reliability assessment of bufferless production …ieomsociety.org/dc2018/papers/266.pdffrom the...

Proceedings of the International Conference on Industrial Engineering and Operations Management Washington

DC, USA, September 27-29, 2018

© IEOM Society International

Reliability Assessment of Bufferless Production System: A

Boolean Algebra and K-Map Approach Firas Sallumi ([email protected]) and Walid Abdul-Kader ([email protected])

Industrial and Manufacturing Systems Engineering

University of Windsor, Windsor (ON) Canada

Abstract In this paper, system reliability is determined based on a K-map (Karnaugh map) and the Boolean algebra

theory. The proposed methodology incorporates the K-map technique for system level reliability, and

Boolean analysis for interactions. It considers not only the configuration of the production line, but also

effectively incorporates the interactions between the machines composing the line. Through the K-map, a

binary argument is used for considering the interactions between the machines and a probability expression

is found to assess the reliability of a bufferless production system. The paper covers three main system

design configurations: series, parallel and hybrid (mix of series and parallel). In the parallel production

system section, three strategies or methods are presented to find the system reliability. These are the Split,

the Overlap, and the Zeros methods. A real-world case study from the automotive industry is presented to

demonstrate the applicability of the approach used in this research work.

Keywords: Series, Series-Parallel, Hybrid, Bufferless, Production Line, Reliability, K-Map, Boolean

algebra

1. IntroductionPresently, the global economy is forcing companies to have their production systems maintain low inventory and

provide short delivery times. Therefore, production systems are required to be reliable and productive to make

products at rates which are changing with the demand. Moreover, many products have a very short life cycle that

results in reducing the time available to develop production systems. So, the designers of production systems need

systematic models to predict system reliability and performance under diverse production scenarios.

Productivity is one of the most important performance measures for a certain machine or system. Usually, productivity

is estimated about both the reliability and the production rate of the production system. The parameters selected for

a machining process at a certain production rate affect the reliability of the machine and consequently its productivity.

The productivity of a system is not only dependent on its reliability, but is also dependent on its configuration. Thus,

the productivity of a production system can be estimated by considering both system design and process planning

simultaneously. To determine production system productivity, the individual machine reliability should be measured.

At the system level, the reliability depends on the system configuration of the production process. Production system

configurations could be serial, such as a traditional transfer line, or parallel, and usually a duplicate of serial lines in

parallel (parallel-series) as required for a slow production processes. Another configuration is the hybrid line, which

consists of a mix of serial and parallel (series-parallel) connected machines that could take any configuration.

Reliability is a major attribute in the design and operation of today’s large and complex production systems. The

integrity of modern production systems is strongly dependent on the availability and reliability of their machines.

However, reliability theory depends heavily on an understanding of failure physics modeling and on techniques of

probability and statistics. Therefore, reliability models play a significant role in system analysis. Typically, reliability

predictions for complex production systems begin with predictions of the probabilities of mission success for the

machines in a system. Predictions are then combined in accordance with a logic model that describes how the

machines interact in the system. The result is a predicted mission success probability for the production system.

Therefore, it is of great importance to have practical models which efficiently predict the reliability of complex

systems, and also give useful design information with respect to individual machines. There are two main benefits of

using a K-map regarding system design. First, the number of machines in the system will present all possible states

the system can have and which of those states are producing. This can be easily found from the system truth table.

Second, by constructing a K-map of the production system, it will make it easier to implement changes on the K-map

and re-project them on the original production system. Great attention should be given to keep the integrity of the

production system intact.

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mailto:[email protected]




Evaluating system reliability is not new using Boolean algebra minimization. In contrast, based on the review of

related literature presented below and to the best of our knowledge, there has been no detailed study found on using

K-map technique to simplify system configuration and find its reliability.

This paper exploits a clear-cut method to find system reliability. A K-map is used by considering all possible cases

that can occur while solving to find the final probability expression. Based on the production system configuration, a

Boolean equation can be generated to model the system. A Boolean variable is assigned to each machine in the system

since any machine can be in one of two states: either working (logic 1, or X) or not working (logic 0, or X’). Depending

on the production system configuration, the product will move from one machine to another. Usually the flow of a

product is in a series configuration of machines, thus the system equation that relates the Boolean variables is

represented by AND operation ((.) or (*) operator). Sometimes there are more than one way for a product to flow; in

this case, the system equation relates the Boolean variables as OR operation represented by a (+) sign. Without using

a K-map and Boolean algebra, any complex hybrid system can be reduced to a single machine by applying the

following two rules successively (Bazovsky, 2004):

Rule 1: When n machines are all in series, the reliability of the system is equivalent to a single machine with

reliability =

n

i

iP1

)(

Rule 2: When n machines are all in parallel, the reliability of the system is equivalent to a single machine

with reliability =

n

i

iP1

)1(1

However, the solution would be simplifying mathematical equations only. The K-map technique provides a complete

visualization of the production system since every (+) means machines in parallel and every (.) means machines in

series. Consequently, any production system can be easily constructed from its final Boolean expression.

