CHAPTER 1
Introduction
4
CHAPTER 1
KEY CONCEPTS
ILLUSTRATED
PROBLEMS
ANSWERED
PROBLEMS
Quantitative Approach to Decision Making
1,2
6,7
Mathematical Models Development
3,4
8,9,10,11,12
Cost, Volume, Profit Analysis
5
13
Marginal Analysis
5
13
Breakeven Analysis
5
13
CONCEPT
REVIEW
1. The quantitative approach to decision making based on the scientific method of problem
solving. This body of knowledge is referred to as management science, operations
research or decision science. It had its early roots in World War II and is flourishing in
business and industry with the aid of computers in general and the microcomputer in
particular.
2. Problem solving is a process designed to better a current state of affairs, It consists of two
phases: (1) decision making; and (2) implementation and evaluation.
3. The decision making process involves: (1) structuring the problem; and (2) analyzing the
problem.
4. Structuring the problem includes: (1) defining the problem; (2) identifying the alternatives;
and (3) choosing the criteria (single or multiple) to be used to evaluate the alternatives.
5. Analyzing the problem consists of: (1) qualitatively and quantitatively evaluating the
alternatives; and (2) making a decision recommendation (choosing an alternative).
6. Evaluating alternatives is often accomplished by experimenting with a model. A model is
a representation of a real object or situation. Generally, experimenting with a model is less
costly, requires less time, and involves less risk.
7. Three forms of models are iconic, analog, and mathematical. Iconic models are physical
replicas (scalar representations) of real objects. Analog models are physical in form, but
do not physically resemble the object being modeled.
INTRODUCTION
5
8. Mathematical models (also called symbolic models), represent real world problems through
a system of mathematical formulas and expressions. They are idealizations of real-life
problems based on key assumptions, estimates, and/or statistical analyses.
9. After the problem has been structured, the steps in the quantitative analysis are:
(1) mathematical modeling; (2) data preparation; (3) model solution (and refinement); and
(4) report generation.
10. Mathematical models relate decision variables (or controllable inputs) with fixed or variable
parameters (or uncontrollable inputs). Frequently mathematical models seek to maximize
or minimize some objective function subject to constraints.
11. If any of the uncontrollable inputs is subject to variation the model is said to be stochastic.
Otherwise the model is said to be deterministic. Generally, stochastic models are more
realistic than deterministic models, but they are more difficult to analyze.
12. The values of the decision variables that provide the mathematically-best output are
referred to as the optimal solution for the model.
13. If a particular decision alternative does not satisfy one or more of the model constraints, the
decision alternative is rejected as being infeasible. If all constraints are satisfied, the
decision alternative is feasible and a candidate for the “best” or recommended decision.
14. Cost/benefit considerations must be made in selecting an appropriate mathematical
model. Frequently a less complicated (and perhaps less precise) model is more
appropriate than a more complex and accurate one due to cost and ease of solution
considerations.
15. Quantitative analysis should be used as one of many input factors for managerial decision
making. It is not a replacement for human decision making.
16. Primary reasons for the use of quantitative analysis are: (1) the problem is complex;
(2) the problem is important and/or the decision must be thoroughly justified; (3) the
problem is new and there is little or no previous experience to rely on; (4) the problem is
repetitive and time-consuming (with varying uncontrollable input values); and (5) what-if
questions need to be answered in an efficient manner.
17. Break-even analysis uses a basic mathematical model of the relationship between a
volume variable and cost, revenue, or profit. The break-even point is the volume that
results in total revenue equaling total cost.
18. Fixed cost is the portion of the total cost that does not depend on the volume variable; this
cost remains regardless of the value of the volume variable. Variable cost is the portion of
the total cost that is dependent on and varies with the volume variable.
19. Marginal cost is the rate of change of the total cost with respect to the volume variable.
Marginal revenue is the rate of change of the total reveune with respect to the volume
variable.
20. Linear programming, integer programming, network models, and simulation are
among the most frequently used quantitative techniques in business.
