# Efficient Blocked Fractional Factorial Designs for Stated Choice Experiments of Size Two and Four

Authors: Emmanuel Amoah; Jakperik Dioggban; Adjabui Michael Jackson
Journal: International Journal of Mathematics and Mathematical Sciences
 (2023)

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## Abstract

Stated choice experiments are increasingly becoming popular due to their ability to optimize information gain with limited resources. Many designs have been developed for the selection of various attributes and their levels to form choice sets. One such design is blocked fractional factorial design (BFFD). Stated choice experiments for symmetric attributes of 4 choice sets of sizes 2 and 4 and 8 choice sets of sizes 2 and 4 were developed using BFFDs. Generators for the stated choice set of sizes 2 and 4 with resolution three, four, and five were developed. The alias structures and confounding effects for the designs were derived, as well as their clear effects if any for estimation. TheA-efficiency was used to compute the efficiencies of the proposed designs since it has better statistical properties. The computed efficiencies for the proposed designs reveal that 4 choice sets of size 4 designs are more efficient. Finally, a practical application of the proposed method was carried out for four choice sets of size 4 using 2V5−1−2 design with attributes and levels of service quality in public transport.


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## Body

## 1. Introduction

Various strategies for constructing and analyzing stated choice (SC) experiments had been proposed in the literature. Most researches focus on the construction of the most desirable SC experiments and experiments with two-level attributes to mixed-level designs ([1–3]). The authors of [4] constructed SC experiments for paired designs, that is, choice sets with two options, for the estimation of first-order interactions for two-level attributes.The authors of [5] developed block fractional factorial designs with two- and three-level designs. The authors of [6] improved on the work of [5] designs through illustrations on the two-level block fractional factorial designs for resolution IV designs with choice set of size four with 2, 4, 16, 32 runs. The work of [5, 6] did not consider situations where the number of attributes for the two-level designs exceeds nine.This paper extends the work of [5] on 2-level block fractional factorial designs for attributes between 4 and 10 inclusive using the techniques of [6] for choice sets of sizes 2 and 4 to 8 choice sets on resolution three, four, and five designs.On the measure of efficiency, theD and A measures of efficiency focus on precise estimation of the model parameters, and the I and G measures of efficiency focus on precise predictions of the response [7]. This research designs focus on the estimation of the model parameters; hence, the D and A measures of efficiency are preferred. But according to [7] while there are many instances where A and D efficiency designs coincide, A efficiency designs tend to have better statistical properties when the A and D efficiency designs differ. Hence, this paper uses the A efficiency to compute the efficiencies of the proposed designs.


## 2. Methodology

### 2.1. Factorial Experiments

Factorial experiment is an experiment in which each complete trial or replicate of the experiment, and all viable combinations of the levels of the attributes are investigated [8].The effects model is(1)yijk=μ+τi+βj+τβij+εijk,τi is the effect of the ith level of the row factor A, βj is the effect of the jth level of the column factor B, τβij represent the effect of the interaction among τi and βj, and εijk is the random error, assumed to be N0,σ2 [8].


### 2.2.2m−n Fractional Factorial Design

If the number of factors (m) is of interest in a study with each at two (2) levels, resources available can only run 1/2n fraction of the complete factorial 2m design. That is, we need to execute a 1/2n partial of a 2m design. This kind of design is known as 2m−n fractional factorial design [5].A2m−n fractional factorial design of a 2m executes is known as a n/2 partial of the 2m design or otherwise known as a 2m−n fractional factorial design, with n<m. We first have to choose n unbiased generators for these designs which consist of the defining relation and their 2m−m−1 generalized interactions [8].


### 2.3. Blocked Fractional Factorial Designs

To block a2m−n FFD in 2m blocks in a b blocking variable, with blocks of length 2m−n−b, where n<m, we choose two generators (that is, the design and block generators). Effects related to these blocking variables, which include aliased effects, are mixed with blocks [5].


