$($c$)$ What are the dual values for each constraint? Interpret each.

constraint $1$

One additional ounce of whole tomatoes will improve profits by $$240\mathrm{.}00.$

One additional ounce of whole tomatoes will improve profits by $$0\mathrm{.}188.$

One additional ounce of whole tomatoes will improve profits by $$0\mathrm{.}125.$

Additional ounces of whole tomatoes will not improve profits.

constraint $2$

One additional ounce of tomato sauce will improve profits by $$240\mathrm{.}00.$

One additional ounce of tomato sauce will improve profits by $$0\mathrm{.}188.$

One additional ounce of tomato sauce will improve profits by $$0\mathrm{.}125.$

Additional ounces of tomato sauce will not improve profits.

constraint $3$

One additional ounce of tomato paste will improve profits by $$240\mathrm{.}00.$

One additional ounce of tomato paste will improve profits by $$0\mathrm{.}188.$

One additional ounce of tomato paste will improve profits by $$0\mathrm{.}125.$

Additional ounces of tomato paste will not improve profits.

$($d$)$ Identify each of the right$-$hand$-$side ranges. $($Round your answers to two decimal places. If there is no upper or lower limit$,$ enter NO LIMIT.$)$

constraint $1$

to $1$

constraint $2$

to $1$

constraint $3$

to $|$Tom$\text{'}$s$,$ Inc., produces various Mexican food products and sells them to Western Foods, a chain of grocery stores located in Texas and New Mexico. Tom's, Inc., makes two salsa products:

Western Foods Salsa and Mexico City Salsa. Essentially, the two products have different blends of whole tomatoes, tomato sauce, and tomato paste. The Western Foods Salsa is a blend of

$50\%$ whole tomatoes, $30\%$ tomato sauce, and $20\%$ tomato paste. The Mexico City Salsa, which has a thicker and chunkier consistency, consists of $70\%$ whole tomatoes, $10\%$ tomato

sauce, and $20\%$ tomato paste. Each jar of salsa produced weighs $10$ ounces.

For the current production period, Tom's, Inc., can purchase up to $275$ pounds of whole tomatoes, $140$ pounds of tomato sauce, and $100$ pounds of tomato paste; the price per pound for

these ingredients is $$0\mathrm{.}96,$$$0\mathrm{.}64,$ and $$0\mathrm{.}56,$ respectively. The cost of the spices and the other ingredients is approximately $$0\mathrm{.}10$ per jar. Tom's, Inc., buys empty glass jars for $$0\mathrm{.}02$ each,

and labeling and filling costs are estimated to be $$0\mathrm{.}03$ for each jar of salsa produced. Tom's contract with Western Foods results in sales revenue of $$1\mathrm{.}64$ for each jar of Western Foods

Salsa and $$1\mathrm{.}93$ for each jar of Mexico City Salsa. Letting

$W=$ jars $of$ Western Foods Salsa

$M=$ jars $of$ Mexico City Salsa

leads to the formulation $($units for constraints are ounces$)$:

Max $1W+1\mathrm{.}25M$

$s.t.$

$5W+7M4,400ozof$ whole tomatoes

$3W+1M2,240ozof$ tomato sauce

$2W+2M1,600ozof$ tomato paste

$W,M0$

The computer solution is shown below.

Optimal Objective Value $=850\mathrm{.}00000$

Variable Value Reduced Cost

W $600\mathrm{.}000000\mathrm{.}00000$

M $200\mathrm{.}000000\mathrm{.}00000$

Constraint Slack$/$Surplus Dual Value

$10\mathrm{.}000000\mathrm{.}12500$

$2240\mathrm{.}000000\mathrm{.}00000$

$30\mathrm{.}000000\mathrm{.}18750$

Variable Objective Allowable Allowable

Coefficient Increase Decrease

$1\mathrm{.}000000\mathrm{.}250000\mathrm{.}10714$

$1\mathrm{.}250000\mathrm{.}150000\mathrm{.}25000$

Constraint RHS Allowable Allowable

Value Increase Decrease

$4400\mathrm{.}000001200\mathrm{.}00000240\mathrm{.}00000$

$2240\mathrm{.}00000$ Infinite $240\mathrm{.}00000$

$1600\mathrm{.}0000060\mathrm{.}00000342\mathrm{.}85714$

$($a$)$ What is the optimal solution, and what are the optimal production quantities?

W

jars

M

profit

$

Enter an exact number.

$($b$)$ Specify the objective function ranges. $($Round your answers to five decimal places.$)$

Western Foods Salsa

to $|$

Mexico City Salsa

to