Tom'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 $285$ pounds of whole tomatoes, $150$ 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 $1$W $+1\mathrm{.}25$M

s$.$t$.$

$5$W $+7$M $<=4,560$ oz of whole tomatoes

$3$W $+1$M $<=2,400$ oz of tomato sauce

$2$W $+2$M $<=1,600$ oz of tomato paste

W$,$ M $>=0$

The sensitivity report is shown below.

Variable Cells

Model Variable Name Final Value Reduced Cost Objective Coefficient Allowable Increase Allowable Decrease

W Western Foods Salsa $520\mathrm{.}0000\mathrm{.}0001\mathrm{.}0000\mathrm{.}2500\mathrm{.}107$

M Mexico City Salsa $280\mathrm{.}0000\mathrm{.}0001\mathrm{.}2500\mathrm{.}1500\mathrm{.}250$

Constraints

Constraint Number Name Final Value Shadow Price Constraint R$.$H$.$ Side Allowable Increase Allowable Decrease

$1$ Whole tomatoes $4,560\mathrm{.}0000\mathrm{.}1254,560\mathrm{.}0001,040\mathrm{.}000560\mathrm{.}000$

$2$ Tomato sauce $1,840\mathrm{.}0000\mathrm{.}0002,400\mathrm{.}0001$E$+30560\mathrm{.}000$

$3$ Tomato paste $1,600\mathrm{.}0000\mathrm{.}1881,600\mathrm{.}000140\mathrm{.}000297\mathrm{.}143$

$($a$)$

What is the optimal solution, and what are the optimal production quantities?

W

$520$

Correct: Your answer is correct.

jars

M

$280$

Correct: Your answer is correct.

jars

profit $

$1394\mathrm{.}20$

Incorrect: Your answer is incorrect.

$($b$)$

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

Western Foods Salsa

$0\mathrm{.}893$

Correct: Your answer is correct.

to

$1\mathrm{.}250$

Correct: Your answer is correct.

Mexico City Salsa

$1\mathrm{.}000$

Correct: Your answer is correct.

to

$1\mathrm{.}400$

Correct: Your answer is correct.

$($c$)$

What are the shadow prices for each constraint? Interpret each.

constraint $1$

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

One additional ounce of whole tomatoes will improve profits by $$560\mathrm{.}00.$

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

Additional ounces of whole tomatoes will not improve profits.

Correct: Your answer is correct.

constraint $2$

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

One additional ounce of tomato sauce will improve profits by $$560\mathrm{.}00.$

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

Additional ounces of tomato sauce will not improve profits.

Correct: Your answer is correct.

constraint $3$

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

One additional ounce of tomato paste will improve profits by $$560\mathrm{.}00.$

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

Additional ounces of tomato paste will not improve profits.

Correct: Your answer is correct.

$($d$)$

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

constraint $1$

Incorrect: Your answer is incorrect.

to

Correct: Your answer is correct.

constraint $2$

Correct: Your answer is correct.

to

Correct: Your answer is correct.

constraint $3$

Incorrect: Your answer is incorrect.

to

Incorrect: Your answer is incorrect.