On the other hand, buffers are not considered in the system equation because they couple machines and there are no

Boolean variables for them as stated in Freiheit et al. (2004). Accordingly, this research work is concerned with

bufferless production systems.

For example, the production system shown in Figure 1 has a Boolean system equation that is a function of the three

Boolean variables that represent its three machines:

F (A, B, C) = A + (B.C) = A + BC

Figure 1: Three-machine production system

From the Boolean expression, F (A, B, C), and without looking at Figure 1, it can be noticed that machine B and

machine C are in series (B.C) and both of them are in parallel with machine A. The system states (minterms in Boolean

algebra) can be determined from the number of machines (in this case; 23 = 8 states). The production system is

producing if the Boolean system equation equals one, i.e., F = 1. This condition will be achieved if either machine (A)

is working, or both machines (B and C) are working, or all three machines are working. Since machines (B) and (C)

are in series, then if either one is not working, the whole branch will not be producing. The corresponding truth table

is shown below.

From Table 1, A is a variable denoting the state of machine A when it is operating, and A’ is the variable when it is

not operating. The same applies for the remaining machines B and C. The minterms m0 –m7 present the possibility

of “Anding” all variables (A, B, and C) in a function, which also represents all possible system states of a production

system.

In this research work, a K-map method is presented to calculate the reliability of the production system. Originally,

the K-map is a graphical representation used to simplify Boolean expressions of digital circuits. The K-map technique

is simple and can be easily applied to production lines regardless of their configurations (series, parallel or hybrid).

The construction of a K-map from a Boolean expression will be explained later in this paper to emphasize the logic

behind using a K-map.

F A

B C

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Table 1: Truth table for production system shown in Figure 1

Decimal

(state #)

Minterms

ID

Minterms or

System States

A B C Output (F)

0 m0 A’B’C’ 0 0 0 0

1 m1 A’B’C 0 0 1 0

2 m2 A’BC’ 0 1 0 0

3 m3 A’BC 0 1 1 1

4 m4 AB’C’ 1 0 0 1

5 m5 AB’C 1 0 1 1

6 m6 ABC’ 1 1 0 1

7 m7 ABC 1 1 1 1

The remainder of this paper is divided in 5 more sections where section 2 covers the literature review, followed by

section 3 on Boolean algebra and K-map techniques. Section 4 presents the design methodology. It also shows the

applicability of this approach on a real-world case study. Sections 5 and 6 cover discussions of the approach and

conclude the paper, respectively.

2. Review of Related Literature Many papers involving reliability calculations using Boolean algebra and a K-map have been reviewed. For this

paper, the most relevant published papers will be discussed. For each paper, there will be a brief explanation of the

technique used and they are chronologically organized according to their date of issue. The aim is to show what

techniques are available and how the proposed methodology is different than those already published.

Case (1977) applies a reduction technique (using Boolean algebra) to obtain a simplified reliability expression

(probability of success) from the canonical form of minterms (having independent variables) such as those generated

from a truth table format. The resulting terms are always mutually exclusive, which allows a simple and direct

transformation to a probability expression.

Bennetts (2004) proposes a method that considers the component reliability parameters as Boolean variables rather

than probabilistic variables and to treat the whole problem as if it were Boolean. A technique has been developed

based on analysing and modifying the Boolean expression prior to the conversion process. The technique was

originally developed as an aid to fault-tree analysis, but it also applies to general problems of reliability assessment.

Koo (1990) conveys the concept of Disjointed Boolean (D/Boolean) algebra. D/Boolean algebra has been in existence

for many years. However, the concept of D/Boolean algebra has not been fully explained to the general public in a

manner comparable to Boolean algebra. D/Boolean algebra is treated as a probability function. This approach gives

the reliability analyst the choice of preserving only essential logic functions, removing restraints of a cumbersome

manipulation of unnecessary logics, and deriving a probability solution of a system. The main theme is to illustrate

how a reliability solution of a moderately complex system can be solved manually. However, for more complex

systems, the use of software is possible and recommended.

Freiheit et al. (2004) examine the importance of the configuration of a production system on productivity, and the

improvements that can be obtained in bufferless parallel-serial configurations. For example, multiple machining lines

in parallel are considered, both with and without crossover between the operations. Combinatorial algebraic models

are developed to quantify the productivity of serial-parallel configurations. It is shown that significant improvements

to productivity can be obtained by placing operations in parallel, and that there is a synergistic improvement to

productivity from having a crossover between the operations.

