CHAPTER 16

APPLICATION OF QUANTITATIVE TECHNIQUES IN PLANNING, CONTROL AND DECISION MAKING – I

Rationale in Using Quantitative Techniques

Businessmen are continually faced with making important decisions which consider a number of variables
namely:

(a) The factors that are relevant to the problem they face
(b) The various alternatives that are available to them
(c) The logical consequences of the possible alternatives
(d) The best alternative in terms of profits and realization of objectives.

Increasingly, management accountants are devoting more attention to providing management and other
interested parties with data they can use in cost control, planning and decision making rather than
emphasizing cost accumulation and determination. Emphasis is now given on the predictive ability of data
rather than solely emphasizing the past. Accountants are now integrating planning and control models
within the accounting system for monitoring actual results against plans to provide feedback for corrective
action.

Accountants must adapt to the change and equip themselves with a background in mathematical methods
if they are to supply the kind of information management currently demands.

The quantitative analysis is a problem-solver which attempts to formulate the decision problem in
mathematical terms.

A mathematical model requires (1) specification of a complete list of variable factors that are relevant to
the problem at hand and (2) specific quantifiable relationships among those variables. The model can help
the decision maker in making a which is compatible with his goals as well as enabling him to consider the
variables which are relevant in making an appropriate decision.

The accountant must concern also himself with the decision-making process since he is the one who must
design the accounting information system. To be effective in this role, the accountant must acquaint
himself with the objectives, assumptions and requirement of the decision models employed. A decision
model is not intended to be the sole basis for making a decision. Rather, it is a tool that the decision maker
uses in addition to other inputs in arriving at a decision. The use of a decision model helps to ensure that
the alternatives are logically evaluated in the light of a specific criterion and explicit assumptions

The more commonly used quantitative models for planning, control and decision making are as follows:

1. Probability

2. Payoff (Decision) Tables

3. Value of Perfect Information

4. Decision Tree
5. Learning Curve

6. Simulation Technique

7. Monte Carlo Technique

8. Sensitivity Analysis

9. Queuing

10. Linear Programming

11. Program Evaluation and Review Technique Critical Path Method (PERT-CPM)

12. Gantt Chart

13. Inventory Modeling

14. Regression. Analysis (discussed in Chapter 11)

15. Present Value Concept (will be discussed in Chapter 19)

An exhaustive treatment of the above-mentioned quantitative techniques is beyond the scope or intent
of this Chapter. Rather a fair understanding and appreciation of the concepts are expected from the
reader.

PROBABILITY

Decision Making under Certainty

Decision making under certainty means that for each decision action there is only one event and therefore
only a single outcome for each action. When an event is certain, there is a 100% chance of occurrence,
hence the probability is 1.0.

Decision Making under Uncertainty

Decision making under uncertainty, which is more common in reality, involves several events for each
action with its probability of occurrence. The decision maker may know the probability of occurrence of
each of the events because of mathematical proofs or the compilation of historical evidence. In the
absence of these two bases, he may resort to the subjective assignment of probabilities.

Management may know enough about the likelihood of each environment to attach probabilities of
occurrence to each alternative. If so, management certainly wants to select the alternative that appears
to produce the largest outcome as long as that alternative does not expose the company to a high
probability of a large loss. The payoffs can be reduced using each alternative to one figure by weighing
the possible payoffs according to the relative probabilities that the various conditions will occur.

Payoff is the value assigned to different outcomes from a decision and may be positive or negative.
Briefly, information is deemed to meet the cost-benefit test if the expected value of a decision (net of the
costs of the information) increases as a result of obtaining additional information. The process in deciding
whether the cost benefit criterion has been met is called information economics.

Assigning Probabilities

Because decision makers normally deal with uncertainty, rather than certainty, they must estimate the
probability of various outcomes. It is necessary to assign probabilities that represent the likelihood of
various events occurring. A probability distribution describes the chance or likelihood of each of the
collectively exhaustive and mutually exclusive set of events. The probability distribution can be based on
past data if management believes that the same forces will continue to operate in the future. Probability
provides a method for mathematically expressing doubt or assurance about the occurrence of a chance
event. The probability of an event varies from 0 to 1.

(a) probability of 0 means the event cannot occur, whereas a probability of 1 means the event is
certain to occur.
(b) A probability between 0 and I indicates the likelihood of the event's occurrence, eg, the probability
that a fair coin will yield heads is 0.5 on any single toss.

Types of Probabilities

a. Objective probabilities are calculated from either logic or actual experience. For example, in rolling dice
one would logically expect each face on a single die to be equally likely to turn up at a probability of 1/6.
Alternatively, the die could be rolled a great many times, and the fraction of times each face turned up
could then be used as the frequency or probability of occurrence.

b. Subjective probabilities are estimates, based on judgment and past experience, of the likelihood of
future events. Weather forecasts often include the subjective probability of rain. In business, subjective
probability can indicate the degree of confidence a person has that a certain outcome will occur, e.g.,
future performance of a new employee,

Basic Terms Used with Probability

(1) Two events are mutually exclusive if they cannot occur simultaneously (e.g heads and tails cannot both
occur on a single toss of a coin).

(2) The joint probability for two events is the probability that both will occur.

(3) The conditional probability of two events is the probability that one will occur given that the other has
already occurred.

(4) Two events are independent if the occurrence of one has no effect on the probability of the other (e.g.,
rolling two dice).

(a) If one event has an effect on the other event, they are dependent.

(b) Two events are independent if their joint probability equals the product of their individual
probabilities.

(c) Two events are independent if the conditional probability of each event equals its unconditional
probability.
Rules in Combining Probabilities

The joint probability for two events equals the probability (Pr) of the first event multiplied by the
conditional probability of the second event, given that the first has already occurred.

Example: If 60% of the students at a university are male, Pr (male) is 6/10. If 1/6 of the male students have
a B average, Pr(B average given male) is 1/6. Therefore, the probability that any given student (male or
female) selected at random, is both male and has a B average is

Pr(male) x Pr(B\male) Primale B)

6/10 X 1/6= 1/10

Pr(male B) is 0.10; that is, the probability that the student is male and has a B average is 10%.

The probability that either one or both of two events will occur equals the sum of their separate
probabilities minus their joint probability.

Example: If two fair coins are thrown, the probability that at least one will come up heads is Pr(coin #1 is
heads) plus Pr(coin #2 is heads) minus Pr(coin #1 and coin #2 are both heads), or

(0.5)+(0.5) (0.5 x 0.5) = 0.75

Example: If in the earlier example 1/3 of all students, male or female, have a B average [Pr(B average) is
1/3], the probability that any given student either is male or has a B average is

Pr(male) + Pr(B avg)- Pr(B male) = Pr(male or has B avg.)

6/10 + 1/3 - 1/10 = 25/30

The term Pr(B male) must be subtracted to avoid double counting those students who belong to both
groups.

The probabilities for all possible mutually exclusive outcomes of a single experiment must add up to one.
Probability Distributions

A probability distribution specifies the values of the variables and their respective probabilities. Certain
standard distributions seem to occur frequently in nature and have proven useful in business.

Discrete distributions include the following:

(1) Uniform distribution. All outcomes are equally likely, such as the flipping of one coin, or even of two
coins, as in the example above.

(2) Binomial distribution. Each trial has only two possible outcomes, e.g., accept or reject, heads or tails.
This distribution shows the likelihood of each of the possible combinations of trial results. It is used in
quality control.

(3) The Bernoulli distribution involves only one trial, whereas the binomial distribution deals with as many
as necessary. Thus, the binomial distribution reduces to the Bernoulli distribution when n is 1.

(4) The hypergeometric distribution is similar to the binomial distribution. It is used for sampling without
replacement.

(a) For finite populations, sampling without replacement removes each item sampled from the
population, thus changing the composition of the population from trial to trial.

(b) For large populations and small samples, the binomial distribution approximates the hypergeometric
distribution and is computationally more convenient.

(5) The Poisson distribution is useful when the event being studied may happen more than once with
random frequency during a given period.

Continuous distributions include the following:

(1) Normal distribution. The most important and useful of all probability distributions, it describes many
physical phenomena. In sampling, it describes the distribution of the sample mean regardless of the
distribution of the population. It has a symmetrical, bell-shaped curve centered about the mean. For the
normal distribution, about 68% of the area (or probability) lies within plus or minus I standard deviation
of the mean, 95.5% lies within 2 standard deviations, and 99% lies within 3 standard deviations of the
mean.

(2) The exponential distribution is related to the Poisson distribution. It is the probability of zero
occurrence in a time period T.

(3) The t-distribution (also known as Student's distribution) is a special distribution used with small
samples of the population, usually less than 30, with unknown population variance.

(a) For large sample sizes (n> 30), the t-distribution is almost identical to the standard normal distribution

(b) For small sample sizes (n<30) for which only the sample standard deviation is known, the t-distribution
provides a reasonable estimate for tests of the population mean if the population is normally distvariance
(c) The t-distribution is useful in business because large samples are often too expensive. For a small
sample, the t-statistic (from a t table) provides a better estimate of the variance than that from a table for
the normal distribution.

(4) The chi-square distribution is used in testing the goodness of fit between actual data and the
theoretical distribution. In other words, it tests whether the sample is likely to be from the population,
based on a comparison of the sample variance and the population variance.

Illustrative Problem 16.1. Decision Making under Uncertainty

Payong Corporation is considering two new colors for their umbrella products Sky Blue and Baby Pink.
Either can be produced using the present facilities Each product requires an increase in annual fixed costs
of P400,000. The products have the same selling price of P100 and the same variable costs per unit of P80.

After studying past experience with similar products, management has prepared the following probability
distribution:

Management would like to know

a. The break-even point for each product.

b. Which product should be chosen, assuming the objective is maximize expected operating income?

Solution:

a Since both products have the same contribution margin per unit of P20 (P100-P80)

break-even point for each product will be the same computed as follows:

Break-even point = P400,000/P20 = 20.000 units

PAYOFF (DECISION) TABLES

Payoff (decision) tables are helpful tools for identifying the best solution given several decision choices
and future conditions that involve risk.

A payoff table presents the outcomes (payoffs) of specific decisions when certain states of nature (events
not within the control of the decision maker) occur.

Example: A dealer in luxury yachts may order 0, 1, or 2 yachts for this season's inventory. The cost of
carrying each excess yacht is P50,000, and the gain for each yacht sold is P200,000. The situation may be
described by a payoff table as follows:
EXPECTED VALUE OF PERFECT INFORMATION

Even though the P210,000 expected value of operating income of the Baby Pink umbrellas is higher than
that of the Sky Blue umbrellas, management and/or owners may resist exposure to the percentage
involved in making a decision under risk. The probabilities associated with which environmental
conditions will actually occur are based on existing information. The company may decide to hire
marketing analysts to obtain additional information on the environmental situation.

The expected value of perfect information is the amount the company is willing to pay for the market
analysts' errorless advice. Assuming that the market analyst could indicate with certainty which condition
would occur, a manager would decide with complete certainty. Of course, "perfect information" is not
perfect in the sense of absolute predictions.

Perfect information is the knowledge that a future state of nature will occur with certainty, i.e., being
sure of what will occur in the future.
The expected value of perfect information (EVPI) is the difference between the expected value without
perfect information and the return if the best action is taken given perfect information.

Example (from the yacht dealer problem on pages 674 to 675): If the yacht dealer were able to poll all
potential customers and they truthfully stated whether they would purchase a yacht this year (i.e., if
perfect information about this year's yacht sales could be purchased), what is the greatest amount of
money the dealer should pay for this information? What is EVPI?

