cost of $$2\mathrm{.}5$ per sweater $($this storage cost is applied to the sweaters stored at the end of each month$).$ However, they have limited storage and they can store at most $20,000$ sweaters per month. At the beginning of month $1,$ they have $5,000$ sweaters available in storage; these can be used to fulfill the demand in month $1.$ All the transactions related to the costs are incurred at the end of the month. In this respect, the cost of the initial inventory at the beginning of month $1$ is already incurred and this cost should not be taken into account for in this problem. On the other hand, the cost of the final inventory at the end of month $5($if there is any$)$ should be taken into account in the calculations.

Warmzies would like to minimize their total costs $($including production and storage costs$)$ over the next five months, while considering the production and storage issues, and meeting all of the demand on time.

We will formulate this problem as a linear program using the following variables $($we will not be concerned about integer number of sweaters in production or storage$)$:

$x_i$ : the number of sweaters produced in month $)=(1,$dots,$5$

$y_i$ : the number of sweaters stored in the inventory at the end of month $)=(1,$dots,$5$

where i refers to months $1,2,$dots,$5.$

Draw the graphical representation reflecting the production. demand, and inventory relations $($clearly label each node in your network and identify$/$label each arc using expressions involving your decision variables and$/$or numerical values$).$

Write down the objective function. The initial inventory of $5,000$ sweaters should be ignored in the objective $($this is a fixed cost, and already accounted for in the previous planning period$).$

Write down your constraints in terms of the decision variables along with a brief explanation for each constraint.