Rahmat et al. (2006) investigate the application of the Boolean truth table modeling method in estimating the

reliability parameters, such as the system’s failure rates and mean time between failures (MTBF) for the

uninterruptible power supply (UPS) systems. All possible state combinations (operating and failed states) of the

major components in the UPS systems were listed and their effects on the overall system were studied.

Kumar et al. (2013) study the reliability of rice mill using Boolean function technique and algebra to logics to assess

the overall performance of the mill.

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Zaitseva and Levashenko (2017) develop a mathematical method for the reliability analysis and evaluation of multi-

state system reliability. They use multiple-valued logic that is a natural extension of Boolean algebra used in reliability

analysis. Colbourn (2010) presented a chapter on the Boolean aspects of network reliability.

The technique used in this paper is based on the K-map and Boolean algebra. It is different from all preceding

techniques because it follows a systematic approach by using three methods (Split, Overlap, and Zeros) to find system

reliability. The Split method is relatively short and it can be applied by forming separate groups of ones to cover all

minterms only once. The Overlap method is used to form overlapped groups of ones to cover as many ones as possible

inside the K-map. However, extra attention should be given to the ones used more than once in the groups. The Zeros

method could be an easy way to find the reliability of the production system when the number of ones is very high

compared to the number of zeros.

3. Boolean Algebra and K-map Basic Identities of Boolean Algebra: The most basic identities of Boolean algebra are listed in Table 2 below. The first nine identities show the relationships

between a single variable X, its complement X’, and the binary constants 0 and 1. The next five identities, 10 through

14, have equivalents in ordinary algebra. The last five identities, 15 through 19, do not apply in ordinary algebra, but

they are useful in manipulating Boolean expressions as stated in Mano and Kime (2004).

The rules listed in Table 2 have been arranged into two columns to demonstrate the property of duality of Boolean

algebra. The property of duality of an algebraic expression is obtained by interchanging OR with AND operations

and replacing 1’s by 0’s and 0’s by 1’s. For example, relation 2 is the dual of relation 1 because OR has been

replaced by AND and 0 by 1. It can be noticed that most of the time the dual of an expression is not equal to the

original expression.

The first eight identities involving a single variable can be easily verified by substituting each of the two possible

values for X. For example, to show that X + 0 = X, let X = 0 to obtain 0+ 0= 0, and then let X = 1 to obtain 1 + 0 =

1. Note that identity 9 states that double complementation restores the variable to its original value. Thus, if X = 0,

X’ = 1, then X’’ = 0 = X.

The commutative laws represented by identities 10 and 11 state that the order in which the variables are written will

not affect the result when using the OR and/or AND operations. The associative laws represented by identities 12

and 13 state that the result of applying an operation over three variables is independent of the order that is taken, that

is:

X + (Y + Z) = (X + Y) + Z = X + Y + Z and X (YZ) = (XY) Z = XYZ

Table 2: Basic Identities of Boolean Algebra, Mano and Kime (2004) and Mano (2005)

1 X + 0 = X 2 X . 1 = X

3 X + 1 = 1 4 X . 0 = 0

5 X + X = X 6 X . X = X

7 X + X’ = 1 8 X . X’ = 0

9 X’’ = X

10 X + Y = Y + X 11 XY = YX Commutative

12 X + (Y + Z) = (X + Y) + Z 13 X (YZ) = (XY) Z Associative

14 X (Y + Z) = XY + XZ 15 X + YZ = (X + Y) (X + Z) Distributive

16 (X + Y)’ = X’ . Y’ 17 (XY)’ = X’ + Y’ DeMorgan’s

18 X + XY = X 19 X (X + Y) = X Absorption

The first distributive law is given by identity 14. The second distributive law, given by identity 15, is the dual of the

ordinary distributive law and does not hold in ordinary algebra. Each variable in an identity can be replaced by a

Boolean expression and the identity still holds. By letting X = A, Y = B, and Z = CD, and applying the second

distributive law, we obtain:

(A + B) (A + CD) = A + BCD

All laws mentioned above are integrated in the process of simplifying the Boolean expression of a K-map to find the

reliability of the production system.

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Identities 16 and 17 in Table 2, (X + Y)’ = X’ . Y’ and (XY)’ = X’ + Y’, are referred to as DeMorgan’s theorem.

This is a very important theorem and it is used to obtain the complement of an expression and of the corresponding

function. DeMorgan’s theorem can be illustrated by means of truth tables that assign all the possible binary values

to X and Y. Table 3 shows two truth tables that verify the first part of DeMorgan’s theorem. In I, we evaluate (X +

Y)’ for all possible values of X and Y. This is done by first evaluating X + Y and then taking its complement. In II,

we evaluate X’ and Y’ and then ANDing them together. The result is the same for the four binary combinations of

X and Y, which verifies the identity of the equation.