If the dealer had perfect knowledge of demand, he/she would make the best decision for each state of
nature. The cost of the other decisions is the conditional cost of making other than the best choice. This
cost may be calculated by subtracting the expected value from the expected value given perfect
information. This difference measures how much better off the decision maker would be with perfect
information. From the payoff table on page 675, we find the expected value of the best choice under each
state of nature.

DECISION TREE

Underlying Concept

A decision tree is an analytical tool used in a problem in which a series of decision has to be made at
various time intervals, with each decision influenced by the information that is available at the time it is
made.

In its simplest form, a decision tree is a diagram that shows the several decisions or acts and the possible
consequences called events of each act. In a more elaborate form, the probabilities and the revenue and
costs of each event's outcome are estimated and these are combined to give an expected value for the
event.

Decision trees provide a systematic framework for analyzing a sequence of interrelated decisions the
managers may make over time. Stemming from the present investment decisions are alternative scenarios
that depend on the occurrence of future events and consequences of those events. Decision tree analysis
encourages the study and understanding of these scenarios.

Advantages of Decision Tree Analysis

Some benefits that may be derived from the use of Decision Tree Analysis are

1. Decision tree is an effective means of presenting the relevant information needed by management in
an investment problem. Such relevant information includes choices, risks, monetary gains, and objectives.

2. Combination of action choices with different events or results of action that chance or other
uncontrollable circumstances partially affect can be better presented and studied.

3. The interactions of the impact of future events, decision alternatives uncertain events and their possible
payoffs can be shown with greater ease and clarity.

4. Data are presented in a manner that enables systematic analysis and better decisions.

Limitations of Decision Tree Analysis

1. A decision tree does not give management the answers to an investment problem.

2. It does not identify all the possible events or does it list all the decisions that must be made on a subject
under analysis.

3. The interactions of such decision with the objective of other parts of the business organization would
be too complicated to compute manually. The use of computers will be suitable when studying the effect
of variations in figures and/or the events involved.

4. Decision tree analysis treats uncertain alternatives as if they were discrete well-defined possibilities. It
should be remembered that uncertain situations depend not only on one variable but on several
independent or partially related variables subject to such chance influences.

Steps in Making a Decision Tree

The requirements for the decision tree preparation are

1. Identification of the points and decision and the alternatives available at each point.

2. Determination of the points of uncertainty and the type or range of alternative outcomes at each point.

3. Estimates of the probabilities of different events or results of actions.

4. Estimates of the costs and gains of various events and actions.

5. Analysis of the alternative values in choosing a course of action.

Because the time between successive decision stages on a decision tree may be long, it would be more
realistic to consider the time value of future earnings and other cash flows. In others the discounted cash
flow approach may be applied on analyzing various investment alternatives. This is discussed extensively
in Chapter 19.

Illustrative Problem 16.2. Preparation of Decision Tree

After several years of supplying gelatin bars to several supermarket chains, the Castillo family decided it
was time to shift to another venture in the light of increasing competition from other gelatin
manufacturers. They would want to introduce a type of gelatin ice cream.

Based on the family accountant's estimates, if the sales are high, the total contribution margin will be
P300,000. If sales are low, the total contribution margin will be P50,000. Fixed costs will be P150,000. The
accountant attaches a probability of 0.5 for high sales and 0.5 for low sales.

The Castillo family can conduct a survey of various health food outlets to determine the true demand for
the new product. The reliability of the survey is such that it will signal high sales 70 percent of the times
when actual sales will be high, and signal low sales 90 percent of the time when actual sales will be low.
The costs of such a survey are P20,000.

Assuming that the Castillo family bases its decisions on expected value:

a) What action will they take without the survey?

b) Should the Castillo family take the survey? What should their decision be?

c) How much will they be willing to pay for perfect information?

LEARNING CURVE

Learning curves reflect the increased rate at which people perform tasks as they gain experience. The time
required to perform a given task becomes progressively shorter, but this technique is only applicable to
the early stages of production or of any new task.

Ordinarily, the curve is expressed as a percentage of reduced time to complete a task for each doubling
of cumulative production. Research has shown learning curve percentages to be between 60% and 80%.
In other words, the time required is reduced by 20% to 40% each time cumulative production is doubled,
with 20% being common.

One common assumption made in a learning curve model is that the cumulative average time per unit is
reduced by a certain percentage each time production doubles.
Example: Given an 80% learning curve model based on the first assumption stated above, the following
performance is expected during the early stages of the manufacture of a new product:

If the average time for 100 units in the example above were 3 minutes per unit, the total time would be
300 minutes. At an average time of 2.4 minutes for 200 units, the total time would be 480 minutes. In
other words, the additional 100 units required only 180 minutes (480-300), or 1.8 minutes per unit.

SIMULATION TECHNIQUES

Simulation is a technique for experimenting with logical and mathematical models using a computer.
Despite the power of mathematics, many problems cannot be solved by known analytical methods
because of the behavior of the variables and the complexity of their interactions, e.g..

a. Corporate planning models

b. Financial planning models

c. New product marketing models

d. Queuing system simulations

e. Inventory control simulations

Experimentation is neither new nor uncommon in business. Building a mockup of a new automobile,
having one department try out new accounting procedures, and test-marketing a new product are all
forms of experimentation. In effect, experimentation is organized trial and error using a model of the real
world to obtain information prior to full implementation.

Models can be classified as either physical or abstract.

(a) Physical models include automobile mockups, airplane models used for wind-tunnel tests, and
breadboard models of electronic circuits.

(b) Abstract models may be pictorial (architectural plans), verbal (a proposed procedure), or logical-
mathematical. Experimentation with logical-mathematical models can involve many time-consuming
calculations. Computers have eliminated much of this costly drudgery and have led to the growing interest
in simulation for management.

The simulation procedure has five steps:

a. Define the objectives. The objectives serve as guidelines for all that follows. The objectives may be to
aid in the understanding of an existing system (e.g., an inventory system with rising costs) or to explore
alternatives (e.g., the effect of investments on the firm's financial structure). A third type of objective is
estimating the behavior of some new system such as a production line. Thus, a simulation can be designed
to ask "what-if" questions, such as whether modifying the actual system will result in better performance.

b. Formulate the model. The variables to be included, their individual behavior, and their
interrelationships must be defined in precise logical mathematical terms. The objectives of the simulation
serve as guidelines in deciding which factors are relevant. Moreover, inputs reflected in the model are of
two kinds: controllable and probabilistic. The former are those subject to the decision-makers' influence,
and the latter involve circumstances beyond their control, such as general economic conditions or the
acts of competitors.

c. Validate the model. Some assurance is needed that the results of the experiment will be realistic. This
assurance requires validation of the model-often using historical data. If the model gives results
equivalent to what actually happened, the model is historically valid. Some risk remains, however, that
changes could make the model invalid for the future.

d. Design the experiment. Experimentation is sampling the operation of a system. For example, if a
particular policy is simulated on an inventory model for two years, the results are a single sample. With
replication, the sample size can be increased and the confidence level raised. The number of runs to be
made, length of each run, measurements to be made, and methods for analyzing the results are all part
of the design of the experiment. The experiments also may take the form of asking "what-if" questions,
that is, varying an input or assumption to ascertain the effect on the results.

e. Conduct the simulation-evaluation results. The simulation should be conducted with care. The results
are analyzed using appropriate statistical methods. These results constitute outcomes that permit
evaluation of the probabilities of real-world performance.

Advantages and Limitations of Simulation

The advantages of simulation are as follows:

a. Time can be compressed. A corporate planning model can show the results of a policy for 5 years into
the future, using only minutes of computer time.

b. Alternative policies can be explored. With simulations, managers can ask what-if questions to explore
possible policies, providing management with a powerful new planning tool.
c. Complex systems can be analyzed. In many cases, simulation is the only possible quantitative method
for analyzing a complex system such as a production or inventory system, or the entire firm.

The limitations of simulation are as follows:

a. Cost. Simulation models can be costly to develop. They can be justified only if the information to be
obtained is worth more than the costs to develop the model and carry out the experiment.

b. Risk of error. A simulation results in a prediction of how an actual system would behave. As in
forecasting, the prediction may be in error.

MONTE CARLO TECHNIQUE

The Monte Carlo technique is often used in simulation to generate the individual values for a random
variable. A random number generator is used to produce numbers with a uniform probability distribution
(equal likelihoods of occurrence). The second step is to transform these numbers into values consistent
with the desired distribution.

The performance of a quantitative model may be investigated by randomly selecting values for each of
the variables in the model (based on the probability distribution of each variable) and then calculating the
value of the solution. If this process is performed a large number of times, the distribution of results from
the model will be obtained.

Example: A new marketing model includes a factor for a competitor's introduction of a similar product
within I year. Management estimates a 50% chance that this event will happen. For each simulation, this
factor must be determined, perhaps by flipping a coin, or by putting two numbers in a hat and selecting
one number. Random numbers between 0 and 1 could be generated. Numbers under 0.5 would signify
introduction of a similar product; numbers over 0.5 would indicate the nonoccurrence of this event.

SENSITIVITY ANALYSIS

Sensitivity analysis describes how sensitive the linear programming optimal solution is to a change in any
one number. Sensitivity analysis answers what-if questions about the effect of change in prices or variable
costs; changes in value; addition or deletion of constraints, such as available machine hours; and changes
in industrial coefficients, such as the labor-hours required in manufacturing in a specific unit.

After a problem has been formulated into any mathematical model, it may be subjected to sensitivity
analysis. This approach is especially useful and significant when probabilities of states of nature and
decision payoffs are derived subjectively rather than by using objectively quantifiable information.

A trial-and-error method may be adopted in which the sensitivity of the solution to changes in any given
variable, parameter, or other assumption is calculated.

(1) The risk of the project being simulated may also be estimated.

(2) The best project may be one that is least sensitive to changes in probabilistic (uncertain) inputs.

(3) A sensitivity analysis may indicate whether expending additional resources to obtain better forecasts
of future conditions is cost justified.
In linear programming problems, sensitivity is the range within which a constraint value, such as a cost
efficient or any other variable, may be changed without changing the optimal solution. Shadow price is
the synonym for sensitivity in that context.

In the application of discounted cash flow methods (e.g., net present value), a sensitivity analysis might
be performed to ascertain the effects of variability of the discount rate or periodic cash flows.

Financial planning models, including those for cash flows and capital budgeting, are other significant
applications of sensitivity analysis. For example, changes in selling, prices or resource costs may affect
available cash and require more or less short-term borrowing.

Still another application is the calculation of the margin of safety in a CVP analysis.

QUEUING

Queuing is often referred to as Waiting line analysis. Queuing can occur in many situations and in a wide
variety of forms. However, every queuing problem involves four basic issues.

1) Input mechanisms

2) Line or queue discipline

3) Service facilities, and

4) Output

The characteristics of these four items describe the queuing model of concern.

With adequate data on the population to be serviced, the queuing and service characteristics, analytic
solution of the queuing problem may be feasible. Many queuing problems are so complex that simulations
must be used to "solve" them. The mathematical deviation of systems is moderately complex even for the
simpler queuing problems and is beyond the scope of this Chapter.

CHAPTER 17

APPLICATION OF QUANTITATIVE TECHNIQUES IN PLANNING, CONTROL AND

DECISION MAKING - II

LINEAR PROGRAMMING

Nature and Applications

Linear programming is mathematical technique that permits the a determination of the best or
optimum use of the available resources, namely, money, personnel, materials, facilities and
time. It is a valuable aid to management because it provides a systematic and efficient
procedure which can be used as a guide in decision making. Managers are often faced with
problems of selecting the most profitable or least costly way to use available resources. The
advantage of linear programming is its applicability to many types of problems and its
usefulness for sensitivity analysis. A major disadvantage is the restrictiveness of its linear
assumptions, for example, that all costs are variable or fixed.

