Table are given in the image:

ASA is a large steel company that produces each of five different types of iron plates at its eight factories. At a recent strategic$-$planning meeting, the management allocated the budgets in Table $1$ to each of the eight factories for the next fiscal year. These budgets were based, in part, on the demands in Table $2$ for the five iron plates, provided by the Sales Department. As vice president of production, you, Mr$.$ Leroy Adams, have been asked to determine a production plan for each of the eight factories. $(50$ points$)$

Table $1.$ The Budget Assigned to Each Factory $($in Thousands of Dollars$)$

Table $2.$ The Demand in Tons for Each Type of Iron Plate

In preparation to formulate such a plan, you asked your associate, Mr$.$ James Aaron, to obtain production costs for each of the five iron plates at each of the eight factories. At your next meeting, Mr$.$ Aaron brought the cost data in Table $3.$ He also pointed out that, as external suppliers, ASA has three subcontractors from which they can purchase precisely the amounts of different types of iron plates at the prices shown in Table $4.$ Mr$.$ Aaron reminded you that contractual obligations require that if a particular subcontractor is chosen, ASA must purchase all of that supplier's five types of iron plates at the indicated prices. You can, of course, choose to purchase from none, one, two, or all three subcontractors. When you relayed this information to Mr$.$ Charles Bentley, the CEO, he approved the use of subcontractors if it would lower the total costs and informed you that the cost for buying the iron plates from the subcontractors would not affect the budgets of any of the factories $($see Table $1).$

As production manager, formulate $($but DO NOT SOLVE$)$ a model on paper to determine a production plan for each of the eight factories and which subcontractor$($s$),$ if any, to buy iron plates from to minimize the total costs while staying within each factory's budget and at least meeting the given demands.

The Table $3.$ The Cost of Producing One Ton of Each Iron Plate Type $\backslash (\backslash$boldsymbol$\{$i$\}\backslash )$ at Factory $\backslash (\backslash$boldsymbol$\{$j$\}\backslash )($in Thousands of Dollars$)$

Table $4.$ The Cost per Ton and the Amount of Each Plate Type Available from Each Subcontractor

$\backslash (\{\}^\{$a$\}\backslash )$ Cost $\backslash ((\backslash$$ $000)\backslash )$ to purchase one ton of plate type $\backslash ($ i $\backslash )$ from a subcontractor.

$\backslash (\{\}^\{$b$\}\backslash )$ For any subcontractor used, all of its available amounts of all five types of iron plates must be purchased at the indicated costs.