Question

2. Salsa Aguilar, a small startup business located in Pennsylvania, produces salsas that are sold through Heisler’s Market, a local grocery store located in McMurray, PA. Salsa Aguilar makes two types of products: Original Salsa and Fuego Salsa. Essentially, the two products have different blends of whole tomatoes, tomato sauce, and chopped vegetables. The Original Salsa is a blend of 50% whole tomatoes, 40% tomato sauce, and 10% chopped onions and jalapenos along with a proprietary spice blend. The Fuego Salsa, has a thicker and chunkier consistency, consists of 70% whole tomatoes, 10% tomato sauce, and 20% chopped onions and jalapenos, along with a proprietary spice blend. Each jar of salsa produced weighs 12 ounces. For the current production period, Salsa Aguilar can purchase up to 200 pounds of whole tomatoes, 150 pounds of tomato sauce, and 85 pounds of chopped onions and jalapenos; the price per pound for these ingredients is $1.00, $0.75, and $0.60, respectively. The cost of the spices and the other ingredients is approximately $0.15 per jar. Salsa Aguilar buys empty glass jars for $0.03 each and labeling and filling costs are estimated to be $0.02 for each jar of salsa produced. Salsa Aguilar’s contract with Heisler’s Market results in sales profit of $3.00 for each jar of Original Salsa and $3.25 for each jar of Fuego Salsa.

  

					Ingredient
		Amount	Price		Cost by jar
	Product mix 12 oz jars	Purchased	per lb	TOTAL	size
Type of Constraint	O                              F	1lb = 16 ounces
Whole tomatoes	50%                       70%	200	$1.00	$200.00	$266.67
Tomato Sauce	40%                        10%	150	$0.75	$112.50	$150.00
Onion & Jalapeno	10%                         20%	85	$0.60	$51.00	$68.00

Letting

O = number of jars of Original Salsa sold

F = number of jars of Fuego Salsa sold

Leads to the formulation of the linear program as follows. The RHS ranges are simply the total ounces of each ingredient and the LHS ranges reflect the quantity in ounces of each ingredient that goes into each jar. Note that the pricing information here is not relevant to our analysis, since we are considering only the net profit for each jar:

               Max       3O + 3.25F

               s.t.

                              6O + 8F                ≤ 3200                Whole tomatoes

                              5O + 1F                ≤ 2400                Tomato Sauce

                              1O + 2F                ≤ 1360                Chopped Onions and jalapenos

                                     O, F               ≥ 0

The computer solution is shown in figure below.

Cell	Name	Final Value	Reduced	Objective	Allowable	Allowable
			Cost	Coefficient	Increase	Decrease
$B$15	Jars Produced O	492.7536232	0	3	10	0,678571429
$C$15	Jars Produced F	28.98550775	0	3.25	0.95	12
Cell	Name	Final Value	Shadow	Constraint	Allowable	Allowable
			Price	R.H. Side	Increase	Decrease
$B$20	Whole tomatoes LHS	3200	0.36231884	3200	2297.1429	200
$B$21	Tomato Sauce LHS	2400	0.17210145	2400	160	1942.857143
$B$22	Onion & Jalapeno LHS	660.8695652	0	1360	1E + 30	699.1304335

1. What is the optimal solution, and what are the total returns based on these optimal production quantities?
2. Specify the objective function ranges
3. What are the dual values (shadow prices) for each constraint? Interpret each.
4. Identify each of the right-hand-side ranges.

Answer 1

a) As seen from sensitivity report, optimal solution:

Jars O = 492.75

Jars F = 28.98

Total Return = 3 * 492.75 + 3.25 * 28.98 = $1,572.435‬

b) Objective coefficient ranges:

For Jar O:

3 + Allowable increase = 3 + 10 = 13

3 - Allowable decrease = 3 - 0.69 = 2.31

Range = 2.31 to 13

For Jar F:

3.25 + Allowable increase = 3.25 + 0.95 = 4.2

3 - Allowable decrease = 3.25 - 12 = -8.75

Range = -8.75 to 4.2

c) Dual Price For constraint of White Tomatoes = 0.362

Dual Price For constraint of Tomato Sauce = 0.17

Dual Price For constraint of Onion and Jalapeno = Not applicable as it is not binding constraint

d) RHS ranges:

For White tomatoes:

3200 + Allowable increase = 3200 + 2297.14 = 5,497.14‬

3200 - Allowable Decrease = 3200 - 200 = 3000

Range = 3000 to 5497.14

For Tomato Sauce:

2400 + Allowable increase = 2400 + 160 = 2560

2400 - Allowable Decrease = 2400 - 1942.85 = 457.15‬

Range = 457.15‬ to 2560

For Onion and Jalapeno:

1360 + Allowable increase = 1360 + infinity = Infinity

1360 - Allowable Decrease = 1360 - 699.13 = 660.87

Range = 660.87‬ to Infinity

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