DeMorgan’s theorem can be applied to production systems theoretically by replacing machines in parallel with

machines in series and vice versa so that system design and configuration can be obtained.

Table 3: Truth Tables to Verify DeMorgan’s Theorem, Mano and Kime (2004)

I X Y X + Y (X + Y)’ II X Y X’ Y’ X’ . Y’

0 0 0 1 0 0 1 1 1

0 1 1 0 0 1 1 0 0

1 0 1 0 1 0 0 1 0

1 1 1 0 1 1 0 0 0

The last two identities 18 and 19 in Table 2, X + XY = X, and X . (X + Y) = X, are referred to as the absorption law.

This is a very useful law used to simplify Boolean expressions. The absorption law can be demonstrated by using

truth tables that assign all the possible binary values to X and Y. Table 4 shows two truth tables that verify both parts

of the absorption law. In I, we evaluate (X + XY) for all possible values of X and Y. This is done by first evaluating

XY and then adding X to it. In II, we evaluate (X + Y) and then ANDing it with X. The result is the same for the

four binary combinations of X and Y, which verifies the absorption law.

Table 4: Truth Tables to Verify the Absorption Law

I X Y XY X + XY II X Y X + Y X . (X + Y)

0 0 0 0 0 0 0 0

0 1 0 0 0 1 1 0

1 0 0 1 1 0 1 1

1 1 1 1 1 1 1 1

K-Map: The map is a diagram made up of squares, with each square representing one minterm of the function. Minterms are

the possibilities of ANDing all the variables in a function. So, for two-variable function, there are four minterms and

for three-variable function there are eight minterms. In general, for the n-variable function, there are 2n minterms.

Since any Boolean function can be expressed as a sum of minterms, it follows that a Boolean function is recognized

graphically in the map from the area enclosed by those squares whose minterms are included in the function.

Alternative algebraic expressions can be derived for the same function by recognizing various patterns (Mano, 2005).

To explain the idea of generating K-maps, a four-variable map will be presented only, since a two-variable map and

three-variable map can be easily understood from the four-variable map, as well as, five-variable and six-variable

maps considered as duplicates of the four-variable map.

Four-Variable Map: A four-variable map is shown in Figure 2a. For the four-variable function, there are sixteen minterms and thus a map

of sixteen squares is provided. Note that the minterms are not arranged in sequence inside the sixteen squares because

only one bit change in value is allowed (0 to 1) or (1 to 0) from one adjacent column to the next, as shown in Figure

2b.

To understand the usefulness of the map for simplifying Boolean functions, the basic property possessed by adjacent

squares must be recognized. Any two adjacent squares in the map differ by only one variable, which is primed in one

square and unprimed in the other. For example, m5 and m7 lie in two adjacent squares. Variable y is primed in m5

and unprimed in m7, whereas the other three variables are the same in both squares. From the claims of Boolean

algebra, it follows that the sum of two minterms in adjacent squares can be simplified to a single AND term consisting

of only three literals. To further clarify this, the sum of m5 and m7 will give:

m5 + m7 = w’xy’z + w’xyz = w’xz (y’ + y) = w’xz , since (y’ + y) = 1.

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a b

Figure 2: Four-variable map

Here the two squares (5 and 7) differ by the variable y, which can be removed when the sum of minterms is formed.

Thus, any two minterms in adjacent squares that are ORed together will result in a removal of the different variable.

Map minimization of four-variable Boolean function can be made by simplifying the Boolean equation through

solving four kinds of situations that can happen and as follows:

1. One minterm alone results in a one term relation of four ANDed variables as shown in case 1 of Figure 3a.

2. Two adjacent minterms that can be combined together to form a three ANDed relation as shown in case 2 of

Figure 3a.

3. Four adjacent minterms that can be combined together to cancel two variables and produce two ANDed

variables as shown in case 3 of Figure 3a.

4. Eight minterms combined together to give one variable as shown in case 4 of Figure 3b.

a b

case 1 (w’x’y’z’) case 2 (wyz) case 3 (wx) case 4 (z’)

Figure 3: Combination cases of four-variable map

Applying the rules above to production systems, case 2 for example, shows that there will be production at the output

if machines w, y, and z are working in series regardless of machine x working or not, i.e., machine x is not affecting

the production of machines w, y, and z in this case.