### 2.4. Stated Choice Experiments with Symmetric Attributes (2m−n−b Designs)

The block fractional factorial design (BFFD) consists of the choice sets and alternatives in the SC experiment, which is the number of blocks. The determined dimension of the block is also the number of alternatives in every choice set, which means the number of choices sets in a SC experiment is2b, that is the number of blocks, and the number of alternatives within the choice sets is 2m−n−b, that is the block dimension. Preference of the BFFD relies upon the following: that is, (1) number of attributes (m), (2) preferred length of choice set (that is, chosen number of alternatives), and then (3) effects which will be established to be clear [6].The number of alternatives is a power of the factor level. This is to be depended on when using a BFFD to come up with a choice set. A choice sets that contain two or four choices can be modeled with two-level BFFD, that is21 and 22 choices, respectively. Therefore, to model a choice set with 21 choices, for example, the required block is 2m−n−b = 2. Also, to model a choice set with 22 choices, the block dimension is 2m−n−b = 4. But if we want to model a choice sets with more than 22 choices, using block size will be 2m−n−b = 8, 16, 32, and so on [6].


### 2.5. Treatments Allocation Using Defining Contrast

This method makes use of the linear combination method:(2)b=λ1y1+λ2y2+…+λkyk,where b is the defining contrast, λi is the exponent appearing on the ith factor in the effect to be confounded, and yi is the level of the ith factor appearing in a particular treatment combination [8].For the2m system, λi = 0 or 1 and yi = 0 (low level) or yi = 1(high level). Treatment combinations that produce the same value of b (mod 2) will be placed in the same block [8].


### 2.6.A-Efficiency

A-optimal designs are said to minimize the average variance of the p parameter estimates [9]. Hence, A-optimal designs minimize the sum of the diagonal elements of the ordinary least squares estimator’s variance-covariance matrix. Hence, A-optimal designs minimize traceX′X−1. This measure associated with the A-optimality criterion is given as follows:(3)A−efficiency=100×1NDtraceX′X−1/p,where p is the number of parameters within the linear model or p is the number of factor effects within the design (columns in X), and ND is the number of design points or is the number of requested runs [9].


## 2.1. Factorial Experiments

Factorial experiment is an experiment in which each complete trial or replicate of the experiment, and all viable combinations of the levels of the attributes are investigated [8].The effects model is(1)yijk=μ+τi+βj+τβij+εijk,τi is the effect of the ith level of the row factor A, βj is the effect of the jth level of the column factor B, τβij represent the effect of the interaction among τi and βj, and εijk is the random error, assumed to be N0,σ2 [8].


## 2.2.2m−n Fractional Factorial Design

If the number of factors (m) is of interest in a study with each at two (2) levels, resources available can only run 1/2n fraction of the complete factorial 2m design. That is, we need to execute a 1/2n partial of a 2m design. This kind of design is known as 2m−n fractional factorial design [5].A2m−n fractional factorial design of a 2m executes is known as a n/2 partial of the 2m design or otherwise known as a 2m−n fractional factorial design, with n<m. We first have to choose n unbiased generators for these designs which consist of the defining relation and their 2m−m−1 generalized interactions [8].


## 2.3. Blocked Fractional Factorial Designs

To block a2m−n FFD in 2m blocks in a b blocking variable, with blocks of length 2m−n−b, where n<m, we choose two generators (that is, the design and block generators). Effects related to these blocking variables, which include aliased effects, are mixed with blocks [5].


## 2.4. Stated Choice Experiments with Symmetric Attributes (2m−n−b Designs)

The block fractional factorial design (BFFD) consists of the choice sets and alternatives in the SC experiment, which is the number of blocks. The determined dimension of the block is also the number of alternatives in every choice set, which means the number of choices sets in a SC experiment is2b, that is the number of blocks, and the number of alternatives within the choice sets is 2m−n−b, that is the block dimension. Preference of the BFFD relies upon the following: that is, (1) number of attributes (m), (2) preferred length of choice set (that is, chosen number of alternatives), and then (3) effects which will be established to be clear [6].The number of alternatives is a power of the factor level. This is to be depended on when using a BFFD to come up with a choice set. A choice sets that contain two or four choices can be modeled with two-level BFFD, that is21 and 22 choices, respectively. Therefore, to model a choice set with 21 choices, for example, the required block is 2m−n−b = 2. Also, to model a choice set with 22 choices, the block dimension is 2m−n−b = 4. But if we want to model a choice sets with more than 22 choices, using block size will be 2m−n−b = 8, 16, 32, and so on [6].


