BUS 190
Homework 4 Solutions
1. Recall the problem about paper supplier Dunder Mifflin from Homework 2. The
problem can be formulated as:
Let
A be the number of reams of paper to buy from Supplier A
B be the number of reams of paper to buy from Supplier B
BUS 190
Homework 5 Solutions
1. Recall the Natures Best Frozen Foods company problem from Homework 3. The complete
formulation is shown below:
Let
S
B
H
V
be
be
be
be
the
the
the
the
number
number
number
number
of
of
of
of
bags
bags
bags
bags
Max
0.22S +
BUS 190
Homework 13 Solutions
1. Suppose that you own a nail salon. The number of clients you serve each week is a random
variable, C, with a binomial distribution. Using the following information, calculate the quantities
below.
P(C65) = .96
P(C53) = .81
BUS 190
Homework 3 Solutions
1. Natures Best Frozen Foods company produces four different mixes of frozen, readytoeat vegetables. The mixes consist of five different vegetables: carrots, mushrooms,
green peppers, broccoli, and corn. The mixes are Stir Fr
BUS 190
Homework 7 Solutions
1. Consider the following integer optimization problem:
Max
10A + 3B
Subject to
6A + 7B 40
3A + B 11
A, B 0 and integer
a)
Solve this problem using Excel. Report the optimal solution and objective value. Turn in your
spreadshe
BUS 190
Homework 2 Solutions
1. You would like to make a nutritious meal of steak and potatoes. The meal should provide at least 50
g of carbohydrates, at least 40 g of protein, and no more than 60 g of fat. A piece of steak contains 5 g
of carbohydrates,
BUS 190
Homework 10 Solutions
1. What are the four laws of probability? Please include both versions of the 4th law.
1.
2.
3.
4.
For any event A, the probability of A is between zero and one. That is, 0 P(A) 1.
If A and B are mutually exclusive events, th
BUS 190
Homework 6 Solutions
1. 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. A jar of
Western Foods salsa uses 5 ounc
More notes on 4th law
What if A and B are independent?
If A occurs, it doesnt affect P (B). So P (B A) = P (B)
Using 4th law,
P (B A) = P (A and B) = P (A) P(B) = P (B)
P (A)
P (A)
These are always the same
for independent events
If A and B are Mutually e
Definition
P (B A) is used to denote conditional probability. It is the probability of event B
given that event A has occurred.
Useful when we have partial information about the outcome of an experiment.
Ex
Roll a die.
Let A be the event that I roll an ev
New TV Show
Youve developed a new show.
One of the networks has purchased your show.
You know the chances of your show being successful are small.
Youd like to know your chance of success.
Your chance of success depends on when your show is introduced: at
You manage a call center.
You have just recruited 100 new trainees.
All 100 take a training class. If they pass they class, they will become fulltime employees.
Each trainee has an 80% chance of passing.
Assume that whether one trainee passes is independ
Ex from last time:
Value
Probability
3
5
6
8
.2
.1
.4
.3
Summary Statistics
Two ways to summarize a random variables distribution are the mean and the standard deviation
Mean
Tells us what value the random variables on average. Tells us where the distribu
Random Variables
When the outcome of an experiment is a number, that number is called a random
variable.
Basically, a random variable is a number that you dont know yet.
Ex
 How many sandwiches Subway sells today
 The numbers on a domino that your oppon
You manage a call center.
You have just recruited 100 new trainees.
All 100 take a training class. If they pass they class, they will become fullt
employees.
Each trainee has an 80% chance of passing.
Assume that whether one trainee passes is independent
BUS 190
Homework 12 Solutions
1. Give examples of five random variables (other than the examples from class).
Many answers are possible.
2. Consider the random variable T which has the following distribution:
Value Probability
3
.5
6
.1
7
.25
10
.1
15
.05
Condo Developer
Deciding how large of a condo development to build.
Were not sure what demand will be when the condos are finished.
Strong
Weak
Small
8
7
Large
20
9
(profit in $million)
We wont know what demand will be when we start construction. Our for
Decision Trees
A tool for making decisions that involve uncertainty.
They find the decision that will be best on average.
New Product Development
Our company is considering investing $5 million in R+D for a new product.
Theres a 50% chance the R+D will le
New product R+D from last week
15
Succeed
Do R+D
project
5
.5
B
5
What value would
make us choose
not to do R+D?
.5
Fail
5
A
Dont
0
B:
15x 5 (1 x) = 20x 5
20x 5 < 5
20x < 5
x < .25
Again, a wide margin of error.
Our original estimate was .5. If our sensi
Solving in Excel
Lets solve the sandwich problem in Excel
Max
S.T
4T + 3C
.6T +.3C 21 (Turkey)
.2C 5 (Bacon)
.4T +.5C 20 (Bread)
T, C 0
First step is to put our formulation in Excel.
Set up a spreadsheet where
 I can type in the number of each sandwich

Business 190
Practice Midterm Exam
Consider the following linear optimization model:
Maximize Profit = x +
Subject to
x + 3y
2x + y
x
x, y
2y
12 (resource A)
18 (resource B)
8 (resource C)
0
Solve the problem graphically, then answer the following qu
We produce chemicals for industrial use.
We produce two products:
 solvent
 adhesive
Were planning production for the upcoming month.
 A major customer needs 125 gallons of solvent
 In total, we must produce at least 350 gallons of the two products to
Possible Solutions
When you solve a linear optimization:
If there is an optimal solution, there will always be a corner point that is optimal.
(The computer uses the simplex algorithm which takes advantage of this.)
Further observations:
When you solve a
Sensitivity Analysis

Sensitivity analysis is performed after you find the optimal solution
It gives you a better understanding of the solution
o Helps you answer whatif questions
 Even longterm strategy question
o Helps you understand the impact of a
Ex 1
Raise price of club sandwich by $1.
Step 1: Is $1 within the allowable increase?
Yes. Allowable increase is 2, which is greater than 1.
Step 2: The profit will change by
($1) x 20 = $20
The profit will increase by $20. It will go from $160 to $180.
E
Sensitivity Analysis of constraint RHS
Repeats the LHS
Repeats the RHS
Constraints
RHS
Excels scientific notation
1x1030 Treat as
Again, there two steps
1. Check if the change is within the allowable increase/decrease.
If not, change spreadsheet + rerun
Transportation Problem
Were manufacturing TVs at 3 plants and shipping them to 4 distribution centers.
Well use some shortcuts to enter this in Excel
o SUMPRODUCT formula
 Multiply and then add together two lists of numbers
(column, row, or rectangle)
o
Transportation Problem
Used to plan the distribution of goods
 Finds the minimum cost way to ship goods from a numbers of original to a number of
destinations
Ex
Dell has 3 plants in the US.
Has to decide for each order which plant will make it.
Consider
Integer Optimization
Linear optimization might give you a fractional answer
 Rounding a fractional answer off might not be the best idea
o The rounded solution might be suboptimal
(i.e. theres a better whole number solution)
o The rounded solution might
Ex
Mixed Integer Optimization
We produce 3 products
 Fuel Additive
 Solvent
 Cleaning Fluid
Made from 3 ingredients
F S
Ingredient 1
.4 .5
Ingredient 2
0 .2
Ingredient 3
.6 .3
C
.6
.1
.3
Revenue
40
(per gallon)
30
50
Available
20
5
21
Theres a setup co