Example: A manufacturer should minimize production costs while satisfying. production

requirements, maintaining required inventory levels, staying within production capacities, and
using available employees. The objective function is the production cost to be minimized; the
constraints are production requirements, inventory levels, production capacity, and available
employees.

This technique may be used to maximize revenue, contribution margin or profit function, or to
maximize a cost function subject to constraints. applications include: Specific

1) Determination of the product mix to maximize contribution margin. 2) Selection of an

investment mix.

3) Determination of materials mix to minimize costs. 4) Assignment of jobs to machines.

5) Determination of transportation routes.

Steps in the Formulation of Linear Program

1. The first step is to define the decision variables.

2. The next step is to express the objective function and constraints in terms of these decision
variables.

Computational Methods of Linear Programming

To solve Linear Programming problems, several solution methods are available and they
include:

1) Graphic Method

2) Algebraic Method 3) Simplex Method

The graphical method, although the easiest technique, is limited to problems with two variables.

The algebraic method is a trial-and-error technique. Pairs of constraints are

solved algebraically to find their intersection. The values of the decision

variables are then substituted into the objective function and compared to find

the best combination.

(1) The basic rule is that the optimal solution will be at the intersection of two or more constraint
equations.

(2) Thus, all intersections can be computed and each solution evaluated in the objective
function to determine which solution is optimal.

The simplex method is the technique most commonly used to solve linear prógramming
problems. It is an algorithm to move from one corner solution to a better corner solution. When a
better solution cannot be found, the optimal solution has been reached.

(1) The simplex method relies on an area of mathematics called matrix algebra. The equations
that form the constraints are arranged in a matrix of coefficients and manipulated as a group
with matrix algebra.

(2) Almost all practical applications of linear programming require the use of computers. Most
computer facilities have a linear programming package that uses the simplex algorithm to find
the optimal solution.

Graphic Method

When a linear programming problem involves only two variables, a two dimensional graph can
be used to determine the optimal solution. The steps are:

1) Determine the objective function.

2) Determine basic relationships.

3) Compute the optimum solution in the situation, especially the constraints as follows:

a) Change inequalities to equalities.

b) Graph the equalities.

c) Identify the correct side for the original inequalities.

d) Identify the area of feasible solution. Feasible solutions are values of decision variables that
satisfy all the constraints simultaneously.

e) Determine the contribution margin at all the corners in the feasible region.

Illustrative Problem 17.1. Linear Programming - Graphic Method applied to Product Mix Problem
A firm produces two products, A and B. Product A requires two hours of grinding and four hours
of polishing. Product B requires five hours of grinding and two hours of polishing. The
manufacturer has 3 grinders and 2 polishers; therefore in a 40-hour week there are 120 hours of
grinding capacity and 80 hours of polishing capacity.

There is a ready market for both products and the contribution margin per unit of Product A and
Product B are P3 and P4, respectively.

To maximize the total contribution margin, the management must decide on

1) the allocation of the available production capacity to Products A and B,

and 2) the number of units of each product to produce.

Solution:

a) Definition of decision variables:

Let A = the number of units of Product A to be produced. B = the number of units of Product B
to be produced.

b) Objective function:

TotalCM = P * 3A + P * 4B

c) Explicit Constraints:

2A + 5B <= 120

4A + 2B <= 80

Implicit Constraint:

A, B >= 0

The LP model is therefore

Maximize: TotalCM = P * 3A + P * 4B

Subject to:

2A + 5B <= 120

4A + 2B <= 80
A, B <= 0

d) Prepare the graph with A axis representing the number of units of product A to be produced
and B axis representing the number of units of product B to be produced.

e) Plot all the constraints on the graph by connecting the points that represent the extremes of
production of each product. These points are:

Grinding:

If overline A = 0 ; B = 24 B =0: A-60

Polishing:

If overline ^ = 0 B = 40 B =0: A = 20
Answer:

Produce 10 units of A and 20 units of B. The total contribution margin associated with this
combination is P110 which is the highest among the four combinations.

Algebraic Method

The combination of A and B that maximizes the objective function will occur at one of the
extreme points or corners of the feasible region.

Simultaneously solving the constraints intersecting at those corners and substituting into the
objective function.

At corner B, the two constraints intersect.

4A + 2B = 80Eql

2A + 5B = 120

Eq 2

4A + 2B = 80

4A + 10B = 240

Eq 3 (Eq 2 * 2 )

8B = 160

B = 20

Substituting the value of B in Eq 1.

4A + 2(20) = 80

4A + 40 = 80

A = 40/4

Simplex Method

A
10

The simplex method is an iterative, stepwise process which approaches an optimum solution in
order to reach an objective function of maximization or minimization. Each iteration or pivot
involves the removal of one old variable from the product mix (the basis) and the addition of one
new variable to find an improved solution - that is, higher profit for maximization problems or
lower of cost for minimization problems. This process is continuously applied until the optimal
solution is reached.

Simplex Method:

Illustrative Problem 17.2. Linear Programming Maximization of Profit

Using the data in Illustrative Problem 18.1 where the linear program has been set

as:

Objective Function

Maximize: Total CM =P 3A +P4B

Subject to:

2A + 5B <= 120

2A + 2B <= 80

A, B >= 0

REQUIRED: Find the best product combination using the simplex method.

Solution:

Step 1. The set of inequalities must be transformed into a set of equations by introducing slack
variables, S_{1} and S_{2} The use of slack variables involves the addition of an arbitrary
variable to one side of the inequality, transforming it into an equality. S_{1} represents the slack
variable or idle capacity for the grinding department while S_{2} represents the slack variable or
idle capacity for the polishing department. The inequalities as equalities are:

2A + SB + S_{1} + 0S_{2} = 120 4A + 2B + O*S_{1} + S_{2} = 80

The objective equation becomes:

Maximize: Total CM = 3A + 4B + 0S_{1} + 0S_{2}

Step 2. Set up the first simplex tableau or first solution as follows:

In this first tableau, the slack variables were introduced into the variable column to find an initial
feasible solution to the problem. However, even if this approach satisfies the constraints, it is
undesirable since the resulting contribution margin (Z) is zero. It is a rule of the simplex method
that the optimum solution for maximization problem has not been reached if the index row still
carries positive values at the completion of the iteration.

Hence, to improve the solution, Table II is set up. To determine the outgoing row from Table I
the following computations are made:

Si shows the lowest positive ratio and should be replaced.

To determine the incoming variable, the variable in the index row of Table I showing the highest
positive value should be chosen.

Step 3. Set up Table II

There are no positive figures anymore in the index row of Table III. This indicates that any
further substitutions will not result in an increase in the contribution margin; hence, the optimum
solution has been obtained.

The optimum strategy is therefore, to produce and sell 10 units of Product A and 20
units of Product B for a contribution margin of P1 10.

Shadow Prices
If a decision maker who has used the LP technique wishes to know whether it pays to add
capacity in hours in a particular department, he can compute for the shadow price. A shadow
price is the opportunity cost or the monetary value of the contribution margin that would be lost
by not adding an additional hour of capacity. To decide in favor of a short-term capacity
expansion, the decision maker must be sure that the shadow price (or opportunity cost) exceeds
the actual price of that expansion.
The steps in computing the shadow prices are:
1) Add one hour (preferably, more than an hour to make it easier to show graphically) to the
constraint under consideration.
2) Resolve the problem and find the maximum CM.
3) Compute for the shadow price which is the difference between CM of
the original LP problem and the CM determined in Step 2.

Using the data in illustrative problem where the best production combination is 10 units of A and
20 units of B, compute the shadow prices of the grinding department of publishing department.

The shadow price of the Grinding Department is also determinable by referring to the Index Row
of Table III in the Solution to Illustrative Problem 17.2. As can be seen, Column S, (slack
variable for Grinding) shows a final figure of P0.625.
The shadow price of the Polishing Department as seen in the same Table II is P0.4375.
Illustrative Problem 17.3. Linear Programming - Graphic Method:

Minimization of cost.
The AZB Products, Inc. produces a chemical solution used for cleaning carpets. This chemical
is made from a mixture of two other chemicals which contain cleaning agent X and cleaning
agent Z. Their product must contain 175 units of agent X and 150 units of agent Z and weigh at
least 100 pounds. Chemical A costs P8 per pound, while chemical B costs P6 per pound.
Chemical A contains one unit of agent X and three units of agent Z. Chemical B contains seven
units of agent X and one unit of agent Z..

REQUIRED:
1. Set up the problem in a linear programming format.
2. Determine the best combination of the ingredients to minimize cost using the graphic method.
Solution:
1. Let A number of pounds of chemical A to be produced. B = number of pounds of
chemical B to be produced.
Objective Function:
Minimize: Total Cost = P8A+ P6B
Subject to:
(1) A+ 7B ≥ 175

(2) 3A+ B≥ 150

(3) A+ B≥ 100
A, B 2 0

If: (1) If A = 0; B = 25 B = 0; A175

(2) If A= 0; B 150 B 0; A 50
(3) If A 0; B -100 B=0; A 100
The basic feasible solutions occur at the corner points labeled a, b, c and d on the graph.
Computation of Total Costs associated with the basic feasible solutions:

PROGRAM EVALUATION AND REVIEW TECHNIQUES

PERT (Program Evaluation and Review Technique) is a systematic procedure for using network
analysis systems for planning, measuring progress against schedule, evaluating changes to
schedule, forecasting future progress, and predicting and controlling costs.

Basic Underlying Concept

Basic to any network analysis technique is the concept of the "arrow diagram" or the "network"
as it is commonly called. The network is a graphic model showing interdependencies between
various project activities by means of a simple charting technique. It is a logic diagram of a
project. In a network, the circles represent events and the lines connecting any two circles
represent activities. PERT Time analysis contains a sequence of arrows showing
interrelationships among activities with time being the basic element in these activities. The
activities are shown from left to right in the necessary order of their accomplishment. Workers
must complete all the activities leading to an event before the event occurs. This is illustrated in
Figure 17.1.
The activities in a project are related to each other in various ways. These relationships are
termed interdependencies. In Figure 17.2, Event 2 and 3 are both dependent on the occurrence
of Event 1. Event 4 cannot occur until activities 1-2, 1-3, and 3-4 have all been performed.

Expected Activity Time

After developing the network diagram, managers make an estimate of the time needed to
complete each activity. The expected activity time is computed using the weighted average of
the shortest time (optimistic time), the most likely (average) time, and the longest time
(pessimistic time). The formula for time estimations using the beta probability distribution is

te = to + 4 tm + tp / 6

te = expected time
to = optimistic time estimate
tn most likely time estimate
tp pessimistic time estimate

Concept of Critical Path

Each activity consumes resources and has a time dimension. In PERT, the activity duration is
specified in terms of three time estimates: most optimistic, most likely, and most pessimistic
durations. In any network, there is always at least one connected path that goes from the start
event to the end event. The path that takes the longest time to reach is called the Critical Path.
In Figure 17.2, the sum of the activity times are as follows:
Path 1-2-4: 4+ 10 = 14 weeks
Path 1-3-4: 12 + 8 = 20 weeks

The critical path of this network is identified as 1-3-4, having a total time requirement of 20
weeks. The critical path provides very vital planning and control information because it can
show that:
1) The project will take 20 weeks to complete.
2) If an activity slips while on the path, the end event will slip by a corresponding amount, thus
increasing the project duration.
3) The activities on this path are the most critical from a schedule
standpoint.
4) The management should focus its attention on the activities that fall on the critical path.