15 14

11 10

12 13

8 9

1

1

1

1

1

1

1

1

00 01 yz

wx

1 1

1

11 10

1

1

1

1

00

01

11

10

1

1

00 01 yz

wx

3 2

7 6

11 10

00

01

11

10

0 1

4 5

15 14

11 10

12 13

8 9

1

1

1 1 1 1

m3

m2

m7 m6

3 2

7 6

m0

m1

m4 m5

0 1

4 5

m15

m14

m11 m10

m12

m13

m8 m9

15 14

11 10

12 13

8 9

15 14

11 10

12 13

8 9

0 1

4 5

00 01 wx

w’x’yz

w’x’yz’

w’xyz w’xyz’

11 10

wxyz

wxyz’

wx’yz wx’yz’

wxy’z’

wxy’z

wx’y’z’ wx’y’z

00

01

11

10

w’x’y’z’

w’x’y’z

w’xy’z’ w’xy’z

yz

1035




As a general rule, to simplify any map, start with the highest combination possible (8 for 3 variables, 16 for 4 variables,

32 for 5 variables and so on). After that, go for the second combination possible (4 for 3 variables, 8 for 4 variables,

16 for 5 variables) and continue this trend until all minterms are covered. In addition, the K-map is considered to lie

on a surface with top and bottom edges, as well as right and left edges, touching each other to form adjacent squares.

For example, m0 and m2 form adjacent squares, as well m0 and m8.

4. Design Methodology

General Method for Reliability Calculations using a K-map and Boolean Algebra:

Series Production System: The block diagram of a series production system is shown in Figure 4. A, B, and C represent three independent

machines. All three machines must work in order for the system to produce.

Figure 4: Three-machine series production system The Boolean algebra expression for the occurrence of the overall event is therefore:

F (A, B, C) = ABC, and the probability expression would be: P (A, B, C) = P(A) P(B) P(C).

This function can be represented by the K-map as shown in Figure 5. It is clear the overall event F(.) requires the

intersection of A, B, and C for its occurrence. In other words, A = B = C = 1. The only possibility for these variables

to have the value of one is m7 as shown in Figure 5.

Figure 5: K-map for three-machine serial production system

For simplicity, m designating minterm is discarded from Figure 5 and all subsequent figures.

Parallel Production System: The block diagram of a parallel production system is shown in Figure 6. A, B, and C represent three independent

machines.

Figure 6: Three-machine parallel production system

The parallel connection indicates that if any machine of the three machines is working the system will produce. There

are three different paths through the system. When machine A is working: m4, m5, m6, and m7 must be equal to 1.

When machine B is working: m2, m3, m6, and m7 must be equal to 1, and when machine C is working: m1, m3, m5,

and m7 must be equal to 1 as shown in Figure 7. To calculate the reliability of the system shown in Figure 7, there

are mainly three methods:

1) Split Method: The split method is considered the shortest approach to find the reliability expression of the production system. It can

be applied by forming groups of ones according to the rules discussed previously by simply covering all minterms

only one time as shown in Figure 8. For Figure 8a, the reliability expression can be found as follows:

F (A, B, C) = A + A’B + A’B’C

P (A, B, C) = P(A) + [1 – P(A)] [P(B)] + [1 – P(A)] [1 – P(B)] [P(C)]

A B C

0 1

4 5

00 01

1

0

BC

A

11 10

3 2

7 6

1

A

B

C

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P (A, B, C) = P(A) + P(B) – P(A) P(B) + [1 – P(A) – P(B) + P(A) P(B)] [P(C)]

P (A, B, C) = P(A) + P(B) + P(C) – P(A) P(B) – P(A) P(C) – P(B) P(C) + P(A) P(B) P(C)

Figure 7: K-map for three-machine parallel production system

For Figure 8b, which is just another way of grouping the ones, but still following the same trend of combinations (4 –

2 – 1), the results stay the same and as follows:

F (A, B, C) = C + BC’ + AB’C’

P (A, B, C) = P(C) + [P(B)] [1 – P(C)] + [P(A)] [1 – P(B)] [1 - P(C)]

= P(C) + P(B) – P(B) P(C) + [P(A)] [1 – P(B) – P(C) + P(B) P(C)]

= P(C) + P(B) – P(B) P(C) + P(A) – P(A) P(B) – P(A) P(C) + P(A) P(B) P(C)

P (A, B, C) = P(A) + P(B) + P(C) – P(A) P(B) – P(A) P(C) – P(B) P(C) + P(A) P(B) P(C).

a b

Figure 8: Split method for group combination of parallel production system

2) Overlap Method: In this method, the groups are overlapped to get the most number of ones a group can have to cover all the ones inside

the K-map (in this case three groups of fours) as shown in Figure 9. However, great attention must be given to

minterms that are used more than one time. These minterms must be subtracted by the number of times they have

been used other than the first time.

Figure 9: Overlap method for the parallel production system

In other words, we have:

Minterm ABC is common into three groups which leads to (-2ABC).

Minterm ABC’ is common into two groups which leads to (-ABC’).

Minterm AB’C is common into two groups which leads to (-AB’C).