6
CHAPTER 1
ILLUSTRATED PROBLEMS
PROBLEM 1
Consider a construction company building a 250-unit apartment complex. The project
consists of hundreds of activities involving excavating, framing, wiring, plastering, painting,
landscaping, and more. Some of the activities must be done sequentially and others can be
done simultaneously. Also, some of the activities can be completed faster than normal by
purchasing additional resources (workers, equipment, etc.).
a) How could a quantitative approach to decision making be used to solve this problem?
b) What would be the uncontrollable inputs?
c) What would be the decision variables of the mathematical model? the objective
function? the constraints?
d) Is the model deterministic or stochastic?
e) Suggest assumptions that could be made to simplify the model.
SOLUTION 1
a) A quantitative approach to decision making can provide a structured way to determine
the minimum project completion time based on the activities' normal times and then
based on the activities' expedited (reduced) times.
b) Normal and expedited activity completion times; activity expediting costs; funds available
for expediting; precedence relationships of the activities.
c) Decision variables--which activities to expedite and by how much, and when to start each
activity; objective function--minimize project completion time; constraints--do not violate
any activity precedence relationships and do not expedite in excess of the funds
available.
d) Stochastic--activity completion times, both normal and expedited, are uncertain and
subject to variation; activity expediting costs are uncertain; the number of activities and
their precedence relationships might change before the project is completed due to a
project design change.
e) Make the model deterministic by assuming normal and expedited activity times are
known with certainty and are constant. The same assumption might be made about the
other stochastic, uncontrollable inputs.
INTRODUCTION
7
PROBLEM 2
Consider a department store that must make weekly shipments of a certain product
from two different warehouses to four different stores.
a) How could a quantitative approach to decision making be used to solve this problem?
b) What would be the uncontrollable inputs for which data must be gathered?
c) What would be the decision variables of the mathematical model? the objective function?
the constraints?
d) Is the model deterministic or stochastic?
e) Suggest assumptions that could be made to simplify the model.
SOLUTION 2
a) A quantitative approach to decision making can provide a systematic way to determine a
minimum shipping cost from the warehouses to the stores.
b) Fixed costs and variable shipping costs; the demand each week at each store; the
supplies each week at each warehouse.
c) Decision variables--how much to ship from each warehouse to each store; objective
function--minimize total shipping costs; constraints--meet the demand at the stores
without exceeding the supplies at the warehouses.
d) Stochastic--weekly demands fluctuate as do weekly supplies; transportation costs could
vary depending upon the amount shipped, other goods sent with a shipment, etc.
e) Make the model deterministic by assuming fixed shipping costs per item, demand is
constant at each store each week, and weekly supplies in the warehouses are constant.
PROBLEM 3
An auctioneer has developed a simple mathematical model for deciding the starting bid
he will require when auctioning a used automobile. Essentially, he sets the starting bid at
seventy percent of what he predicts the final winning bid will (or should) be. He predicts the
winning bid by starting with the car's original selling price and making two deductions, one
based on the car's age and the other based on the car's mileage. The age deduction is
$800 per year and the mileage deduction is $.025 per mile.
a) Develop the mathematical model that will give the starting bid (B) for a car in terms of the
car's original price (P), current age (A) and mileage (M).
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CHAPTER 1
b) Suppose a four-year old car with 60,000 miles on the odometer is up for auction. If its
original price was $12,500, what starting bid should the auctioneer require?
c) The model is based on what assumptions?
SOLUTION 3
a) The expected winning bid can be expressed as:
P - 800(A) - .025(M)
The entire model is:
B = .7(expected winning bid) or
B = .7(P - 800(A) - .025(M)) or
B = .7(P) - 560(A) - .0175(M)
b) B = .7(12,500) - 560(4) - .0175(60,000) = $5460.
c) The model assumes that the only factors influencing the value of a used car are the
original price, age, and mileage (not condition, rarity, or other factors). Also, it is
assumed that age and mileage devalue a car in a linear manner and without limit. (Note,
the starting bid for a very old car might be negative.)
PROBLEM 4
A firm manufactures two products made from steel and has just received this month's
allocation of b pounds of steel. It takes a1 pounds of steel to make a unit of product 1 and it
takes a2 pounds of steel to make a unit of product 2. Let x1 and x2 denote this month's
production level of product 1 and product 2 respectively.