The activities that do not lie on the critical path have varying amount of "slack" time associated
with them. Slack means the length of time by which a particular activity can slip without having
any delaying effect on the end event. In Example 2, activities 1-2 and 2-4 have combined slack
of 6 weeks. Schedules in these activities can be "slipped" six weeks without delaying the
scheduled deadline for Event 4. Note that if this six-week slippage did occur, path 1-2-4 would
also become critical. Slack also introduces flexibility into the network because it serves as a
buffer for events not located in the critical path. When time lags appear on the critical path,
managers can transfer materials, labor and equipment to the problem areas. Managers however
must be alert to the effect of these transfers on other paths particularly if they would create new
problems on other paths.

Cost Estimating

Network scheduling systems calculate the start and end dates for the activities (or work
packages) from the estimated times (ie., activity durations) and the start of the beginning event.
Once the schedule is determined, each work package's cost is estimated. When this estimating
process is completed, a project time-phased budget is developed.

Illustrative Problem 17.4. Preparation of PERT-CPM Network

Following are a set of activities for a project, their predecessor restrictions, and three time
estimates as well as expected completion time.
2. The critical path is denoted on the flowchart by the arrows connecting 1-4-6-7 with total
estimated time of 19 weeks.
3. The following formula is used to compute the mean of the Beta probability distribution, which
is the expected time:

The expected time represents the average time an activity would require if it were repeated a
large number of time.

Crashing the Network

Information about slack time allows management not only to continually monitor a project's
status but also serve as a guide in rescheduling tasks to shorten the overall project's completion
time. Activity time may also be reduced by hiring more labor, overtime or acquiring more
equipment.

Crashing refers to the efforts designed to complete the project ahead of schedule. However,
when managers use crashing, the variable costs of the project will also increase. Crashing the
network means finding the minimum cost of completing the project in minimum time to achieve
an optimum trade-off between time and cost. Determining the appropriate trade-off is referred to
as PERT-Cost analysis."

PERT-Cost Network

PERT-Cost analysis is an extension of PERT-Time, and essentially a study of the cost-

effectiveness of crashing.

A simplified approach in applying PERT-Cost Analysis includes the following

steps:

1. Each activity is assigned two projected time estimates,

a) a normal time, and
b) a crash time.
2. Compute the incremental crash cost for each activity that can be crashed using the following
formula.
Crash Cost - Normal Cost / Number of Days Saved
3. Set up the "Stage Table" which summarizes the status of the project.
4. To find alternative trade-offs, crash the critical activity (usually in the critical path) whose
incremental cost is the least.

Illustrative Problem 17.5. PERT- Cost Network

Assume that for a special project of ABC Manufacturing Co. the total normal cost is P316,000
and the PERT-Time inputs are shown in the network below:
The above table shows the first of the time-versus-cost trade-offs which is to crash none of the
activities and finish the project on schedule in the anticipated 67 days at a cost of P316,000.

Management wants to determine whether the project's completion time can be shortened at the
least possible incremental cost. Thus, a PERT-Cost Analysis is made producing the following
network.

The network shows the activity time on the upper sides of the arrows, the normal time is written
first, followed by its crash time. If an activity cannot be crashed, only the normal time appears.
Written below the arrows are the incremental costs per day of those activities that can be
crashed.

If management wishes to reduce total project time, it must do so along the critical path. To find
the next trade-off, crash the critical activity whose incremental cost is the least.- This cost/time
trade-offs should always be considered along the critical path.
The activity to crash is the one on Path Z whose incremental cost is the least, which is activity 1-
5 with an incremental cost of P2,000 per day. Crashing that activity reduces the time of Path Z
by 16 days and adds an additional P32,000 to the project's cost. However, in the meantime,
Path Y has turned critical with a time of 59 days.

The above stage table also indicates that the final trade-off is to crash 2 activities (1-5 and 1-4)
and finish the minimum crash time of 55 days at a minimum total crash cost of P358,400
(P318,400+ P32,000+ P10,400). It will also be noted that the activity with the lowest incremental
cost (3 -7) was not crashed and the project's minimum crash time has been attained without
crashing all the activities.

Variation in Activity Time

In the discussion thus far made, nothing has been said about variability in connection with
activity time. The expected activity times have been treated as though they were certain, which
of course they are not. The expected activity times are estimates only, and as such they are
subject to error.
The expected activity time (te) is computed on the basis of a 1-4-1 weighted average of these
subjective time estimates: an optimistic estimate (to), a most likely estimate (tm), and a
pessimistic estimate (tp). This is presented in the following formula:
te to +4 tm + tp 6

The above formula is based on a probability function called the beta distribution. Derived from
the same distribution is a measure of the standard deviation of activity time as follows:

Standard Deviation (o) Pessimistic Estimate - Optimistic Estimate / 6

Thus, as assumed earlier in Example 4, if the three time estimates for Activity 1 2 are:

If the actual time of an activity will be normally distributed about the expected activity time (te)
with standard deviation σ then the confidence intervals can be assigned to the time estimate.
The "Table of Areas from Mean to the Number of Standard Deviations from Mean for a Normal
Distribution" can be used for this purpose. A table reflecting the areas under the standard
normal curve is provided in Figure 17.3.

Figure 17.3
Table of Areas from Mean to the Number of Standard Deviations (2) from Mean for a Normal
Distribution
Therefore, the chances are 68.2 in 100 that the time of activity 1 - 2 will be between 9 and 13
days and 95.4 chances in 100 that the time of activity 1 - 2 will be between 7 and 15 days.

Variation Along a Path

It is important to know what variability can be expected in the time along an entire path
(particularly the critical path) as is the time of a given activity.

To estimate the variability in activity time along a network path, the time variances of the
individual activities are added and the square root is taken. The formula in computing the
Standard Deviation of Time along a PERT network is as follows:
Thus, the standard deviation along the critical path is

sigma path = sqrt 58.78 or 7

It is important that the expected variation along all network paths should be computed and as
the project develops, activity time should be closely monitored. Any radical deviation from
expectancy should immediately trigger corrective or remedial actions.

Accountant's Role in PERT

Certain PERT tasks belong specifically to the accountant such as determining costs, allocating
them properly and keeping management informed of the shifting trade-offs. Rarely, however
would the PERT iterations be done manually, since they are more effectively performed by a
computer. accountant should understand the rationale of PERT, know what inputs are
Nevertheless, the needed, know how to interpret results and be aware of the benefits as well as
its limitations.

Benefits and Limitations of PERT

Benefits of PERT

1. Extensive planning required by PERT and knowing about the project's tasks and their
interrelationship makes one better equipped to control

and coordinate the project. 2. PERT can be adjusted to changing conditions.

3. By measuring and tracking slack, PERT helps to keep the project on schedule; and

4. As an information generator PERT is invaluable, providing continual feedback to

management on the project progresses.

Limitations of PERT
1. Reliable cost data are difficult to obtain, particularly for jobs that are done infrequently or
projects that are being undertaken for the first time.

2. Inclinations of supervisors to overstate their needs and cost requirements are inflated in order
to maintain budgets.

3. Time estimates are more likely to be overly conservative - motivated in part by the realization
that when ones' performance is being rated, it is better to have slack than to miss a deadline.

GANTT CHART

A Gantt chart is a bar chart with time shown on the horizontal axis and the duration of the task
represented as a bar running from the starting date to the ending date. Gantt charts are
considered very useful tools in planning projects and recording progress toward goal. The chart
also shows (1) how expected performance of a specific task compares with actual performance,
(2) which tasks should be in progress on a specific date, and (3) how close to completion a task
should be given on a given date.

A Gantt chart is also a control technique because it allows the comparison of actual production
with scheduled production to identify variations and initiate corrective action.

Steps in Preparing a Gantt Chart

The preparation of a Gantt chart involves the following steps:

1. Identify and determine the sequences of activities.

2. Schedule the work by periods.
3. Prepare the chart using the Y-axis to represent the tasks or activities workers must perform
while the X-axis represents the time available for work.
4. Plot the lines representing the work to be done.
a) Horizontal broken line is used to represent work scheduled by periods within the
departments.

b) Use short vertical broken lines to represent work unfinished or carried over from previous
periods..

e) The large shaded bar line is a summation of the individual horizontal broken lines and
represents cumulative work to be performed.

Figure 17.4 illustrates a Gantt Chart.

The above chart divides time into two weeks for each month. schedules Task A for the first and
third weeks of October, the middle of Management November, and the last week of December.
The four horizontal lines show this scheduled work for Task A. When added, these lines
represent four weeks of cumulative work; the heavy horizontal line indicates this work. On
examination, the chart shows that management could schedule additional jobs during the
second and fourth weeks of January. The workers have no backlog of work to complete on Task
A, because there are no short vertical lines as there are for Task B.

Advantage/Disadvantage of Gantt Charts

The major advantage of the Gantt chart is its simplicity. It forces the planner to think ahead and
define logical activities. Gantt charts show a visual display of planned utilization of facilities so
managers can make appropriate changes to obtain better use of resources. After managers
study the horizontal broken lines illustrating when work is scheduled, they can plan additional
tasks for the time periods represented by breaks in the horizontal lines. The heavy bold line
representing cumulative work assists managers in computing total work-hours required for each
task as well as in scheduling repairs and maintenance. Gantt charts also alert managers to
areas in which large variations in planned and actual performance exist so they can reallocate
resources.

The major disadvantage is that interrelationship among activities are not shown. Several special
methods have been developed to show these on a Gantt chart but they are feasible only for
simple relationships.

INVENTORY MODELING

The objective of inventory decisions is usually to minimize the total relevant costs while
maintaining the quantities of inventories needed for smooth operations.

Inventory Costs
Some costs increase with the quantities of inventory on hand while other costs.. decrease.

Carrying Costs increase with the quantity of inventory on hand. The two classes of carrying
costs are:

1. Out-of-pocket costs which include such items as insurance on the value of the inventory,
inventory taxes, rent, obsolescence, record keeping costs, etc.

2. Cost of capital which is the opportunity cost of having funds in inventory rather than in other
earning assets.

Ordering Costs decrease with the quantity of inventory on hand. These include costs involved in
requisition, purchase order, receipt of goods, placing of goods in inventory and processing of
payment to suppliers. Given a constant usage, the greater the inventory on hand, the less
frequently one must order; thus, the lower the ordering costs.

The inventory costs can be represented graphically as follows:

The Economic Order Quantity (EOQ) Model

Note from the graph (Figure 17.5) that the minimum total cost occurs at the point where the two
cost functions are equal. This coincidence occurs in the most basic EOQ problem but may not
be generalized to more complex problems. The EOQ or optimal Q [labeled (Q)] answers the
question how much to order and may be computed by the equation:

ELS = sqrt((2Annualusage(units) * Set - up * cos ts)/(Carrying * cos tperunit))

The EOQ model can also be used to compute the optimal (least cost) production run or
economic production run (lot) size. Thus:

matrix EOQ\\ * Q^ * matrix = sqrt 2Annualusage(units) * Cos t dot s per order Carrying cost per
unit matrix
The annual costs associated with inventory may therefore be computed as follows:

Annual Ordering Costs =

Annual Usage (demand) / EOQ x Cost per order

Annual Carrying Costs =

Order Size / 2 x Carrying costs per unit
Reorder Point (ROP)

ROP is the quantity level that automatically triggers a new order. This answers the questions -
When to order and may be computed as follows:

ROP Average lead time usage + safety stock Average lead time usage: Lead time x Average
usage per unit of time

A safety stock of 250 has been chosen for Material X. The lead time is 6 days and the annual
usage is 25,000 units.

