Chapter 1
Introduction
Learning Objectives
1. Develop a general understanding of the management science/operations research approach to decision
making.
2. Realize that quantitative applications begin with a problem situation.
3. Obtain a brief introduction to quantitative techniques and their frequency of use in practice.
4. Understand that managerial problem situations have both quantitative and qualitative considerations
that are important in the decision making process.
5. Learn about models in terms of what they are and why they are useful (the emphasis is on mathematical
models).
6. Identify the step-by-step procedure that is used in most quantitative approaches to decision making.
7. Learn about basic models of cost, revenue, and profit and be able to compute the breakeven point.
8. Obtain an introduction to the use of computer software packages such as The Management Scientist and
Microsoft Excel in applying quantitative methods to decision making.
9. Understand the following terms:
model infeasible solution
objective function management science
constraint operations research
deterministic model fixed cost
stochastic model variable cost
feasible solution breakeven point
1-1
,Chapter 1
Solutions:
1. Management science and operations research, terms used almost interchangeably, are broad
disciplines that employ scientific methodology in managerial decision making or problem
solving. Drawing upon a variety of disciplines (behavioral, mathematical, etc.), management
science and operations research combine quantitative and qualitative considerations in order to
establish policies and decisions that are in the best interest of the organization.
2. Define the problem
Identify the alternatives
Determine the criteria
Evaluate the alternatives
Choose an alternative
For further discussion see section 1.3
3. See section 1.2.
4. A quantitative approach should be considered because the problem is large, complex, important,
new and repetitive.
5. Models usually have time, cost, and risk advantages over experimenting with actual situations.
6. Model (a) may be quicker to formulate, easier to solve, and/or more easily understood.
7. Let d = distance
m = miles per gallon
c = cost per gallon,
2d
Total Cost = c
m
We must be willing to treat m and c as known and not subject to variation.
8. a. Maximize 10x + 5y
s.t.
5x + 2y 40
x 0, y 0
b. Controllable inputs: x and y
Uncontrollable inputs: profit (10,5), labor hours (5,2) and labor-hour availability (40)
1-2
, Introduction
c.
Profit: $10/unit for x
$ 5/ unit for y
Labor Hours: 5/unit for x
2/ unit for y
40 labor-hour capacity
Uncontrollable Inputs
Production Quantities Max 10 x + 5y Projected Profit and
s.t. check on production
x and y
time constraint
10 x + 5 y 40
Controllable x 0 Output
Input y 0
Mathematical
Model
d. x = 0, y = 20 Profit = $100
(Solution by trial-and-error)
e. Deterministic - all uncontrollable inputs are fixed and known.
9. If a = 3, x = 13 1/3 and profit = 133
If a = 4, x = 10 and profit = 100
If a = 5, x = 8 and profit = 80
If a = 6, x = 6 2/3 and profit = 67
Since a is unknown, the actual values of x and profit are not known with certainty.
10. a. Total Units Received = x + y
b. Total Cost = 0.20x +0.25y
c. x + y = 5000
d. x 4000 Kansas City Constraint
y 3000 Minneapolis Constraint
e. Min 0.20x + 0.25y
s.t.
x+ y = 5000
x 4000
y 3000
x, y 0
1-3
, Chapter 1
11. a. at $20 d = 800 - 10(20) = 600
at $70 d = 800 - 10(70) = 100
b. TR = dp = (800 - 10p)p = 800p - 10p2
c. at $30 TR = 800(30) - 10(30)2 = 15,000
at $40 TR = 800(40) - 10(40)2 = 16,000
at $50 TR = 800(50) - 10(50)2 = 15,000
Total Revenue is maximized at the $40 price.
d. d = 800 - 10(40) = 400 units
TR = $16,000
12. a. TC = 1000 + 30x
b. P = 40x - (1000 + 30x) = 10x - 1000
c. Breakeven point is the value of x when P = 0
Thus 10x - 1000 = 0
10x = 1000
x = 100
13. a. Total cost = 4800 + 60x
b. Total profit = total revenue - total cost
= 300x - (4800 + 60x)
= 240x - 4800
c. Total profit = 240(30) - 4800 = 2400
d. 240x - 4800 = 0
x = 4800/240 = 20
The breakeven point is 20 students.
14. a. Profit = Revenue - Cost
= 20x - (80,000 + 3x)
= 17x - 80,000
17x - 80,000 = 0
17x = 80,000
x = 4706
Breakeven point = 4706
b. Profit = 17(4000) - 80,000 = -12,000
Thus, a loss of $12,000 is anticipated.
c. Profit = px - (80,000 + 3x)
= 4000p - (80,000 + 3(4000)) = 0
4000p = 92,000
p = 23
1-4
Introduction
Learning Objectives
1. Develop a general understanding of the management science/operations research approach to decision
making.