## 2.5. Treatments Allocation Using Defining Contrast

This method makes use of the linear combination method:(2)b=λ1y1+λ2y2+…+λkyk,where b is the defining contrast, λi is the exponent appearing on the ith factor in the effect to be confounded, and yi is the level of the ith factor appearing in a particular treatment combination [8].For the2m system, λi = 0 or 1 and yi = 0 (low level) or yi = 1(high level). Treatment combinations that produce the same value of b (mod 2) will be placed in the same block [8].


## 2.6.A-Efficiency

A-optimal designs are said to minimize the average variance of the p parameter estimates [9]. Hence, A-optimal designs minimize the sum of the diagonal elements of the ordinary least squares estimator’s variance-covariance matrix. Hence, A-optimal designs minimize traceX′X−1. This measure associated with the A-optimality criterion is given as follows:(3)A−efficiency=100×1NDtraceX′X−1/p,where p is the number of parameters within the linear model or p is the number of factor effects within the design (columns in X), and ND is the number of design points or is the number of requested runs [9].


## 3. Proposed Design and Block Generators

### 3.1. Design and Block Generators form = 3, 4, and 5 with 4 Choice Sets of Size 2

#### 3.1.1. Generators and Defining Relations

LetMNP and MNOQ be the generators of the 25−2 design, in Table 1. The generalized interactions are formed by taking all 2-way products as follows:(4)MNPMNOQ=M2N2OPQ=OPQ.Table 1
Eight-run design and block generators form = 3, 4, and 5 with 4 choices set of size 2.
mnbDesign generatorsBlock generatorsDefining contrastIBlock contrastsFractionsAliasing effectsConfounding effectsNo. of runsClear effects302…b1=MNb2=MOMN,MO,NO2III3…MN,MO, NO8All 3 main effects412P≡MNOb1=MNb2=MOI≡MNOPMN,MO,NO2IV4−1M≡OPM≡NPMO,NP,MN,OP,NO,MP8All 4 main effects522P≡MN, Q≡MOb1=MOb2=NOI≡MNP≡MNOQ≡OPQMO,NO,MN2III5−2N≡MPM≡NPP≡MN≡OQN≡MPQ≡OPO≡PQM≡MNN≡MOMO,NQ,NO,MQ,MN,P,OQ8NoneNote. The rest of the designs are illustrated in Table 1.That is, the treatment defining relation is given byI≡MNP≡MNOQ≡OPQ because the length of the shortest “word” is three, and this design is resolution III, denoted as 2III5−2 fractional factorial design. The detailed results are contained in Table 1.


#### 3.1.2. Alias Structures

Considering25−2 design in Table 1, with design relation:(5)I≡MNP≡MNOQ≡OPQ.The aliased structure is determined by multiplying the effects with the design relation.Let consider the main effects,MMNOQ≡M2NOQ≡NOQ and MOPQ≡MOPQ. Similarly, we can obtain the other aliases effects as given as follows.Hence, the main effects and two-factor interactions are aliased as follows:(6)M≡NP≡NOQ≡MOPQ,Q≡MNPQ≡MNO≡OP,N≡MP≡MOQ≡NOPQ,O≡MNOP≡MNQ≡PQ,P≡MN≡MONPQ≡OQ,MNP≡OPQ≡MNOQ.


#### 3.1.3. Determining the Confounding of Effects

Consider the 8-run of25−2 design in four blocks in Table 1, with the treatment defining contrast subgroup:(7)I≡MNP≡MNOQ≡OPQ.The generalized block defining contrast subgroup is formed by taking all 2-way products as follows:MONO≡MNO2≡MNThat is, the block-defining contrast subgroup is given byMO, NO, and MN.By multiplying each defining word byMO, NO, and MN, we have(8)MNPMO≡NOP,MNPNO≡MOP,MNPMN≡P,MNOQMO≡NQ,MNOQNO≡MQ,MNOQMN≡OQ,OPQMO≡MPQ,OPQNO≡NPQ,OPQMN≡MNOPQ.Considering only the block-defining generators, it might be concluded that the block effectsb1 and b2 are only confounded with MO, NO, and MN because the treatment-defining relation is also aliased with the effects NOP,NQ,MPQ,MOP,MQ,NPQ,P,OQ, and MNOPQ. Therefore, b1 and b2 are confounded with nine effects:(9)NOP,NQ,MPQ,MOP,MQ,NPQ,P,OQ,andMNOPQ.