If there are 220 working days per year, what is the reorder point of Material X?

Illustrative Problem 17.7. Reorder Point Computation

Lead time is the time interval between placing an order and receiving delivery.
Safety Stock

Safety stock is the quantity of goods that are carried as a protection against possible stockouts.
Two costs must be considered in establishing an optimal safety stock policy:
1) the cost to carry safety stock and
2) the cost of a stockout.

The cost of carrying safety stock is the same as the cost of carrying working inventory while the
stockout costs are costs usually expressed in terms of the costs of alternative sources of supply,
loss of customers or goodwill and shutting down of operations over the stockout period.
 CHAPTER 18

RELEVANT COSTS FOR NON-ROUTINE DECISION MAKING

Managers must constantly make decisions. In making these decisions, they must estimate how
each decision could affect operating income.

The management accountant's role in this process is to supply information on changes in costs
and revenues to facilitate the decision process. How does the accountant decide which
information to present?

Managers often select the course of action that maximizes expected operating income over the
period affected by the decision. To do this, they analyze relevant information. Relevant
information is the expected future data that differ among alternative courses of action.

In decision making, revenue and costs are often the key factors. These revenues and costs of
one alternative must be compared against the revenues and costs of other alternatives as one
step in the decision making process. The problem is that some costs associated with an
alternative may not be relevant to the decision to be made. A relevant cost can be defined as a
cost that is applicable to a particular decision in the sense that it will have a bearing on which
alternative the manager selects.

The Decision Making Process

Decision making is the process of studying and evaluating two or more available alternatives
leading to a final choice. This selection process is not automatic; rather, it is a conscious
procedure. Intimately involved with planning for the future, decision making is directed toward a
specific objective or goal.

Although there are innumerable variables or factors that exist and should be considered in
making decisions in the real word, in textbook problems only a few variables that affect decision
results will be taken into account. Therefore, an organized and systematic approach may be
helpful to managers in making decisions.

The steps are outlined as follows:

1. Define strategies: business goals and tactics to achieve them.

2. Identify the alternative choices or courses of action.
3. Collect and analyze the relevant data on the choices.
4. Choose the best alternative to achieve goals.

Consideration should also be given not only to quantitative analysis but also major qualitative
issues in applying the above steps.

Identifying Relevant Costs

Any cost that is avoidable is relevant for decision purposes.

Avoidable cost can be defined as a cost that can be eliminated (in whole or in part) as a result of
choosing one alternative over another in a decision-making situation. All costs are considered
avoidable, except:

1. Sunk costs
2. Future costs that do not differ between the alternatives at hand.

Relevant costs are expected future costs which differ between the decision alternatives. These
are costs that will be increased or decreased as a result of a decision.

Under the concept of relevant cost, decision-making process involves the following analytical
steps:

1. Determine all costs associated with each alternative being considered.

2. Drop those costs that are sunk or historical.
3. Drop those costs that do not differ between alternatives.
4. Make a decision based on the remaining costs. These costs will be the future differential or
avoidable costs, and hence the costs relevant to the decision to be made.

Sunk or historical costs are never relevant in decisions because they are not avoidable and
therefore they must be eliminated from the manager's decision framework. Depreciation relating
to the book value of old equipment is not relevant in decision making. However, it is not correct
to assume that depreciation of any kind is irrelevant in the decision making process.

Depreciation is irrelevant in decision on / y if it relates to a sunk cost. Depreciation on a new

machine is relevant because, the investment in the new machine has not yet been made and
therefore it does not represent depreciation of a sunk cost.
The resale of disposal value of an existing asset will be relevant in any decision that involves
disposing of the asset.

Lastly, any future cost that does not differ between the alternatives in a decision situation is not
a relevant cost so far as that decision is concerned.

Opportunity costs are the profits lost by the diversion of an input factor from one use to another.
They are the net economic benefit given up when an alternative is rejected. They are relevant
when a company is considering eliminating one activity and using plant facilities advantageously
in another activity. Usually formal accounting systems do not record opportunity costs because
such costs do not involve cash receipts or outlays and only data concerning the alternative
selected are recorded. However, these rejected alternatives do have significance in decision
making. For example, a single proprietor has foregone the opportunity to earn a salary
elsewhere by owning a company. In deciding to own a business, the proprietor weighs the
salary that would have been earned if he worked elsewhere.

Out-of-pocket costs involve either an intermediate or near-future cash outlay; they are usually
relevant to decisions. Frequently, variable costs fall into this classification. For example, the
direct materials needed to fill additional orders are both relevant and out-of-pocket cost while
depreciation on the existing manufacturing facilities is not. Out-of-pocket costs are important in
decision making because management should determine whether a proposed project would, at
the minimum return is initial cash outlay.

Consider the following example:

On December 31, 2018, Company A completed the construction of a ner P900,000 machine.
On January 3, 2019, a salesman from an equipment supplier offered to sell the company an
P800,000 machine that can replace the next five years (the life of both machines). The machine
built by Company A has no salvage value. Which costs are relevant?

The relevant costs in this example are the P800,000 potential outlay for the new machine and
the resulting operating savings of P200,000 per year. The book value of the old machine is
irrelevant. However, it is used in determining the gain or loss on disposal for tax purposes and
the effect of taxes on cash flow is a relevant cost. For Company A, a 32% tax rate on the loss of
P900,000 could mean a tax benefit of P288,000; it is the tax benefit that would be the relevant
amount.

Illustrative Problem 18.1. Identification of Relevant and Irrelevant Costs

Rosal Company owns a rice milling machine that was purchased three years ago for P250,000
with five years remaining life. Its present book value is P156,250 and resale value is P100,000.
The company is contemplating replacing this machine with a new one which will cost P500,000
and have a five-year useful life with no salvage value. The new machine will generate the same
amount of revenue as the old one but will substantially decrease the variable operating costs.
Based on normal sales volume of 20,000 units, the annual sales and operating costs of the old
machine and the proposed replacement are estimated as follows:

At first glance, it appears that the new machine will provide an increase in net income of
P81,050 annually. The book value of the old machine however, is a sunk cost and is not
relevant to this decision. In addition, sales and fixed costs (insurance taxes, salaries, etc.) are
also not relevant since they do not differ between the two alternatives being considered. If the
irrelevant costs, taxes and time value of money can be disregarded, the alternatives can be
analyzed as follows:

The above computation will indicate that it would be a good move to buy the new machine
because it would result to a net cash flow of P350,000 for the 5-year period.

Approaches in Analyzing Alternatives in Nonroutine Decision Making

The two commonly used approaches in evaluating alternative courses of action are
1) Incremental or Differential analysis approach
2) Total Project Analysis approach or Comparative Statements approach.

Incremental, Differential, or Relevant Cost analysis contrasts choices by comparing differential

revenues, differential costs and differential contribution margins. It has the advantage of
showing only relevant amounts. All sunk and nondifferential items are disregarded.

The following steps are followed in using this approach:

1. Gather all costs associated with each alternative.
2. Drop the sunk costs and non-differential costs.
3. Select the best alternative based on the remaining cost data.

Total Project Analysis approach shows all the items of revenue and cost data (whether they are
relevant or not) under the different alternatives and compares the net income results.
Comparative income statements under this approach are prepared in a Contribution format.
These approaches are illustrated in the following case problem on Special Sales Order.

Illustrative Problem 18.2. Special Sales Order

Parton, Inc., a manufacturer of rattan baskets, ordinarily sells regular baskets for P32.00 each.
At the beginning of the year 2019, an exporter has offered Parton P350,000 for 50,000 baskets
or P17.50 per basket. This sale will not affect regular business in any way. Furthermore, it will
not change fixed costs nor require additional variable selling and administrative expense and it
will put to use idle manufacturing capacity. Parton's manufacturing product cost of a basket is
P20 of which P12 is variable cost.

Based on the computations, the company may decide to drop Product C.

As shown above, overall net income will be P142,800. This is slightly higher than the present
overall income of P134,800 with no increase in total assets required. However, management
should consider other factors, such as the future sales of product C and whether the increased
sales of Product A and B will continue or would occur without eliminating Product C operations.

Sell Now or Process Further

In some industries, a number of end products are produced from a single or common raw
material input. For example, in the meat-packing industry, a great variety of end products - ham,
bacon, spare ribs, pork roasts, and so on are produced from a common input are referred to as
joint products. Firms that produce several end products from a common input are faced with the
problem of deciding how the joint product cost of that input is going to be divided among the
joint products. Joint product costs is used to describe those manufacturing costs that are
incurring is producing the joint products up to the split-off point. The split-off point is that point in
the manufacturing process at which the joint product can be recognized as separate products.

Joint product costs are irrelevant in decisions regarding what to do with a product from the split
off point forward because they have already been incurred and therefore are sunk costs.

Costs incurred after the split-off point for the benefit of only one particular product are called
separable costs. They are relevant costs in the sell-or process-further decision.

In sell-or-process-further decision, it will always be profitable to continue processing a joint

product after the split-off so long as the incremental revenue from such processing exceeds the
incremental processing costs.

Assume that three products are derived from a single raw material input. Cost and revenue data
relating to the products are presented before (along with an analysis of which products should
be sold at the split-off point and which should be processed further.)

Illustrative Problem 18.5. Sell Now or Process Further

Special Sales Pricing

Managers often must evaluate whether a special order should be accepted, or if the order is
accepted, the price that should be charged. A special order is a one time order that is not
considered part of the company's ongoing business. Managers may be asked to consider
accepting a special order for their product at a reduced price to make use of the excess, or idle
facilities. Such orders are worth considering, provided they will not affect regular sales of the
same product.

Illustrative Problem 18.6. Accept or Reject a Special Order

Utilization of Scarce Resources

Choosing which products to manufacture and sell is a common managerial decision. Managers
are routinely faced with the problem of deciding how scarce resources are going to be utilized.
For example, a department store has a limited amount of flow space and therefore cannot stock
every product that may be available. A small CPA firm, due to a shortage of personnel may
have to choose between performing work for client A or for client B. A manufacturing firm has a
limited number of machine hours and a limited number of direct labor-hours at its disposal.

When capacity becomes pressed because of a scarce resource, the firm is said to have a
constraint. Because of the constrained scarce resource, the company cannot fully satisfy
demand, so the manager must decide how the scarce resource should be used. Fixed costs are
usually unaffected by such choices, so the manager should select the course of action that will
maximize the firm's total contribution margin. This is based on the assumption that the product
choices as short-run decisions because we have adopted the definition that in the short run,
capacity is fixed, while in the long-run, capacity can be changed.

Contribution in Relation to Scarce Resources

To maximize total contribution margin, a firm should not necessarily promote those products
that have the highest contribution margins per unit. With a single constrained resource, the
important measure of profitability is the contribution margin per unit of scarce resource used.

In this case, the contribution margin of each product is the same, and assuming that the
production time per unit is uniform, the profit-volume relationship is the same regardless of the
mix of products produced and sold. For instance, Aeon I can sell 25,000 hard rolls or 25,000
soft rolls or any combination of the two products totaling 25,000 to break-even per month.

Suppose that Aeon's capacity is limited to 720 machine hours per month and the machines may
be used to produce either 300 hard rolls per machine-hour or 500 soft rolls per machine-hour.
Which product should Aeon produce to maximize its profit?

This problem could be analyzed as follows:

1) Determine the contribution margin per machine-hour of each product.
2) Rank the product lines according to contribution margin per machine hour.
3) Consider market limitations if any.
4) Compute the units to be produced based on the profitability ranking made.