Denote by p1 and p2 the unit profits for products 1 and 2, respectively. The manufacturer
has a contract calling for at least m units of product 1 this month. The firm's facilities are
such that at most u units of product 2 may be produced monthly.
a) Write a mathematical model for this problem.
b) Suppose b = 2000, a1 = 2, a2 = 3, m = 60, u = 720, p1 = 100, p2 = 200. Rewrite the
model with these specific values for the uncontrollable inputs.
c) The optimal solution to (b) is x1 = 60 and x2 = 626 2/3. If the product were engines,
explain why this is not a true optimal solution for the "real-life" problem.
SOLUTION 4
a) The total monthly profit
= (profit per unit of product 1) x (monthly production of product 1)
+ (profit per unit of product 2) x (monthly production of product 2)
= p1x1 + p2x2.
INTRODUCTION
9
The total amount of steel used during monthly production
= (steel per unit of product 1) x (monthly production of product 1)
+ (steel per unit of product 2) x (monthly production of product 2)
= a1x1 + a2x2.
This quantity must be less than or equal to the allocated b pounds of steel:
a1x1 + a2x2 < b.
The monthly production level of product 1 must be greater than or equal to m:
x1 > m.
The monthly production level of product 2 must be less than or equal to u:
x2 < u.
The production level for product 2 cannot be negative:
x2 > 0.
Thus, the model is:
MAXIMIZE p1x1 + p2x2
s.t.
a1x1 + a2x2 < b
x1
>m
x2 < u
x2 > 0
b) Substituting, the model is:
MAXIMIZE 100x1 + 200x2
s.t.
2x1 +
x1
3x2 < 2000
> 60
x2 < 720
x2 >
0
c) One cannot produce and sell 2/3 of an engine. Thus the problem is further restricted by
the fact that both x1 and x2 must be integers. They could remain fractions if it is
assumed these fractions are work in progress to be completed the next month.
PROBLEM 5
Ponderosa Development Corporation (PDC) is a small real estate developer operating in
the Rivertree Valley. It has seven permanent employees whose monthly salaries are given
in the table below:
10
CHAPTER 1
Employee
President
VP, Development
VP, Marketing
Project Manager
Controller
Office Manager
Receptionist
Monthly Salary
$10,000
6,000
4,500
5,500
4,000
3,000
2,000
PDC leases a building for $2,000 per month. The cost of supplies, utilities, and leased
equipment runs another $3,000 per month. PDC builds only one style house in the valley.
Land for each house costs $55,000 and lumber, supplies, etc. run another $28,000 per
house. Total labor costs are figured at $20,000 per house. The one sales representative of
PDC is paid a commission of $2,000 on the sale of each house. The selling price of the
house is $115,000.
a) Identify all costs and denote the marginal cost and marginal revenue for each house.
b) Write the monthly cost function c(x), revenue function r(x), and profit function p(x).
c) What is the monthly profit if 12 houses per month are built and sold? Solve algebraically
and using a spreadsheet.
d) What is the breakeven point for monthly sales of the houses? Solve algebraically and
using a spreadsheet.
SOLUTION 5
a) The monthly salaries total $35,000 and monthly office lease and supply costs total
another $5,000. This $40,000 is a monthly fixed cost. The total cost of land, material,
labor, and sales commission per house, $105,000, is the marginal cost for a house. The
selling price of $115,000 is the marginal revenue per house.
b) c(x) = variable cost + fixed cost = 105,000x + 40,000
r(x) = 115,000x
p(x) = r(x) - c(x) = 10,000x - 40,000
c) p(12) = 10,000(12) - 40,000 = $80,000 monthly profit.
Spreadsheet with formulas
1
2
3
4
5
6
7
8
9
A
PROBLEM DATA
Fixed Cost
Variable Cost Per Unit
Selling Price Per Unit
MODEL
Sales Volume
Total Revenue
Total Cost
Total Profit (Loss)
B
$40,000
$105,000
$115,000
=B4*B6
=B2+B3*B6
=B7-B8
INTRODUCTION
Spreadsheet with solution
1
2
3
4
5
6
7
8
9
A
PROBLEM DATA
Fixed Cost
Variable Cost Per Unit
Selling Price Per Unit
MODEL
Sales Volume
Total Revenue
Total Cost
Total Profit (Loss)
B
$40,000
$105,000
$115,000
12
$1,380,000
$1,300,000
$80,000
d) r(x) = c(x) or 115,000x = 105,000x + 40,000. Solving, x = 4.