Minterm A’BC is common into two groups which leads to (-A’BC).

For Figure 9, the reliability expression can be found as follows:

1

1 1

0 1

4 5

00 01

1

0

BC

A

1

1

1 1

3 2

7 6

11 10

0 1

4 5

00 01

1

0

BC

A

3 2

7 6

11 10

1 1

1 1 1 1

0 1

4 5

00 01

1

0

BC

A

3 2

7 6

11 10

1

1

1

1 1

1 1

0 1

4 5

00 01

1

0

BC

A

3 2

7 6

11 10

1 1 1 1

1 1

1 1

1 1

1 1

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F (A, B, C) = A + B + C – 2ABC – ABC’ – AB’C – A’BC

P (A, B, C) = P(A) + P(B) + P(C) – 2P(A) P(B) P(C) – P(A) P(B) [1 – P(C)]

– P(A) P(C) [1 – P(B)] – P(B) P(C) [1 – P(A)]

= P(A) + P(B) + P(C) – 2P(A) P(B) P(C) – P(A) P(B) + P(A) P(B) P(C)

– P(A) P(C) + P(A) P(B) P(C) – P(B) P(C) + P(A) P(B) P(C)


3) Zeros Method: The Zeros method is another technique of finding the reliability of the production system that works the same way as

the ones, but the whole function is inverted (instead of F, F’ is used). Sometimes it could be easier to use the zeros

(m0 in this case) instead of the ones to calculate the reliability as shown in Figure 10.

Figure 10: Zeros method for the parallel production system

From Figure 10, the reliability expression can be found as follows:

F’ (A, B, C) = A’B’C’

P (A, B, C) = 1 – P’(A) P’(B) P’(C)

= 1 – [1 – P(A)] [1 – P(B)] [1 – P(C)]

= 1 – [1 – P(A) – P(B) + P(A) P(B)] [1 – P(C)]

= 1 – [1 – P(A) – P(B) + P(A) P(B) – P(C) + P(A) P(C) + P(B) P(C)

– P(A) P(B) P(C)]

= 1 – 1 + P(A) + P(B) – P(A) P(B) + P(C) – P(A) P(C) – P(B) P(C) + P(A)P(B)P(C)


Hybrid Production System: The meaning of the word hybrid here is a series-parallel combination of production machines that can function as one

system. The following case study explains how a K-map can be applied to find the reliability of a hybrid production

system consisting of an automated paint system in a medium-duty truck assembly company located in South-Western

Ontario, Canada.

Case Study: The automated paint system consists of five stages as shown in Figure 11. In Stage 1, the Scuff / Sand Booth (S), is

utilized for scuffing and sanding of cabs and plastic parts. In Stage 2, the Prep Booth (P), is utilized for solvent wipe

and tacking of cabs and plastic parts. Stage 3 is composed of Base Coat 1 and Base Coat 2 booths. Stage 4 is the

Clear Coat Booth, (CC). Stage 5 is the Oven (V) designed as a single pass enclosure with an air seal at the entrance

and exit. The block diagram of the paint system is shown in Figure 11. The production system has a hybrid

configuration consisting of five stages in series with one stage consisting of two parallel units. S, P, BC1, BC2, CC,

and V represent six independent booths or units.

The hybrid connection shown in Figure 11 indicates that the system paints the cabs and hoods if booths S, P, BC1,

CC, and V or S, P, BC2, CC, and V are working. The system paints if both combinations occur as well. The paint

system has six units and so there will be sixty-four minterms distributed over four maps of sixteen squares as shown

in Figure 12. It should be mentioned that there is no need to construct a truth table to find the final reliability expression

as long as the user can construct the K-map from the configuration of the production system.

1

1 1

0 1

4 5

00 01

1

0

BC

A

1

1

1 1

3 2

7 6

11 10

0

1038




It can be noticed that when booth S, P, CC, oven V or both BC1 and BC2 are not operational the whole paint system

is down. This will be translated in the K-map as zeros in squares (0 – 15), (16 – 31), and (48 – 63). On the other

hand, squares (32 – 38), (40 – 42), and (44 – 46) are also zeros because there will be no production for the same past

rules mentioned. Now, in order for the system to paint, all the ones must lie in the map of SP = 11 and column CCV

= 11. Likewise, booth BC1 or BC2 or both are functioning which translate to minterms m39, m43 and m47 are ones as

shown in Figure 12. To solve this problem, the split and the overlap methods are applied as shown in figures 12 and

13 presented below.

