Question

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 280 pounds of whole tomatoes, 130 pounds of tomato sauce, and 100 pounds of tomato paste; the price per pound for these ingredients is $0.96, $0.64, and $0.56, respectively. The cost of the spices and the other ingredients is approximately $0.10 per jar. Tom's, Inc., buys empty glass jars for $0.02 each, and labeling and filling costs are estimated to be $0.03 for each jar of salsa produced. Tom's contract with Western Foods results in sales revenue of $1.64 for each jar of Western Foods Salsa and $1.93 for each jar of Mexico City Salsa.LettingW = jars of Western Foods SalsaM = jars of Mexico City Salsaleads to the following formulation:Max 1W + 1.25Ms.t.5W + 7M ? 4480 Whole tomatoes3W + 1M ? 2080 Tomato sauce2W + 2M ? 1600 Tomato pasteW, M ? 0The sensitivity report is shown in figure below.Optimal Objective Value = 860.00000Variable Value Reduced CostW 560.00000 0.00000M 240.00000 0.00000Constraint Slack/Surplus Dual Value1 0.00000 0.125002 160.00000 0.000003 0.00000 0.18750Variable ObjectiveCoefficientAllowableIncreaseAllowableDecreaseW 1.00000 0.25000 0.10714M 1.25000 0.15000 0.25000Constraint RHSValueAllowableIncreaseAllowableDecrease1 4480.00000 1120.00000 160.000002 2080.00000 Infinite 160.000003 1600.00000 40.00000 320.00000What is the optimal solution, and what are the optimal production quantities?WMProfit $ Specify the objective coefficient ranges. Round your answers to three decimal places. If there is no lower or upper limit, then enter the text "NA" as your answer.Variable Objective Coefficient Rangelower limit Upper limitWestern Foods SalsaMexico City SalsaWhat are the shadow prices for each constraint? Round your answers to three decimal places.Constraint Shadow PriceWhole tomatoesTomato sauceTomato pasteInterpret each shadow price.The shadow price is the value that the will change by if you increase the constraint by one unit.Whole tomatoes have a shadow price of which means that if we add one pound of whole tomatoes the value of the objective function would .Tomato sauce has a shadow price of which means that if we add one pound of tomato sauce the value of the objective function would .Tomato paste has a shadow price of which means that if we add one pound of tomato paste the value of the objective function would .Constraints with a shadow price of 0 means that .Identify each of the right-hand-side ranges. If there is no lower or upper limit, then enter the text "NA" as your answer.Constraint Right-Hand-Side Rangelower limit upper limitWhole tomatoesTomato sauceTomato paste

Answer 1

The optimal solution involves producing 560 jars of Western Foods Salsa and 240 jars of Mexico City Salsa for maximum profit. The objective coefficient ranges and shadow prices indicate how changes in costs or availability of ingredients would affect the objective function (profit). Whole tomatoes and tomato paste have positive shadow prices, indicating profit sensitivity to these ingredients, while tomato sauce shows no impact.

Step-by-step explanation:

The optimal solution for Tom's, Inc., according to the sensitivity report provided, consists of producing 560 jars of Western Foods Salsa (W) and 240 jars of Mexico City Salsa (M) to maximize profit. This production plan results in a total profit of $860.

The objective coefficient range for W (Western Foods Salsa) is from $0.893 to $1.25, and for M (Mexico City Salsa), it is $1.00 to $1.40. These ranges signify the limits within which the price per unit can change without affecting the optimal solution.

The shadow prices for the constraints are as follows:

• Whole tomatoes: $0.125, implying that adding one more pound of whole tomatoes would increase the objective function (profit) by $0.125.
• Tomato sauce: $0, meaning adding tomato sauce won't change the profit since it isn't a binding constraint.
• Tomato paste: $0.1875, meaning that adding one more pound of tomato paste would increase the objective function by $0.1875.

The right-hand-side range for the constraints is as follows:

• Whole tomatoes: The range is between 4320 and 5600 pounds.
• Tomato sauce: No lower limit (NA) and an infinite upper limit, since the tomato sauce constraint has slack.
• Tomato paste: The range is between 1280 and 1640 pounds.

Answer 2

Step-by-step explanation:

Objective of the problem is to maximize profit.

Each jar is of 10 ounces.

Western Food Salsa (W) contains:

50% whole tomatoes, i.e. 50%*10 = 5 ounces;

30% tomato sauce, i.e. 30%*10 = 3 ounces;

20% tomato paste, i.e. 20%*10 = 2 ounces.

Mexico City Sales (M) contains:

70% whole tomatoes, i.e. 70%*10 = 7 ounces;

10% tomato sauce, i.e. 10%*10 = 1 ounces;

20% tomato paste, i.e. 20%*10 = 2 ounces.

Note: 1 pound = 16 ounces for below calculations

Maximum available quantity of Whole Tomatoes = 280*16 = 4480 ounces

Maximum available quantity of Tomato Sauce = 130*16 = 2080 ounces

Maximum available quantity of Tomato Paste = 100*16 = 1600 ounces

Profit for each jar of W = 1.64 - (0.96*5/16 + 0.64*3/16 + 0.56*2/16 + 0.10 + 0.02 + 0.03) = $ 1.00

Profit for each jar of M = 1.93 - (0.96*7/16 + 0.64*1/16 + 0.56*2/16 + 0.10 + 0.02 + 0.03) = $ 1.25

a. Refer first table in Sensitivity report, Optimal solution is following

W = 560 jars

M = 240 jars

Profit = $ 860

b. Refer third table in Sensitivity report, Objective coefficient range is as follows

Variable Objective coefficient range

Lower limit Upper limit

W 0.89286 1.25

M 1 1.4

Lower limit is obtained by subtracting allowable decrease from the objective coefficient and upper limit is obtained by adding allowable increase to objective coefficient

c. Refer third table in Sensitivity report, Shadow prices (or Dual value) of each constraint are following

Whole tomatoes = 0.12500

Tomato sauce = 0.00000

Tomato paste = 0.18750

Interpret each shadow price

i. The shadow price is the value that the objective function will change...

ii. 0.125, increase by $ 0.125

iii. 0, not change

iv. 0.188, increase by $ 0.188

v. it is a non-binding constraint

d. Refer fourth table of sensitivity report, Right hand side ranges of the constraints are following

Variable Righ Hand Side Range

Lower limit Upper limit

Whole tomatoes 4480-160 = 4320 4480+1120 = 5600

Tomato sauce 2080-160 = 1920 2080-inifinite = infinite

Tomato paste 1600-320 = 1280 1600+40 = 1640

cheers i hope this helped !!!