Determining the break-even point using Excel’s Goal Seek tool
Step 1: Select the Tools pull-down menu
Step 2: Choose the Goal Seek option
Step 3: W hen the Goal Seek dialog box appears:
Enter B9 in the Set cell box
Enter 0 in the To value box
Enter B6 in the By changing cell box
Click OK
1
2
3
4
5
6
7
8
9
A
PROBLEM DATA
Fixed Cost
Variable Cost Per Unit
Selling Price Per Unit
MODEL
Sales Volume
Total Revenue
Total Cost
Total Profit (Loss)
B
$40,000
$105,000
$115,000
4
$460,000
$460,000
$0
11
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CHAPTER 1
ANSWERED PROBLEMS
PROBLEM 6
Zipco Printing operates a shop that has five printing machines. The machines differ in
their capacities to perform various printing operations due to differences in the machines'
designs and operator skill levels. At the start of the workday there are five printing jobs to
schedule. The manager must decide what the job-machine assignments should be.
a) How could a quantitative approach to decision making be used to solve this problem?
b) What would be the uncontrollable inputs for which data must be collected?
c) Define the decision variables, objective function, and constraints to appear in the
mathematical model.
d) Is the model deterministic or stochastic?
e) Suggest some simplifying assumptions for this problem.
PROBLEM 7
Zizzle Company is a new small local company that is about to manufacture Zizzle
briefcases in three styles. The company wants to determine how to use its resources most
efficiently to get the product mix that will maximize its profits.
One manager has advocated hiring an outside consulting firm to analyze sales
potentials and customer preferences. He suggests doing a complete time and motion study
of the production process and analyzing the potential for acquiring additional manpower and
material resources. In short, he is advocating an extremely accurate but complex study.
A second manager suggests a simplified model using "best guess", rough
approximations and simplifying assumptions as a starting point. The data for this model can
be obtained in a short period of time, and the model can be solved in-house at a fraction of
the cost of the more complex model.
a) Which manager's advice would you follow? Explain.
b) Would your answer change if Zizzle Company were a large national conglomerate which
plans to market tens of thousands of briefcases?
INTRODUCTION
13
PROBLEM 8
A client of an investment firm has $10,000 available for investment. He has instructed
that his money be invested in three stocks so that no more than $5,000 is invested in any
one stock but at least $1,000 is invested in each stock. He has further instructed the firm to
use its current data and invest in a manner that maximizes his expected overall gain during
a one-year period. The stocks, the current price per share, and the firm's projected stock
price a year from now are summarized in the following table.
Stock
James Industries
QM Inc.
Delicious Candy Co.
Current
Price
$25
$50
$100
Projected Price
1 Year Hence
$35
$60
$125
a) Let sj = the number of shares of stock j purchased for j = 1 (James), 2 (QM),
3 (Delicious). Formulate a mathematical model for this problem using these decision
variables.
b) Let xj = the number of dollars invested in stock j. Reformulate this mathematical model
in terms of these decision variables instead of those used in part (a).
c) If both models were solved using a quantitative approach to decision making, how would
you expect the results of the models to compare?
PROBLEM 9
Bank Guard Company provides security service for banks and savings and loan
companies during business hours. The number of guards supplied is a function of the
average number of people in the facility. One guard is provided for an average of 25
customers.
Let L = the average number of customers in the facility, Cg = the hourly cost
per guard, N = the number of guards required, and C = the daily cost for guard service
(based on an 8-hour day).
a) Develop a mathematical model for N in terms of L. Then develop a mathematical model
for C in terms of N and Cg. Finally, express C in terms of L and Cg.
b) If night guard service costs a flat $Cn, develop a mathematical model for 24 hour guard
service. Discuss the results with Cg = $10 per hour, Cn = $100 per night and L = 50.
Discuss the results with L = 65.
14
CHAPTER 1
PROBLEM 10
Comfort Plus Inc. (CPI) manufactures a standard dining chair used in restaurants. The
demand forecasts for quarter 1 (January-March) and quarter 2 (April-June) are 3700 chairs
and 4200 chairs, respectively. CPI has a policy of satisfying all demand in the quarter in
which it occurs.