2. Realize that quantitative applications begin with a problem situation.
3. Obtain a brief introduction to quantitative techniques and their frequency of use in practice.
4. Understand that managerial problem situations have both quantitative and qualitative considerations
that are important in the decision making process.
5. Learn about models in terms of what they are and why they are useful (the emphasis is on mathematical
models).
6. Identify the step-by-step procedure that is used in most quantitative approaches to decision making.
7. Learn about basic models of cost, revenue, and profit and be able to compute the breakeven point.
8. Obtain an introduction to the use of computer software packages such as The Management Scientist and
Microsoft Excel in applying quantitative methods to decision making.
9. Understand the following terms:
model infeasible solution
objective function management science
constraint operations research
deterministic model fixed cost
stochastic model variable cost
feasible solution breakeven point
1-1
,Chapter 1
Solutions:
1. Management science and operations research, terms used almost interchangeably, are broad
disciplines that employ scientific methodology in managerial decision making or problem
solving. Drawing upon a variety of disciplines (behavioral, mathematical, etc.), management
science and operations research combine quantitative and qualitative considerations in order to
establish policies and decisions that are in the best interest of the organization.
2. Define the problem
Identify the alternatives
Determine the criteria
Evaluate the alternatives
Choose an alternative
For further discussion see section 1.3
3. See section 1.2.
4. A quantitative approach should be considered because the problem is large, complex, important,
new and repetitive.
5. Models usually have time, cost, and risk advantages over experimenting with actual situations.
6. Model (a) may be quicker to formulate, easier to solve, and/or more easily understood.
7. Let d = distance
m = miles per gallon
c = cost per gallon,
2d
Total Cost = c
m
We must be willing to treat m and c as known and not subject to variation.
8. a. Maximize 10x + 5y
s.t.
5x + 2y 40
x 0, y 0
b. Controllable inputs: x and y
Uncontrollable inputs: profit (10,5), labor hours (5,2) and labor-hour availability (40)
1-2
, Introduction
c.
Profit: $10/unit for x
$ 5/ unit for y
Labor Hours: 5/unit for x
2/ unit for y
40 labor-hour capacity
Uncontrollable Inputs
Production Quantities Max 10 x + 5y Projected Profit and
s.t. check on production
x and y
time constraint
10 x + 5 y 40
Controllable x 0 Output
Input y 0
Mathematical
Model
d. x = 0, y = 20 Profit = $100
(Solution by trial-and-error)
e. Deterministic - all uncontrollable inputs are fixed and known.
9. If a = 3, x = 13 1/3 and profit = 133
If a = 4, x = 10 and profit = 100
If a = 5, x = 8 and profit = 80
If a = 6, x = 6 2/3 and profit = 67
Since a is unknown, the actual values of x and profit are not known with certainty.
10. a. Total Units Received = x + y
b. Total Cost = 0.20x +0.25y
c. x + y = 5000
d. x 4000 Kansas City Constraint
y 3000 Minneapolis Constraint
e. Min 0.20x + 0.25y
s.t.
x+ y = 5000
x 4000
y 3000
x, y 0
1-3
, Chapter 1
11. a. at $20 d = 800 - 10(20) = 600
at $70 d = 800 - 10(70) = 100
b. TR = dp = (800 - 10p)p = 800p - 10p2
c. at $30 TR = 800(30) - 10(30)2 = 15,000
at $40 TR = 800(40) - 10(40)2 = 16,000
at $50 TR = 800(50) - 10(50)2 = 15,000
Total Revenue is maximized at the $40 price.
d. d = 800 - 10(40) = 400 units
TR = $16,000
12. a. TC = 1000 + 30x
b. P = 40x - (1000 + 30x) = 10x - 1000
c. Breakeven point is the value of x when P = 0
Thus 10x - 1000 = 0
10x = 1000
x = 100
13. a. Total cost = 4800 + 60x
b. Total profit = total revenue - total cost
= 300x - (4800 + 60x)
= 240x - 4800
c. Total profit = 240(30) - 4800 = 2400
d. 240x - 4800 = 0
x = 4800/240 = 20
The breakeven point is 20 students.
14. a. Profit = Revenue - Cost
= 20x - (80,000 + 3x)
= 17x - 80,000
17x - 80,000 = 0
17x = 80,000
x = 4706
Breakeven point = 4706
b. Profit = 17(4000) - 80,000 = -12,000
Thus, a loss of $12,000 is anticipated.
c. Profit = px - (80,000 + 3x)
= 4000p - (80,000 + 3(4000)) = 0
4000p = 92,000
p = 23
1-4