#### 3.1.4. Clear Effects

Hence, from the above illustrations, the remaining 6 (= 8 − 2) degrees of freedom that do not involve the block effects, each of them is aliased with a two-factor or higher interactions, detailed can be seen in alias structures previously, whereM≡NP≡NOQ≡MOPQ and so on. Hence, the five main effects are not clear. Among the two-factor interactions, NQ, MQ, OQ, NO, MN, and MO are confounded with the block effect b1 and b2 and are not eligible for estimation. Hence, none of the effects is clear.


#### 3.1.5. Blocked Fractional Factorial Design for 4 Choice Sets of Size 2

The blocked fractional factorial design2III5−2−2 is an eight-run design. First, the design is constructed by choosing the column for P to be equal to column M plus column N (mod 2); that is, every entry in column P is sum of column M and column N (mod 2). Hence, the generators of this 2-level design are P=MN and Q=MO. If y1...y5 represents the five factors, then P=MN and Q=MO are equivalent to y4=y1+y2 (mod 2) and y5=y1+y3 (mod 2), respectively. The factor level for factor y4 is obtained by adding the corresponding treatment combinations of factor y1 and y2; likewise, factor y5 is obtained by adding the corresponding treatment combinations of factor y1 and y3.To illustrate this for treatment combination1, with the factor level for factor y1 = 0 and y2 = 0, factor y4 = 0 + 0 (mod 2) = 0 and y5 = 0 + 0 (mod 2) = 0. Also, for treatment combination a, with the factor level for factor y1 = 1 and y2 = 0, factor y4 = 1 + 0 (mod 2) = 1 and y5 = 1 + 0 (mod 2) = 1 and so on.To obtain 4 choice sets of size four, we select two effects to be confounded with the block. Suppose we choosey1y3 and y2y3 to be confounded with the blocks. These effects will have two block defining contrast, that is,(10)b1=y1y3,b2=y2y3.All the treatment combinations are evaluated usingb1 and b2. λ1=λ2=λ3=1.I: b1 = 1(0) + 0(0) + 1(0) = 0 (mod 2)b2 = 0(0) + 1(0) + 1(0) = 0 (mod 2)a: b1 = 1(1) + 0(0) + 1(0) = 1 (mod 2)b2 = 0(1) + 1(0) + 1(0) = 0 (mod 2)b: b1 = 1(0) + 0(1) + 1(0) = 0 (mod 2)b2 = 0(0) + 1(1) + 1(0) = 1 (mod 2)ab: b1 = 1(1) + 0(1) + 1(0) = 1 (mod 2)b2 = 0(1) + 1(1) + 1(0) = 1 (mod 2)c: b1 = 1(0) +0(0) + 1(1) = 1 (mod 2)b2 = 0(0) + 1(0) + 1(1) = 1 (mod 2)ac: b1 = 1(1) + 0(0) + 1(1) = 0 (mod 2)b2 = 0(1) + 1(0) + 1(1) = 1 (mod 2)bc: b1 = 1(0) + 0(1) + 1(1) = 1 (mod 2)b2 = 0(0) + 1(1) + 1(1) = 0 (mod 2)abc: b1 = 1(1) + 0(1) + 1(1) = 0 (mod 2)b2 = 0(1) + 1(1) + 1(1) = 0 (mod 2)Hence, from treatment combinations above, all treatment combinations with level 0 inb1 and b2 form block 1 which is an inherent choice set one, and also all treatment combinations with level 1 in b1 and level 0 in b2 form block 2, which also form choice set two. Also, all treatment combinations with level 0 in b1 and level 1 in b2 form block 3, which is an inherent choice set three, and finally, all treatment combinations with level 1 in b1 and level 1 in b2 form block 4, which also form choice set four.Therefore, from treatment combinationsI and abc forms choice set 1, a and bc form choice set 2, b and ac form choice set 3, and ab and c form choice set 4. Hence, a choice set of 4 and size 2 is formed.