The chair contains an upholstered seat that can be produced by CPI or a
subcontractor. The subcontractor currently charges $12.50 per seat, but has announced a
new price of $13.75 effective April 1. CPI can produce the seat at a cost of $10.25.
Seats that are produced or purchased in quarter 1 and used to satisfy demand in
quarter 2 cost CPI $1.50 each to hold in inventory.
a) What are the controllable inputs?
b) Develop a mathematical model for the total cost (objective) function.
PROBLEM 11
Continuing with problem 10, consider the following additional information. CPI's seatproducing capacity is 3800 seats per quarter. CPI cannot hold more than 300 seats in
inventory from quarter to quarter.
a) Complete the mathematical model started in problem 10 by modeling the constraints.
b) Is the model stochastic or deterministic?
c) How would adding a third quarter to the problem change the model?
PROBLEM 12
A retail furniture store has set aside 800 square feet to display its new 18th Century
Collection of sofas and chairs. Considering aisle space, it is estimated that each sofa
utilizes 50 sq. ft. and each chair utilizes 30 sq. ft. At least five sofas and at least five chairs
are to be displayed.
a) Write a mathematical model representing the store's constraints.
b) Suppose the profit on sofas is $200 and on chairs is $100. On a given day, the
probability that a displayed sofa will be sold is .03 and that a displayed chair will be
sold is .05. Mathematically model each of the following objectives:
1) Maximize the total pieces of furniture displayed.
2) Maximize the total expected number of daily sales.
3) Maximize the total expected daily profit.
INTRODUCTION
15
PROBLEM 13
Universal Computer is considering producing the new UC15 computer. Its components
cost $630 and it will take 1/2 hour to assemble and 1/4 hour to pack. Skilled workers
assemble the computer and make $16 per hour. Unskilled laborers pack the computers
and make $4 per hour. Packaging material cost $11 per computer.
The computer will be produced in a plant located on Harbor Blvd. and leased for
$3,100 per month. Utilities, supplies, insurance, etc. are expected to run another $1,900
per month regardless of the quantity of computers produced. Based on current market
conditions, Universal assumes it can sell at $700 each all units it produces.
a) Identify the variable and fixed costs for Universal. Let x = the monthly production
quantity of UC15s. Express the monthly cost function in terms of x.
b) Express the revenue function in terms of x.
c) What is the breakeven point?
d) What is the marginal cost of a computer?
e) What will be the monthly profit with a production schedule for UC15 computers of 500
per month?
16
CHAPTER 1
TRUE/FALSE
___ 14. The optimal solution to a mathematical model is always the policy that should be
implemented by the company.
___ 15. A problem to decide monthly shipping patterns is dependent on the amount of
product available at the factory. This amount can be modeled by a normal
distribution. Thus, this is a stochastic mathematical model.
___ 16. Microcomputers have made most quantitative approaches to decision making more
accessible to even moderate size firms.
___ 17. The three most commonly used quantitative approaches to decision making are
statistical analyses, simulation, and linear programming.
___ 18. A company seeks to maximize profit subject to limited availability of man-hours. Manhours is a controllable input.
___ 19. Past data might be used to test the validity of a mathematical model.
___ 20. A toy train layout designed to represent an actual railyard is an example of an analog
model.
___ 21. A feasible solution is one that satisfies at least one of the constraints in the problem.
___ 22. The terms stochastic and deterministic have the same meaning in quantitative
approaches to decision making.
___ 23. If you are deciding to buy either machine A, B, or C with the objective of minimizing
the sum of labor, material and utility costs, you are dealing with a multicriteria
decision.
___ 24. If your decision alternatives for dealing with increased product inventory are: (1) build
another warehouse, (2) reduce the production rate, and (3) increase the sales rate,
you are dealing with a multicriteria decision.
___ 25. Model development should be left to quantitative analysts; the model user's
involvement should begin at the implementation stage.
___ 26. The volume that results in marginal revenue equaling marginal cost is called the
break-even point.
___ 27. In quantitative analysis, the optimal solution is the mathematically-best solution.
___ 28. Problem solving is narrower in scope than decision making.