Figure 11: Block diagram for an automated paint system

The split method is used to combine two minterms m43 and m47 in one group and one minterm m39 alone. The Boolean

function and the reliability of the system can be found as:

F (S, P, BC1, BC2, CC, V) = S P CC V BC1 + S P CC V BC1’ BC2

P (S, P, BC1, BC2, CC, V) = P(S) P(P) P(CC) P(V) P(BC1)

+ P(S) P(P) P(CC) P(V) P(BC2)[1 – P(BC1)]

= P(S) P(P) P(CC) P(V) P(BC1) + P(S) P(P) P(CC) P(V) P(BC2)

– P(S) P(P) P(CC) P(V) P(BC1) P(BC2)

P (S, P, BC1, BC2, CC, V) = P(S) P(P) P(CC) P(V) [P(BC1) + P(BC2) – P(BC1) P(BC2)].

Based on historical data: P(S) = P(P) = 0.99, P(V) = 0.97 & P(BC1) = P(BC2) = P(CC) = 0.87. Thus, the reliability

of the system is: P (S, P, BC1, BC2, CC, V) = 0.8131 or 81.31%

To prove the same results, the overlap method is used as shown in Figure 13. As described earlier, by combining two

minterms m39 and m47 into one group, and minterms m43 and m47 into another group, the Boolean function and the

reliability of the system can be found as follows:

F (S, P, BC1, BC2, CC, V) = S P CC V BC1 + S P CC V BC2 – S P CC V BC1 BC2

P (S, P, BC1, BC2, CC, V) = P(S) P(P) P(CC) P(V) P(BC1) + P(S) P(P) P(CC) P(V) P(BC2)

– P(S) P(P) P(CC) P(V) P(BC1) P(BC2)

SP = 00 SP = 01

15 14

11 10

12 13

8 9

3 2

7 6

0 1

4 5

00 01 11 10

00

01

11

10

16

00 01 11 10

00

01

11

10

20

28

24

17

21

29

25

19

23

31

27

18

22

30

26

16

15 14

11 10

12 13

8 9

CC V CC V

BC1 BC2 BC1 BC2

Scuff

Sand

Booth

Prep.

Booth

B.C. 1

Booth

B.C. 2

Booth

C.C.

Booth Oven

S P CC V

BC1

BC2

Cabs &

Hoods

Unpainted

Cabs &

Hoods

Painted

1039




SP = 11 SP = 10

Figure 12: K-Map for the paint system using split method

SP = 00 SP = 01

SP = 11 SP = 10

Figure 13: K-Map for the paint system using overlap method

15 14

11 10

12 13

8 9

3 2

7 6

0 1

4 5

00 01 11 10

00

01

11

10

16

00 01 11 10

00

01

11

10

20

28

24

17

21

29

25

19

23

31

27

18

22

30

26

16

15 14

11 10

12 13

8 9

15 14

11 10

12 13

8 9

16

0 01 11 10

00

01

11

10

60

56

49

53

61

57

51

55

63

59

50

54

62

58

48

15 14

11 10

12 13

8 9

00 01 11 10

00

01

11

10

32 33

36 37

35 34

38

44 45

40 41

46

42

CC V CC

V

CC V CC

V

BC1 BC2 BC1 BC2

BC1 BC2 BC1 BC2

1

1

1

1

15 14

11 10

12 13

8 9

16

00 01 11 10

00

01

11

10

52

60

56

49

53

61

57

51

55

63

59

50

54

62

58

48

15 14

11 10

12 13

8 9

00 01 11 10

00

01

11

10

32 33

36 37

35 34

38

44 45

40 41

46

42

CC V CC V

BC1 BC2 BC1 BC2

1

1

1

52

1040




5. Discussion Finding the reliability of the production system using a K-map and Boolean algebra has many advantages because this

method is considered a direct solution technique. As it can be investigated from the above example, some of the

advantages of this technique are summarized below:

Advantages:

1) Finding reliability expression for a production system using a K-map presents a novel approach of looking

at the system from a graphical point of view without dealing with long mathematical equations.

2) Regarding system design, a K-map presents all possible states the system can have and which of those states

are producing. Also, it can be easily found from the system truth table.

3) Constructing K-map of any production system will make it easier on the designer to implement changes on

the K-map and re-project them on the original production system. As every square of the K-map represents

one state of the production system, the designer during the design process can add one to any square in the

K-map and re-evaluate the system for reliability improvement. The added “one” is considered a producing

state, which can be achieved by changes in production line configuration. This is an easy technique that can

be done graphically or manually. This important feature makes K-map very unique to designer imagination.

4) The K-map and Boolean algebra manipulation have the advantage over other methods of being an exact

solution. There are no approximations in the procedure.

5) A Boolean equation can be easily transformed into a K-map, and generally, the probability expression can

be straightforwardly found from the constructed K-map. This feature is quite important when dealing with

complicated hybrid production systems.

6) The K-map and Boolean approach provides complete freedom in assigning availabilities and production rates

to each machine in the system, but this is beyond the scope of this paper.