Contribution margin per machine hour:

Hard rolls P2.000 (P4.00 x 500 units)
Soft rolls P * 1.2(P * 4 * 300units)

2. Ranking:
1. Soft rolls
2. Hard rolls

3. & 4. Since there is no market limitation, the company should produce 360,000 soft rolls (720
hours x 500 units) and generate P1,440,000 contribution margin (360,000 x P4) and P440,000
operating profit (P1,440,000 P1,000,000 fixed costs).

If only hard rolls are produced, Aeon would generate P864,000 (216,000 x P4) contribution
margin and incur P136,000 loss (P864,000-P1,000,000 fixed cost).

The Problem of Multiple Constraints

If a firm is operating under several scarce resource constraints, what should it do? Constraints
may refer to limited availability of raw materials, limited direct labor-hours available, limited
capital available for investments and many more. As more constraints and products are added,
solving product mixes becomes more complex. Although it is possible to solve these problems
by hand, they are typically solved by computer. The optimal proper combination of product "mix"
can be found by use of a quantitative method known as linear programming which was covered
in quantitative techniques in Chapter 17.

Shut down or Continue operations

Illustrative Problem 18.8. Shut down or continue operations

The ABC Company, now operating below 50% of its practical capacity expects that the volume
of sales will drop below the level of 5,000 units per month. An operating statement prepared for
the monthly sales of 5,000 units shows the following:

Management is concerned with the fact that a further drop in sales volume will create a loss.
This concern has been intensified by the sales manager's opinion that the selling price of the
company's product will soon have to be adjusted to meet the increasing pressure of
competition. Since all costs, as the president puts it, have been cut to the bones, management
has under consideration a recommendation that operations be suspended until favorable
conditions can be attained and a better selling price can be set.

Before making their final decision, the company executives must recognize that not all of the
non-variable costs will be eliminated by a temporary closing of the plant. Key personnel cannot
be discharged lest they seek employment elsewhere; a skeleton staff must be maintained;
maintenance costs of building and equipment will continue, taxes and insurance premiums must
be paid during the shut-down period. As a first step, an estimate of the shutdown costs must be
made.

Assume that a conservative estimate of costs if plant operations are suspended indicates a
shut-down cost of P2,000 per month. Since there is no immediate possibility of profit under
present conditions, the problem of the company is the possibility of minimizing the loss.

REQUIRED:

Determine if the company should shut down temporarily or continue operations.

Analysis:

The decision to continue operations or shut down will depend upon the expected sales of the
company in comparison with the shutdown point of 3,000 units computed as follows:

If the expected demand exceeds 3,000 units but below 5,000 units (break-even volume)
operating loss will be lesser than shutdown loss and therefore the company can continue
operations. If expected demand is less than 3,000 units, the company should discontinue
operations on a temporary basis until favorable conditions prevail.

Pricing Products and Services

Some businesses have no pricing problems at all. They may be making a product for which a
market price already exists. Under these circumstances, no price calculations are necessary
because every firm charges whatever is the prevailing market price. This usually is true for basic
raw materials such as farm products, minerals, etc.

In many situations however, the firm is faced with the problem of selling its own prices. The
pricing decision can be critical because
1. the prices charged for a firm's products largely determine the quantities customers are willing
to purchase and
2. the prices should be high enough to cover all the costs of the firm.

Cost-Plus Pricing
The most basic approach in pricing decision is that the price of the product or service should
cover all the costs that are traceable to the product and service, variable as well as fixed. If
revenues are not sufficient to cover these traceable costs, then the firm would be better off
without the product or service. In addition to the traceable costs, all products and services must
assist in covering the common costs of the organization. These common costs may include
general factory, selling and administrative costs. And of course, the selling price should not only
cover all the costs of the organization but also provide a return on invested capital.

In practice, the most common approach to pricing of products is to use some type of cost-plus
pricing formula. The formula is expressed as follows:

Products however, may be costed in at least two different ways:

1. By the absorption approach where the cost base is defined as the cost to manufacture one
unit and therefore excludes all selling general and administrative expenses.

2. By the contribution approach where cost base consists of all the variable costs associated
with a product including variable selling, general and administrative expenses (SGA).

In both approaches, expenses not included in the cost base are provided for through the
markup which should be high enough to provide the company also with a satisfactory profit
margin.

Illustrative Problem 18.9. Pricing Decision

Assume that Knox Company is in the process of setting a selling price on a product that has just
undergone some modifications in design. The following cost estimates for the redesigned
product have been provided by the Accounting Department:

The costs above are based on an anticipated volume of 10,000 units produced and sold each
period. The company uses cost-plus pricing, and it has the policy of obtaining target selling
prices by adding a mark.p of 50% of unit manufacturing cost or by adding a markup of 100% of
variable costs.
Assuming that the company uses absorption costing approach to cost-plus pricing, how much
will the target selling price for one unit of product be?

Determining the Markup Percentage

To facilitate the computation of selling price, formulas can be used to determine the appropriate
markup percentage assuming that the desired Return on Investment (ROI) and unit sales
volume are given.

Under the absorption approach to cost-plus pricing:

Under the contribution approach to cost-plus pricing:
Target costing

This pricing approach is used when company will already know what price should be charged
and the problem will be to produce the product that can be marketed profitably. Target costing is
the process of determining the maximum allowable cost for a new product and then developing
a sample that can be profitably manufactured and distributed for that maximum target cost
figure. The target cost is computed as follows:

Target cost = Anticipated selling price - Desired Profit

Illustrative Problem 18.10. Target Costing

The target cost per unit is computed as follows:

Karate Auto Supply, Inc., is a producer and distributor of auto supplies. The company desires to
enter a rapidly growing market for long-life batteries that is based on a newly discontinued
technology. Management believes that to be fuxy competitive, the new battery that the company
is planning can not be priced at more than P1,300. At this price, management is confident that
the company can sell 12,500 batteries per year. The batteries would require permanent
investment of P5,000,000 and the desired ROI is 20%. Compute the target cost of one battery.
 CHAPTER 19
CAPITAL BUDGETING DECISIONS

Implementing long-range plans usually requires capital expenditures. Plans for expansion may
call for new production facilities or new products. Since all firms have limited capital, a manager
must often choose between several competing investments and his skill in selecting
investments ultimately determines how well an organization performs over the long run.

Capital Budgeting Defined

Capital budgeting is the process of deciding whether or not to commit resources to projects
whose costs and benefits are spread over several time periods. It involves;

1) the preparation of annual budget for capital investment

2) the assessment of funding capacities and
3) the allocation of resources to renewal and expansion projects which most clearly conform
with the company's priorities.

Capital budgeting is used to describe actions relating to the planning and financing capital
outlays for such purposes as the purchase of new machinery, the modernization of plant
facilities or the introduction of new product lines. Capital budgeting is an investment concept,
since it involves a commitment of funds now in order to receive some desired return in the future
in the form of additional cash. inflows or reduced cash outflows.

Characteristics of a Capital Investment Decision

Capital expenditures are long-term commitments of resources to realize future benefits and
budgeting for them is one of the most important areas of managerial decision. They deserve
penetrating analysis and attention of top management because of the following reasons:

1) Substantial amount of funds are required in capital projects.

2) Because of the length of time spanned by a capital investment decision, the element of
uncertainty becomes more critical.
3) The effect of managerial errors will be difficult to reverse.
4) Plans must be made well into an uncertain future.
5) Success or failure of the company may depend upon a single or relatively few investment
decisions.

One of the most difficult steps involved in the decision-making process relates to the
identification of costs relevant to the problem. Because the alternatives lie in the future, the only
costs which are relevant are future costs. Historical costs arising from past decisions are sunk
costs and so cannot affect future alternatives. Such considerations as monetary advantage of
an alternative, its effect on employee relations, company image and relations with other
companies are usually evaluated in choosing from among the alternatives.

Categories of Capital Investments

The two general types of capital investment decisions are:

A. Independent capital investment projects or Screening decisions

these are projects which are evaluated individually and reviewed against predetermined
corporate standards of acceptability resulting in an "accept" or "reject" decision. Examples are:

1. Investment in long-term assets such as property, plant and equipment.

2. New product development.
3. Undertaking a large scale advertising campaign.
4. Introduction of a computer.
5. Corporate acquisitions (such as purchase of shares in subsidiaries or affiliates).

B. Mutually exclusive capital investment projects or Preference decisions

these are projects which require the company to choose from among specific alternatives. The
project to be acceptable must pass the criteria of acceptability set by the company and be better
than the other investment alternatives. Examples are:

1. Replacement against renovation of equipment or facilities.

2. Rent or lease against ownership of facilities."
3. Manual bookkeeping system against computerized system.
4. Preventive maintenance against periodic overhaul of machineries
5. Purchase of machinery from an outside supplier against assembly of the machinery by the
company's own staff.

Elements of Capital Budgeting

The elements or factors to be considered in evaluating capital investment proposals are:

1. The net amount of the investment.

2. The operating cash flows or returns from the investment.
3. The minimum acceptable rate of return on the investment.

Net Initial Investment or Project Cost

Net investment represents the initial cash outlay that is required to obtain future returns or the
net cash outflow to support a capital investment project. This may be computed as follows:

Initial cash outlay

Add: Additional cash outlay related to the asset
Additional working capital
Total
Less: Cash inflow arising from sale of old asset being replaced
Avoidable costs
Net Investment

In certain cases, the net investment is the sacrifice of an inflow of cash, that is, the opportunity
cost that arises when a benefit is rejected. An example is when a company has in its possession
fixed assets no longer used in operation and are awaiting disposal through sale. If it should
happen that these assets can be put to good use on a proposed capital project rather than be
disposed of, then the estimated project cost or investment should include the net amount to be
realized from the sale of the assets.

Illustrative Problem 19.1. Determination of Net Initial Investment

The management of Maingat Company plans to replace a sorting machine that was acquired
several years ago at a cost of P60,000. The machine has been depreciated to its residual value
of P10,000.

A new sorter can be purchased for P96,000. The dealer will grant a trade-in allowance of
P16,000 on the old machine. If a new machine is not purchased, Maingat Company will spend
P10,000 to repair the old machine. Gains and losses on trade-in transactions are not subject to
income taxes. The cost to repair the old machine can be deducted in computing income taxes.
Income taxes are estimated at 40% of the income subject to tax. Additional working capital
required is P50,000.
REQUIRED: Compute the net initial investment in this project.

Solution: Maingat Company

Net Cash Returns

The cash returns are the inflows of cash expected from a project reduced by the cash cost that
can be directly attributed to the project. This is computed as follows:

Annual incremental revenue from the project

Less: Incremental cash operating costs
Annual cash inflow before taxes
Less: Taxes
[Tax rate (Annual cash inflow before taxes - Depreciation)]
Annual net cash inflow after taxes

Or

Annual incremental revenue from the project

Less: Incremental cash operating costs
Annual cash inflow before taxes
Less: Incremental depreciation
Net income before taxes
Less: Income taxes
Net income after taxes
Add: Incremental depreciation
Annual net cash inflow after taxes

Some projects however are expected to produce an inflow of cash but will yield returns in the
form of cash savings. This is determined as follows:

Annual cash operating costs (if the old asset or method is used)
Less: Annual cash operating costs (if the new asset or method is used)
Annual cash savings before taxes
Less: Taxes
Annual cash savings after taxes

or

Cash operating costs (if the old asset or method is used)

Less: Annual cash operating costs (if the new asset or method is used)
Cash savings before taxes
Less: Incremental depreciation
Increase in income before taxes
Less: Income taxes
Increase in income after taxes.
Add: Incremental depreciation
Net cash savings after taxes

Illustrative Problem 19.2. Determination of Annual Cash Returns

Alalay Company is considering the acquisition of a machine which will cost P120,000. It has an
expected useful life of five years at the end of which its scrap value will be P20,000. The
company expects to be able to generate annual cash flow before taxes of P40,000. Estimated
income tax rate is 30%. What is the annual cash flow after taxes on this investment?

