The manager of a supermarket meat department finds she has 200 lb of round steak, 800 lb of chuck steak, and 150 lb of pork in stock on Saturday morning. She will use these to make three products: hamburger meat, picnic patties, and meatloaf. The demand for each of these items always exceeds the supermarket supply. Therefore, she will be able to sell off all of however much of her current meat ingredients as she wants to. Hamburger meat must consist of at least 20 percent round steak and at least 50 percent chuck steak by weight. Picnic patties must consist of at least 20 percent pork and at least 50 percent chuck steak by weight. Meatloaf must consist of at least 10 percent round steak, 30 percent pork, and 40 percent chuck. In each product, the remainder is an inexpensive filler, and the supermarket has an unlimited supply of it. The manager wants to know how much of each product she should produce so that the total amount of leftover meat ingredients that need to be carried over can be minimized. Solve this problem using Linear Programming.

a. Production quantity of hamburger meat: ______
b. Production quantity of picnic patties: _
c. Production quantity of meatloaf: _
d. The total quantity of leftover meat: _____

You must also remember that a pound of hamburger meat and picnic patties both have 30% of a filler. So 1,120 pounds of meat can really produce 1,120 / 0.7 = 1,600 pounds of processed meat products.

Explanation:

200 lb of round steak

800 lb of chuck steak

150 lb of pork

hamburger meat:

20% of round steak

50% of chuck steak

picnic patties:

20% of pork

50% of chuck steak

meatloaf:

10% of round steak

30% of pork

40% of chuck steak

x₁ = round steak

x₂ = chuck steak

x₃ = pork

hamburger = 0.2x₁ + 0.5x₂

patties = 0.5x₂ + 0.2x₃

meatloaf = 0.1x₁ + 0.4x₂ + 0.3x₃

you have to maximize the total amount of meat used

M = 0.3x₁ + 1.5x₂ + 0.8x₃

x₁ ≤ 200

x₂ ≤ 800

x₃ ≤ 150

x₁ ≥ 0

x₂ ≥ 0

x₃ ≥ 0

using solver, the solution is:

850 pounds of hamburger meat and 750 pounds of picnic patties

170 pounds of round steak will be used
800 pounds of chuck steak
150 pounds of pork

You must also remember that a pound of hamburger meat and picnic patties both have 30% of a filler. So 1,120 pounds of meat can really produce 1,120 / 0.7 = 1,600 pounds of processed meat products.