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Linear programming: Please use the graphical model to solve this problem.
A Nutritionist wants to put together a Lunch menu for certain Nursing Homes’ long term occupants. The menu
is to include two items,
A and B. Suppose that each ounce of A provides 2 units of vitamin C and 2 units of iron and each ounce of B
provides 1 unit of vitamin C and 2 units of iron. Suppose the cost of A is 4¢/ounce and the cost of B is 3¢/ounce.
Since Item B was purchased under difficult negotiation contact, at least 1 ounce of it must be present per
serving. If the lunch menu must provide at least 8 units of vitamin C and at least 10 units of iron, how many
ounces of each item should be provided in order to meet the iron and vitamin C requirements to minimize
cost? What will this lunch cost?
What is the objective function?
What are the limitations (inequalities)?
Please use graph paper below to show interactions among limitations
What is the best combination of number of ounces of food A and B that should be considered to minimize Cost?
What is the minimum cost?
Linear programming: Please use the graphical model to solve this problem.
A Nutritionist wants to put together a Lunch menu for certain Nursing Homes’ long term occupants. The menu
is to include two items, A and B. Suppose that each ounce of A provides 2 units of vitamin C and 2 units of iron
and each ounce of B provides 1 unit of vitamin C and 2 units of iron. Suppose the cost of A is 4¢/ounce and the
cost of B is 3¢/ounce. Since Item B was purchased under difficult negotiation contact, at least 1 ounce of it
must be present per serving. If the lunch menu must provide at least 8 units of vitamin C and at least 10 units
of iron, how many ounces of each item should be provided in order to meet the iron and vitamin C
requirements to minimize cost? What will this lunch cost?
What is the objective function?
What are the limitations (inequalities)?
Please use graph paper below to show interactions among limitations
What is the best combination of number of ounces of food A and B that should be considered to minimize Cost?
What is the minimum cost?
Chapter 7 Linear Programming: (Graphical Method)
Definition:
Linear Programming (LP) is a mathematical technique to determine an optimum
solution to a problem with several variables that are subject to constraints
(restrictions or limitations). “LP is a technique that helps in resource allocation
decisions”. Resources can be product such as Machineries, Furniture, Food,
Clothes…or services such as Scheduling decisions or Investment decisions.
FYI: Even though it is called Linear Programming, Programming has very little to
do with this…This is a mathematical technique.
Requirements for LP
• Maximize or minimize an objective function (Usually Minimize cost)
• We Must operate under some limitations called constraints…(We do not
have unlimited resources)
• There must be alternative ways of allocating resources when producing
more than I good.
• Mathematical relationships are linear…Thus implying proportionality and
additivity.
Linear model: ax1 + bx2 + cx3 + …nxn = k
o Proportionality: if producing 1 item takes 5 hours, producing 10
items will take 50 hours.
o Additivity: If the production of 1 item generates $1000.00 in profit,
the production of a second item generates $1250.00 in profit, the
total profit for both items is the sum of the profit of each individual
item ($2250.00)
• Certainty: We assume that the number of objectives and constraints are
known and constant during a given period.
• Divisibility: We must assume that all solutions need no to be whole
numbers. Solutions could be fractional and could just mean that the work is
in progress…
• Non Negative variable: It is very difficult to imagine negative values of
tangible items…We cannot produce negative Chairs…
Formulating a Linear Programming
Linear programming is frequently used in business to seek maximum profit,
revenues… or minimum cost. The first step in solving linear programming
problems is to set up a function that represents cost, profit, or some other
quantity to be maximized or minimized subject to the constraints of the problem
(Objective Function). Then outline the constraints with a system of linear
inequalities. The solution of these systems of inequalities is called the feasible
region. Graph the inequalities to determine the coordinates of the vertices of the
region. Evaluate the function at each vertex. The largest and smallest of those
values are the maximum and minimum values of the function, respectively.
Example:
Jules is in the business of constructing dog houses. A small dog house requires 8
square feet of plywood and 6 square feet of insolation. A large dog house requires
16 square feet of plywood and 3 square feet of insolation. Jules has only 48
square feet of plywood and 18 square feet of insolation available for this project.
If a small dog house sells for $15 and a large dog house for $20, then how many
dog houses of each type should be built to maximize revenue and satisfy the
constraints?
Let x be the number of small dog houses and y be the number of large dog
houses.
Objective: maximize revenue by selling each small dog house at $15 and each
large dog house at $ 20… Objective function: Revenues = 15 x + 20 Y
Constraints:
Plywood limitation: 8x + 16 y ≤ 48➔ simplifies to x + 2y ≤ 6
Insolation limitation: 6x + 3 y ≤ 18➔ simplifies to 2x + y ≤6
Since we cannot have negative tangible items: x ≥ 0 and y ≥ 0
We can now mathematically set up the word problem as follows:
Maximize revenue = 15 x + 20 Y
Subject to: x + 2y ≤ 6
2x + y ≤6
x≥0
y≥0
Graphical solution to a linear Programming Problem
The graphical method is easier to use when there are 2 decision variables but
could give us imaginative ideas of how a much larger system could be dealt with.
Graphically, we are going to use our Cartesian coordinates system to find a
solution. We are going to be restricted in the first quadrant due to the fact than
we have a nonnegative assumption we must respect.
Feasible solution: our feasible solution is going to be a system of inequalities
characterized by the interactions of constraints…This will be in the first quadrant.
An example of a feasible region (Maximizing problem)
An example of a feasible region (Minimizing problem)
Let’s take a closer look at problem # 1; we will graph the following inequalities
and identify the feasible region in class.
x + 2y ≤ 6
2x + y ≤6
x≥0
y≥0
Corner point solution
The optimal solution is going to be at a corner of the polygon generated by the
graphical interaction of constraints. Looking at our graph, we are going to solve
our objective function by evaluating it using all corner points; the combination
with largest output will maximize the revenue.
x
y
Revenue=15 x + 20 y
0
3
60
3
0
45
2
2
70
0
0
0
Application of Linear Programming Graphical method
1. A farmer wanted to raise geese and pigs. She wants to raise no more than
16 animals, including no more than 10 geese. She spends $15 to raise a
goose and $ 45 to raise a pig, and she has $540 for the project. Find the
maximum profit she can make if each goose produces a profit of $7 and
each pig a profit of $20.
2. A manager in an office needs to purchase new filling cabinet. He knows that
Ace cabinet cost $40 each, requires 6 square feet of floor space, and holds
8 cubic feet of files. On the other hand, each Excello cabinet cost $ 80,
requires 8 square feet of files, and holds 12 cubic feet. His budget permits
him to spend no more than $560 on file, while office has a room for no
more than 72 square feet of cabinets. The manager desires the greatest
storage capacity within the limitations imposed by funds and space. How
many of each type of cabinet should he buy?
3. A manufacturer of refrigerators must ship at least 100 refrigerators to its
two West Cost warehouses. Each warehouse holds a maximum of 100
refrigerators. Warehouse A holds 25 refrigerator already, an warehouse B
has 20 refrigerator on hand. It cost $12 to ship a refrigerator to warehouse
A and $10 to ship one to warehouse B. Union rules requires that a least 300
workers be hires. Shipping a refrigerator to A requires 4 workers, while B
requires 2 workers. How many refrigerators should be shipped to each
warehouse to minimize cost? What is the minimum cost?
Please Bring your computers next class, we are going to use MS Excel to solve
LPs…We are going to look at LPs involving 3 or more decision variables. Make
sure you have the “solver” option active. (Option➔ add-Ins➔ Analysis
toolpak➔go➔check solver)
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