Illustrative Problem 19.3. Determination of Annual Cash Savings

The Visayan Division of Marlow Supply Company has been considering a new production
method that can reduce materials costs by an estimated amount of PS2,000 a year. The new
method is also expected to result in an annual savings of labor and overhead methods is
estimated at P40,000 a year over a period of ten years. Income taxes are estimated at 30% of
income before income taxes. What are the annual net returns (or savings) expected from the
new production method?

Solution: Marlow Supply Company

Minimum or Lowest Acceptable Rate of Return

The minimum or lowest acceptable rate of return or opportunity cost may equal the average rate
of return that the company will earn from alternative investment opportunities or the cost of
capital which is the average rate of return that the firm must pay to attract investment fund. The
cost of capital according to source may be computed as follows:

1. Cost of Debt: Interest rate (1- Corporate Tax Rate)

2. Cost of Preference Shares:

Dividends per share / Market value per share of Preference Shares

3. Cost of Ordinary Shares:

a. Stock price-based
Expected Cash Dividends per share / Current price per share of Ordinary Shares + Dividend
growth rate

b. Book-value based
Next year's Projected Earnings per share / Current price per share of Ordinary Shares

4. Cost of Retained Earnings - same as cost of ordinary equity

This is used when dividend growth rate is not known.

To compute for the overall or weighted average cost of capital, multiply the cost of each type of
capital by their respective weights (percentage of each source to the firm's total capital
structure) and add up the individual weighted cost of capital.

Illustrative Problem 19.4. Computation of Weighted Average Cost of Capital

The following information on Bettina Corporation's capital structure is available from the latest
financial statement:
Process of Capital Budgeting

The capital budgeting can be divided into seven significant phases:

1. Finding Investment Opportunities

Many capital expenditures proposals can be identified during the strategic or long-term planning
process. Since the long-term profitability of most companies depends on the nature and quality
of their capital investments, these investment opportunities should be carefully analyzed and
evaluated.

2. Collect Relevant Information about Opportunities

To effectively evaluate any investment opportunity, the expected cash flows from the project
must be estimated and the total cash outlay necessary to place the investment in operative form
must be determined. A plan for implementing the opportunity must be developed and other
nonfinancial information must be assembled.

3. Select Discount Rate

Before the cash flow can be evaluated, the discounted (cost of capital) must be established if
the discounted cash flow approach is to be applied (These techniques will be discussed later).

4. Financial Analysis of Cash Flows

The techniques of capital budgeting are applied to the estimated cash flows developed in the
second phase.

5. Decision
Many factors, quantitative as well as qualitative, should be given consideration before the final
decision is made as to the selection of a particular investment. They will include among others,
relationship of this opportunity to other aspects of the company operations and long term goals,
the timing of the cash flows, the availability of funds for investment purposes, the impact on the
financial structure of the company, social impact of the opportunity, and legal ramifications.

6. Project Implementation
Once the decision has been made to invest funds, more detailed plans for making the project
operational are developed.

7. Project Evaluation and Appraisal

This last phase involves the assessment of how effective the investment actually is. The
evaluation may be in the form of continuous monitoring of the project, so that corrective action
can be taken. Regardless of how difficult this may be, it is important that not only the
effectiveness of the project be determined, but that the overall decision-making process be
appraised for possible improvement.

The stages just described give us only some indication or summary from, of the capital
investment process. The way they are implemented depends on the nature of the project, size
of the capital outlay and length of time to place the project in operation.

Categories of Project Cash Flows

This section of the Chapter outlines a method of estimating cash flows for investment projects.
The major categories of cash flows for a project are as follows:

Cash Inflows:
1. Periodic cash inflows from operations, net of taxes
2. Investment tax credit
3. Proceeds from sale of old asset being replaced, net of taxes
4. Avoidable costs, net of taxes
5. Return of some working capital invested in the project*
6. Cash inflow from salvage of the new long-term asset at the end of its useful life. This will be
net of tax consequence.*

Cash Outflows:

7. Acquisition cost of purchasing and installing assets (e.g., new equipment or machinery)
8. Additional working capital
9. Other cash flows such as severance payments, relocation costs, restoration costs and similar
costs.

The end of a project's life will usually result in some cash flows. These cash flows are referred to
as disinvestment flows.

A brief discussion of the foregoing items follows:

1. Periodic net cash returns or cash inflows from operations, net of taxes.
To generate positive periodic operating cash flows is usually the primary reason for acquiring
long-term assets. These positive flows may result from such revenue generating activities as
new products or they may stem from cost-saving programs.

2. Investment-Tax Credit
The investment tax credit allows a credit against a company's income tax liability based on the
cost of an acquired asset. If the present income tax laws allow investment tax credit, it would
reduce the cost of making investments by giving companies a credit against their corporate
income taxes equal to (say) 10% of the cost of assets.

3. Proceeds from sale of old assets being replaced, net of taxes

If an old equipment is to be sold, the proceeds from such sale is treated as a reduction from
cost of initial investment. If the old asset is sold at a gain, the incremental income tax should be
deducted from the proceeds. If the old asset is sold at a loss that is, its book value exceed the
selling price, the tax savings will be added to the proceeds.

4. Avoidable costs, net of taxes

In some instances, purchase of new asset may result to the avoidance of incurring expenses to
repair the old asset. The avoidable repairs cost, net of incremental tax will be treated as a
deduction in computing the cost of initial investment.

5. Return of working capital

When a project ends, there are usually some leftover inventory, cash or other working capital
items that were used to support operations. These working capital items are then freed for use
elsewhere and treated as a cash inflow.
6. Cash inflow from salvage of the new long-term asset at the end of its useful life
Ending a project will usually require disposal of its assets. In some cases, more money is spent
in disassembling the assets and disposing these than is gained from their sale. Any net outflows
from the disposal of a project's assets become tax deductions in the year of disposal. The net
salvage value of an asset is listed as a cash inflow at the time it is expected to be realized. If the
net salvage value of the asset is negative, then it is listed as cash outflow also at the time it is
expected to be incurred.

7. Acquisition costs of purchasing and installing assets:

These acquisition costs represent the primary outflows for most capital investments. They are
listed as cash outflows in the years in which they are incurred.

8. Additional working capital

Many projects require fund for working capital needs (for example, to build up inventories,
additional cash balance to handle increased level of activities). These cash flows often occur
before the project is in operation.

SCREENING CAPITAL INVESTMENT PROPOSALS

Several methods are available for the evaluation of alternative capital investment proposals.
One method may be used exclusively or in combination with another. The most commonly used
methods of evaluating capital investment projects are:

A. Non-discounted cash flow (unadjusted) approach

1. Payback period
2. .Accounting rate of return (book value rate of return)
3. Payback reciprocal

B. Discounted cash flow (time-adjusted) approach

1. Net present value
2. Discounted rate of return or internal rate return
3. Profitability index
4. Discounted payback period

Payback Period

Payback period (also known as payoff and payout period), measures the length of time required
to recover the amount of initial investment. It is the time interval between time of the initial outlay
and the full recovery of the investment.

When the periodic cash flows are uniform, payback period is computed as follows:

(Net Investment)/(Annual cash returns)

When the periodic cash flows are not uniform, payback period is computed by cumulating the
estimated annual cash inflows and determining the point in time at which they equal the
investment outlay.

Decision Rule:

The desirability of the project is determined by comparing the project's payback period against
the maximum acceptable payback period as predetermined by management. The project with
shorter payback period than the maximum will be accepted. In short:

If: PB period <= Maximum allowed PB period; Accept

If: PB period > Maximum allowed PB period; Reject

Advantages of Payback period method:

1. It is easy to compute and understand.

2. It is used to measure the degree of risk associated with a project.
3. Generally, the longer the payback period, the higher the risk.
4. It is used to select projects which provide a quick return of invested funds.

Disadvantages of the payback period method:

1. It does not recognize the time value of money.

2. Ifignores the impact of cash inflows after the payback period.
3. It'does not distinguish between alternatives having different economic lives.
4. The conventional payback computation fails to consider salvage value, if any.
5. It does not measure profitability only the relative liquidity of the investment.
6. There is no necessary relationship between a given payback and investor wealth
maximization so an investor would not know what an acceptable payback is.

Bail-out Period
In conventional payback computations, investment salvage value is usually ignored. An
approach which incorporates the salvage value in payback computations is the "Bail-out period.
This is reached when the cumulative cash earnings plus the salvage value at the end of a
particular year equals the original investment.

Accounting Rate of Return or Simple Rate of Return

Simple Fate of return or Accounting rate of return (ARR) also known as book value rate of
return, measures profitability from the conventional accounting standpoint by relating the
required investment to the future annual net income. This is computed as follows:

AAR = Average Annual Net Income / Initial Investment or Average Investment

or, if a cost reduction project is involved, the formula becomes

AAR = Cost savings - Depreciation on new equipment / Initial Investment or Average

Investment

Decision Rule:

Under the ARR method, choose the project with the highest rate of return. Accept the project if
the ARR is greater than the cost of capital. Thus:

If: ARR >= Required rate of return; Accept

If: ARR< Required rate of return: Reject

Advantages of using the ARR:

1. It is easily understood by investors acquainted with financial statements.

2. It is used as a rough preliminary screening device of investment proposals.

Disadvantages of using the ARR:

1. It ignores the time value of money by failing to discount the future cash inflows and outflows
2. It does not consider the timing component of cash inflows.
3. Different averaging techniques may yield inaccurate answers.
4. It utilizes the concepts of capital and income primarily designed for the purposes of financial
statements preparation and which may not be relevant to the evaluation of investment
proposals.

Illustrative Problem 19.7. Determination of Accounting Rate of Return

Discounted Cash Flow Techniques

Under the discounted cash flow decision criterion, also frequently called the present-value
approach, cash outlays and cash inflows are both discounted back to the present period using
an appropriate discount rate. The variations in the DCF techniques are as follows:

1) Net present value or excess present method.

2) Discount rate of return or internal rate of return.
3) Profitability index.
4) Discounted payback period.

The Table for Present Value of P1 is shown in Appendix A-1 and the Table for Present Value of
an Annuity of P1 in Arrears is presented in Appendix A-2.

Advantages of the DCF Method:

1. It is more reliable because the time value of money is taken into account.
2. Income over the entire life of the project is considered.
3. It is more objective and relevant because it focuses on cashflow.

Disadvantages of the DCF Method are:

1. It is not easily understood.

2. It is more complex and difficult to apply.
3. It requires detailed long-term forecasts of incremental cash flow data.
4. It requires pre-determination of the cost of capital or the discount rate to be used.