#### 3.1.6. Design Efficiency

The result can be seen in Table2, which shows 60.0–71.4% A-efficiency for all the designs.Table 2
Design efficiency form = 3, 4, and 5 with 4 choice sets of size 2.
DesignsA-efficiency (%)2III3−0−260.02IV4−1−266.72III5−2−271.4


### 3.2. Design and Block Generators for m = 4, 5, 6, 7, 8, and 9 with 4 Choice Sets of Size 4

#### 3.2.1. Generators and Defining Relations

LetMNOPQ be the generator of the 25−1 design in Table 3. The treatment defining relation is given by I≡MNOPQ, and because the length of the shortest “word” is five, this design is resolution V, denoted as 2V5−1 fractional factorial design. The detailed results are contained in Table 3.Table 3
Sixteen-run design and block generators form = 4, 5, 6, 7, 8, and 9 with 4 choice sets of size 4.
mnbDesign generatorsBlock generatorsDefining contrastIBlock contrastFractionsAliasing effectsConfounding effectsNo. of runsClear effects402…b1=MOPMO,NOP,2IV4…MOP,16All 4 main effects plus all 2 factor interactions exceptMN which is confounded in the block effect.b2=NOPMNPNOP,MN512Q≡MNOPb1=MNI≡MNOPQMN,MO,2V5−1MN≡OPQMN,MO,16All 5 main effects plus all 2 factor interactions exceptMN, MO, and NO.MO≡NPQNO,OPQ,b2=MONOM≡NOPQNPQ,MPQP≡MNOQ622Q≡MNOb1=MOPI≡MNOQ≡MNPR≡OPQRMOP,2IV6−2MN≡OQ≡PRMN,16All 6 main effectsR≡MNPb2=NOPNOP,MNMO≡NQOQ,PRNO≡MQMP≡NRNP≡MROP≡QRPQ≡QR732Q≡MNOb1=MN, b2=MOI≡MNOQ≡MNPR≡MOPS≡OPQR≡NPQS≡NORSMN,MO,NO2IV7−3MN≡OQ≡PRMN,OQ,16All 7 main effectsMO≡NQ≡PSPR,MO,NO≡MQ≡RSNQ,PS,R≡MNPMP≡NR≡OSNO,MQ,NP≡MR≡QSS≡MOPOP≡MS≡QRRS.OR≡NS≡PQ842Q≡MNOb1=MNI≡MNOQ≡MNPR≡MOPS≡NOPT≡OPQR≡NPQS≡MPQT≡NORS≡MORT≡MNSTMN,MO,NO2IV8−4MN≡ST≡OQ≡PRMN,ST,16All 8 main effectsR≡MNPMO≡RT≡NQ≡PSOQ,PR,NO≡RS≡MQ≡PTMO,RT,S≡MOPb2=MOMP≡QT≡NR≡OSNQ,PS,NP≡QS≡MR≡OTNO,RS,T≡NOPOP≡QR≡MS≡NTMQ,PTPQ≡QR≡NS≡MT952Q≡MNb1=NOb2=NPI≡MNQ≡MOR≡MPS≡NOPT≡NOPQU≡NOQR≡NPQS≡MOPQT≡MOPU≡OPRS≡MNPRT≡MNOST≡MNPQRU≡QTU≡MNOQSUNO,NP,2III9−5MN≡QNO,QR,16NoneR≡MOMO≡PU≡RPT,NP,MP≡OU≡SS≡MPMQ≡NQS,OT,OPNQ≡OR≡PS≡MT≡NOPM≡OROP,RS,MS≡PU≡NOPQQT≡UNTQU≡TNote. The rest of the designs are illustrated in Table 3.


#### 3.2.2. Alias Structures

Considering25−1 design in Table 3 with design relation:(11)I≡MNOPQ.The alias structure is obtained similarly by multiplying the treatment effects by the design relation. Hence, the main effects are aliased as follows:(12)M≡NOPQ,N≡MOPQ,O≡MNPQ,P≡MNOQ,Q≡MNOP,OP≡MNQ,NQ≡MOP,MQ≡NOP,OQ≡MNP,NO≡MPQ,PQ≡MNO,MP≡NOQQ,NP≡MOQ.