K-map technique is very easy to deal with to find the reliability of the production system. However, as the system

gets longer, calculations become lengthy since the number of system states is (2number of machines). In this case, the use

of decomposition techniques is possible.

6. Conclusion This paper addressed the reliability assessment of a production line by considering serial, parallel and hybrid or

series-parallel configurations. Following the proposed methodology, the production system can be broken down

into its independent states and a K-map can then be constructed by illustrating every possible combination of the

independent events, that together, can bring a complete product to the end of the production line. The drawn K-map

filled with “one” in every square represents an intersection of the corresponding variables. Then, all ones can be

combined together according to the procedure described earlier to obtain the overall probability function as a sum of

mutually exclusive terms, each of which is a product of independent probabilities. The K-map method is

“universal” in the sense that it is a uniform technique that can be applied equally well to all probability

combinations, regardless of whether the configuration of the production system is simple or complicated. The real-

world case study taken from the automotive industry demsonstrate the applicability and the practicality of the

modeling approach on an unbuffered series-parallel production line.

However, the K-map technique is a way of solving the reliability problem graphically. So, even without any

mathematical equation, a person can find the reliability of the system using one of the three methods mentioned in the

paper. Moreover, the K-map technique gives the final reliability expression in one statement, and in particular, with

the split and the zeros methods by just replacing the algebraic terms with probability values. Another feature

considered as an advantage of using a K-map is that it can give the analyst a sense of how the machines, forming a

production system, interact among themselves within the system itself. In other words, because the K-map takes all

possible states the system can have, the analyst can, other than solving the reliability problem, discover more possible

solutions or scenarios to improve or modify the original configuration of the production system.

References Bazovsky, I., “Reliability Theory and Practice,” Dover Publications Inc., First Edition, 2004.

Bennetts, K.G., “Analysis of Reliability Block Diagrams by Boolean Techniques,” IEEE Transactions on Reliability,

Vol. R-31, No. 2, June 1982.

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Case, T., “A Reduction Technique for Obtaining a Simplified Reliability Expression,” IEEE Transactions on

Reliability, Vol. R-26, No. 4, October 1977.

Colbourn, C., “Boolean Models and Methods in Mathematics, Computer Science, and Engineering,” Part of

Encyclopedia of Mathematics and its Applications, Chapter 17, Editors: Yves Crama and Peter L. Hammer, June

2010, ISBN: 9780521847520.

Freiheit, T., M. Shpitalni and S. J. Hu, “Productivity of Paced Parallel-Serial Manufacturing Lines With and Without

Crossover,” Journal of Manufacturing Science and Engineering, Vol. 126, May 2004.

Koo, D.Y., “D/Boolean Applications in Reliability Analysis,” IEEE Proceedings of Annual Reliability and

Maintainability Symposium, 1990.

Kumar, M., Jasbir Singh, J., and Avtar, R., “Reliability Analysis of Rice Mill by Boolean Function Technique,”

International Journal of Engineering Trends and Technology, Vol. 4 (5), May 2013.

Mano, M., M., and Kime, C.R., “Logic and Computer Design Fundamentals”, Prentice Hall, Third Edition, 2004.

Mano, M., M., “Digital Design,” Prentice Hall, Third Edition, 2005.

Rahmat,M. K., S. Jovanovic and K. L. Lo, “Reliability Estimation of Uninterruptible Power Supply Systems: Boolean

Truth Table Method,” IEEE Telecommunications Energy Conference, 28th Annual International, pp 1-6, September

2006.

Zaitseva, E., and Levashenko, V., “Reliability analysis of multi-state system with application of multiple-valued

logic,” International Journal of Quality & Reliability Management, Vol. 34 (6), 2017.

Biographies Firas Sallumi holds Industrial & Electrical Engineering degrees, and a master of applied science in Industrial

Engineering from the University of Windsor. He has vast knowledge in quality assurance, reliability assessment, root

cause failure analysis, lean manufacturing, and lean maintenance (preventive, predictive, condition-based, and

reliability-centered maintenance). Keen on safety, planning, organizing, eliminating waste, & continuous

improvement. Manufacturing experience in food & beverage packaging, pharmaceutical, automotive, DC power

supplies, and steelmaking.

Walid Abdul-Kader is a professor of Industrial Engineering in the Faculty of Engineering at the University of

Windsor, Windsor, Canada. He holds a PhD degree in Mechanical Engineering from Université Laval, Québec City,

Canada. He completed his bachelor’s degree from Université du Québec à Trois-Rivières, Canada, and master’s degree

from École polytechnique de Montréal, Canada. His research interests relate to performance evaluation of reverse

logistics and manufacturing/remanufacturing systems prone to accidental failure and repair. His publications have

appeared in many leading national and international journals and conferences proceedings.

1042

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