Net Present Value (NPV)

Net Present Value is the excess of the present value of cash inflows generated by the project
over the amount of the initial investment. This is computed as follows:

Present value of cash inflows computed based on minimum desired discount rate
Less: Present value of investment
Net Present Value

Decision Rule:

For independent project proposal, accept it if NPV is positive or zero and reject if NPV is
negative. If the NPV is positive, it means that the project will earn a return greater than the
discount rate also known as the hurdle rate. If the projects do not meet the hurdle rate, they
should be rejected because the funds that would be invested in them can earn a higher rate in
some other investment. In short:

If: NPV .>=0 Accept

If NPV < 0 Reject

Illustrative Problem 19.8. Net Present Value Application, Uniform Cash Inflows

ABC wants to invest in a machine costing P80,000 with a useful life of six years and no salvage
value. The machine will be depreciated using the straight-line method and is expected to
produce annual cash inflow from operations, net of income taxes, of P22,000. The present value
of an ordinary annuity of P1 for six periods at 10% is 4.355. The present value of PI for six
periods at 10% is 0.564. Assuming that ABC wants a minimum rate of return of 10%, what is the
net present value of this proposed investment? Is the proposal acceptable?

Solution: ABC Company

Yes. The ABC Company should invest in the new machine because it could earn more than the
minimum return that they desire as indicated by the positive net present value in the above
schedule.

Illustrative Problem 19.10. Application of Net Present Value Method

Under a special licensing arrangement, Santos Company has an opportunity to market a new
product in Southern Luzon for a five-year period. The product would be purchased from the
manufacturer, with Santos Company responsible for all costs of promotion and distribution. The
licensing arrangement could be renewed at the end of the five-year period at the option of the
manufacturer. After careful study, Santos Company has estimated that the following costs and
revenues would be associated with the new product:
Discounted Rate of Return

Discounted Rate of Return, also known as internal rate of return (IRR) and time-.. adjusted rate
of return, is the rate which equates the present value of the future cash inflows with the cost of
the investment which produces them. It is also the equivalent maximum rate of interest that
could be paid each year for the capital employed over the life of an investment without loss on
the project.

Steps in the Computation of the Discounted Rate of Return

A. Cash inflows are evenly received:

If the cash returns or inflows are evenly received during the life of the project, the computational
procedures are as follows:
1. Compute the Present Value Factor by dividing Net Investment by Annual Cash Returns.
2. Trace the PV factor in the Table for Present Value of PI received annually using the life of the
project as point of reference.
3. The column that gives the closest amount to the PV factor is the "Discounted rate of return".
4. To get the exact Discounted rate of return, interpolation is applied.

B. Cash inflows are not evenly received:

The steps in computing for the discounted rate of return are:

1. Compute the Average Annual Cash Returns by dividing the sum of the returns to be received
during the life of the project by the total 'economic life of the project.
2. Divided Net Investment by the Average Annual Cash Returns to get
the Present Value Factor.
3. Refer to the Table for Present Value of P1 received annually to determine the rate that will
give the closest factor to the computed present value factor.
4. Using the rate obtained in Step No. 3, refer to the Table for Present Value of P1. If the returns
are increasing, use a discount rate lower than the rate obtained in Step No. 3, if the returns are
decreasing, use a higher rate. Compute the present value of the annual cash returns.
5. Add the present value of the annual returns and compare with the Net Investment.
6. If the result in Step No. 5 does not give equality of present value of returns and net
investment, try at another rate.
7. Interpolate to get the exact discounted rate of return.

Decision Rule:

Accept the proposed investment if DCR or IRR is equal to or greater than minimum desired rate
of return or cost of capital. Reject the proposal if IRR is lower than the minimum desired rate of
return. In short:

If: IRR >= Required rate of return; Accept

If IRR < Required rate of return; Reject

Illustrative Problem 19.11. Discounted or Internal Rate of Return Computation, Uniform Cash
Returns

An investment of P50,000 will yield an average annual cash return of P7,500 a year for a period
of 10 years. What is the discounted rate of return?
Payback Reciprocal

Payback reciprocal is the rate of recovery of investment during the payback period. When a
project is at least twice the payback period and the annual cash flows are approximately equal,
the payback reciprocal may be used to estimate the discounted rate of return. A project with an
infinite life would have a discounted rate of return exactly equal to its payback reciprocal.

To compute for the payback reciprocal, the following may be used.

Paybackreciprocal = (Annual Cash Inflows)/(Net Investment)

Or

1 / Payback Period

Profitability Index

The Profitability Index, (also known as present value index, benefit-cost rate, desirability index),
is the ratio of the total present value of future cash inflows to the initial investment. The index
expresses the present value of cash benefits as to an amount per peso of investment in a
project and is used as a measure of ranking projects in a descending order of desirability. This
is computed as follows:

PV index = PV of Cash Inflows / PV of Net Investment

Decision Rule:

The higher the profitability index, the more desirable the project. Projects with index of less than
1 are rejected. Thus:

If: PV Index >= 1; Accept

PV Index < 1; Reject

Illustrative Problem 19.13. Profitability Index

Company XYZ has P200,000 funds available for investment. It is considering the following
projects:

3. The company should invest in Projects B and C for the following reasons:
a) The PV indexes of Projects B and C are higher than Project A.
b) The combined net present value of Projects B and C is higher than that of Project Ac
c) The company can afford to invest in both A and B..

Discounted Payback Period

A method that recognizes the time value of money in a payback context is the discounted
payback method. This is used to compute the payback in terms of discounted cash flows
received in the future. That is, the periodic cash flows are discounted using an appropriate cost
of capital rate. The payback period is computed using the discounted cash flow values rather
than the actual cash flows.

Illustrative Problem 19.14. Discounted Payback Period

A project requiring an investment of P70,000 is expected to generate the following cash inflows:
Preference Decisions - The Ranking of Investment Projects

Preference decisions come after screening decisions and attempt to resolve the question of
"How do the investment proposals, all of which have been screened and provide an acceptable
rate of return, rank in terms of preference?". The preference decisions are much more difficult to
make than screening decisions because investment funds are usually limited and that some or
many other profitable investment opportunities may have to be foregone.

Basically, either the internal rate of return method or the net present value method can be used
in making preference decisions.

Internal Rate of Return Method

The preference rule when using the internal rate of return method to rank. competing investment
projects is:

"The higher the internal rate of return, the more desirable the project.

This method is widely used for two main reasons, namely

1) No additional computations need to be made beyond those already performed in making the
initial screening decisions.
2) The ranking data are easily understood by management.

Net Present Value Method

The net present value method can be used to rank competing investment projects if the projects
are of equal size, that is, investment funds required are the same. If the competing projects
require different amount of funding, it may be necessary to compute the profitability index.
Profitability index is computed by dividing the present value of the cash inflows by the
investment required.

The preference rule to rank competing investment projects using the profitability index is

"The higher the profitability index, the more desirable the project."

Comparing the Preference Rates

If an independent project is being evaluated, then the NPV and IRR criteria always lead to the
same accept/reject decision.

For mutually exclusive projects (choosing among acceptable alternative) especially those that
differ in scale (project size) and/or timing a conflicts of ranking may arise. That is, the IRR
method may favor one alternative over another while the NPV method may indicate otherwise. If
conflicts arise, the NPV method should be used. The NPV method assumes the cash flows will
be reinvested at the firm's cost of capital while the IRR method assumes reinvestment at the
project's IRR. Because reinvestment at the cost of capital is generally a better (closer to reality)
assumption, the NPV is superior to the IRR.

The profitability index is conceptually superior to the internal rate of return as method of making
preference decisions. The reason is that the profitability index will always give a correct
indication as to the relative desirability of alternatives, even if the alternatives have different lives
and different pattern of earnings. On the other hand, if lives are unequal, the internal rate of
return method can lead the manager to make incorrect decisions.

Comparing Projects with Unequal Lives

In previous examples, replacement decisions involved comparing two mutually exclusive

projects: retaining the old asset versus buying a new one. It is also assumed that the new
equipment had a life equal to the remaining life the old equipment. However, if we were deciding
between two mutually exclusive alternative with significantly different lives, an adjustment would
be necessary. This problem may be dealt with using any one of these procedures,

1. The replacement chain method and

2. The equivalent annual annuity method (EAA)

Replacement Chain (Common Life) Approach

This method compares project of unequal lines which assumes that each project can be
repeated as many times as necessary to reach a common life span. The Net Present Values
(NPVs) over this life span are then compared, and the project with the higher common life NPV
is chosen.

Observations and Analysis

(a) Based on the NPV, Project N appears to the better project. This analysis is, however,
incomplete and the decision to choose it, may not be correct.

(b) If Project M is chosen, there will be an opportunity to make a similar investment in 3 years
and if cost and revenue conditions continue, this investment will also be profitable. If Project N is
chosen, there is no second investment opportunity.

(c) To make a proper comparison of Project M and Project N, the replacement chain (common
life) approach could be applied.

For Project M, add in a second project to extend the overall life of the combined project to 6
years.

Assuming that Project M's investment cost and annual cash inflows will not change if the project
is repeated in 3 years (Year 1: P70,000, Year 2: P130,000 and Year 3: P120,000) cost of capital
will remain at 12%. The new NPV of this project will be P88,240 and IRR will be at 25.2%.

(d) Since the P88,240 extended NPV of Project M over the common life of 6 years is greater
than the P64910 NPV of Project N, Project M should be selected.
Although the above illustrative case shows why an extended analysis is necessary if there are
mutually exclusive project with different lines, the arithmetic is generally more complex in
practice. However, even for mutually exclusive projects it is not always appropriate to extend the
analysis to a common life. This should only be done if there is a high probability that the projects
will actually be repeated at the end of their initial life.

Equivalent Annual Annuity (EAA) Approach

Another procedure, known as the equivalent annual annuity (EAA) method may also be used in
evaluating mutually exclusive projects with different lives.

Equivalent Annual Annuity (EAA) Method is a method which calculates the annual payments a
project would provide if it were an annuity. Generally, when comparing projects of unequal lives,
the one with the higher equivalent annual annuity should be chosen.

To illustrate how this procedure is applied, let us assume the same data for the two projects,
Project M and Project N on page.

1. It is noted that
Project M's NPV =P51.550
Project N^ prime sNPV =p64.910

The Expected Net Cash Flows for Projects N and M are computed as follows:
The analyst however, should always consider a potential serious weakness inherent in this type
of unequal life analysis. These are:

(1) If inflation is expected, the static conditions built into the analysis would be invalid;
(2) Replacements that occur down to the road would probably employ new technology which in
turn might change the cash flows.
(3) Estimating the lines of a series of projects is different and is often just a speculation.

Given all the uncertainties in the estimation process, such projects would, for all practical
purposes be assumed to have the same life while the cash flow estimation is complicated, the
concepts involved are exactly the same in the two approaches.

Inflation and Capital Budgeting

Does inflation affect a capital budgeting analysis? The answer is a qualified yes in that inflation
affects the number that are used in the analysis but does not affect the results of the analysis of
certain conditions are satisfied:

This is best illustrated using the following data:

Marvex Corporation, wants to purchase a new equipment that costs P360,000. The equipment
would provide annual net cash flows from operations of P200,000 and it would have a three
year life with no salvage value. For each of the next three years, the company expects a 10%
inflation rate in the cash flows associated with the new machine. If the company's real cost of
capital is 12% or market based cost of capital of 23.2%, should the equipment be purchased.

Analysis:

When performing a new present value analysis, the following should be observed,

1. If a "market-based cost of capital" is used to discount cash flows, then the cash flow should
be adjusted upwards to reflect the effects of inflation in forthcoming period.

2. If the "real cost of capital" is used in the analysis, there is no need to adjust the cash flows
upward since the inflationary effects have been taken out of the discount rate.
It will be noted that the net present value obtained in Solution B, where inflation is explicitly
taken into account is the same within rounding error to the obtained in Solution A where the
inflation effects were not considered. This result may seem surprising but it is logical because
we have adjusted both the cash flow and the discount rate so that they are consistent and these
adjustments cancel each other out across the two solutions.