ion by Willian

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Welch's, a major producer of grape juice, packages a few families of the products on each line. In order to prepare the master production schedule, the planner first aggregates the demand and the required safety stock of each product in each family during the next four weeks. She also aggregates the initial inventory of each product to obtain the family initial inventory. Then, given the production hours available during the regular time and overtime, the production rates and the setup time for each family, and some other cost data, she uses a linear programming software to determine the production quantity of each family during the next four weeks in order to minimize the total cost of holding inventory, setup time, and overtime, such that after satisfying the demand, the ending inventories each week are greater than or equal to the required safety stocks.

On a particular line, four families are packaged. One of the families has three products in it$.$ The forecast demand $($in $1,000$ cases$),$ the safety stock, and the beginning inventory for each product during the next four weeks are as follows:

$\backslash$table$[[$Product $1,6\mathrm{.}7,6\mathrm{.}78,6\mathrm{.}78,6\mathrm{.}78],[$Product $2,0\mathrm{.}5,1\mathrm{.}48,1\mathrm{.}48,1\mathrm{.}48],[$Product $3,2\mathrm{.}0,1\mathrm{.}88,1\mathrm{.}88,1\mathrm{.}88],[$Safety Stock,Week $1,$Week $2,$Week $3,$Week $4],[$Product $1,17\mathrm{.}99,19\mathrm{.}03,17\mathrm{.}12,16\mathrm{.}73],[$Product $2,0\mathrm{.}93,0\mathrm{.}98,3\mathrm{.}16,3\mathrm{.}60],[$Product $3,3\mathrm{.}02,3\mathrm{.}16,5\mathrm{.}72,6\mathrm{.}23]]$