#### 3.2.3. Determining the Confounding of Effects

Consider the 16-run of25−1 design in four blocks in Table 3, with the treatment defining contrast subgroup(13)I≡MNOPQ.The generalized block defining contrast subgroup is formed by taking all 2-way products as follows:MNMO≡M2NO≡NOThat is, the block defining contrast subgroup is given byMN, MO, and NO.By multiplying the defining word byMN, MO, and NO, we have(14)MNOPQMN≡M2N2OPQ≡OPQ,MNOPQMO≡M2NO2PQ≡NPQ,MNOPQNO≡MN2O2PQ≡MPQ.Considering the block defining generators, it might be concluded that the block effectsb1 and b2 are only confounded with MN, MO, and NObecause of the treatment defining relation is also aliased with the effects OPQ, NPQ, and MPQ. Therefore, b1 and b2 are confounded with three effects: OPQ, NPQ, and MPQ.


#### 3.2.4. Clear Effects

Hence, from the above illustrations, the remaining 14 (= 16 − 2) degrees of freedom do not involve the block effects, each of the 5 main effects is aliased with 4-factor interactions, and the 7 two-factor interactions effects are aliased with three-factor interaction, refer to alias structures above, whereM≡NOPQ,OP≡MNQ,NQ≡MOP, and so on. The 5-factor interaction is chosen as a generator. Hence, all 5 main effects and all two main effects are clear, except MN, MO, and NO which are confounded with the block effect b1 and b2 and are not eligible for estimation.


#### 3.2.5. Blocked Fractional Factorial Design for 4 Choices Set of Size 4

The blocked fractional factorial design2V5−1−2 is a sixteen-run design. Hence, the generator of this 2-level design is =NMNOP. If y1,...,y5 represents the five factors, then Q=NMNOP is equivalent to y5 = y1+y2+y3+y4 (mod 2). The factor level for factor y5 is obtained by adding the corresponding treatment combinations of factor y1, y2, y3, and y4.To illustrate this for treatment combinationI, suppose the level for factors y1 = 0, y2 = 0, y3 = 0, and y4 = 0, and y5 = 0 + 0+0 + 0 = 0 (mod 2). Also, for treatment combination a, with the level for factors y1 = 1, y2 = 0, y3 = 0, y4 = 0, y5 = 1 + 0+0 + 0 = 1 (mod 2), and so on.To obtain 4 choice sets of size four, we select two effects to be confounded with the block. Suppose we choosey1y2 and y1y3 to be confounded with the blocks. These effects will have two block defining contrast, that is,(15)b1=y1y2,b2=y1y3.All the treatment combinations are evaluated usingb1 and b2. λ1 = λ2 = λ3 = 1I: b1 = 1(0) + 1(0) + 0(0) = 0 (mod 2)b2 = 1(0) + 0(0) + 1(0) = 0 (mod 2)a: b1 = 1(1) + 1(0) +0(0) = 1 (mod 2)b2 = 1(1) + 0(0) + 1(0) = 1 (mod 2).Similarly, we can obtainb1 (mod 2) and b2 (mod 2) values for other treatments. Hence, from the values above, all treatment combinations with level 0 in b1 and b2 form block 1 which is an inherent choice set one, and also all treatment combinations with level 1 in b1 and level 1 in b2 form block 2, which also form choice set two. Also, all treatment combinations with level 1 in b1 and level 0 in b2 form block 3, which is an inherent choice set three, and finally, all treatment combinations with level 0 in b1 and level 1 in b2 form block 4, which also form choice set four.


#### 3.2.6. Design Efficiency

The result can be seen in Table4, which shows 66.7–81.8 percent A-efficiency for all the designs.Table 4
Design efficiencies’ design and block generators form = 4, 5, 6, 7, 8, and 9 with 4 choice sets of size 4.
DesignsA-efficiency (%)2IV4−0−266.72V5−1−271.42IV6−2−275.02IV7−3−277.72IV8−4−280.02III9−5−281.8


### 3.3. Design and Block Generators form = 5, 6, 7, and 8 with 8 Choices Set of Size 2

#### 3.3.1. Generators and Defining Relations

In Table5, let MNOQ, MNPR, and MOPS be the generators of the 2IV7−3 design. The generalized interactions are formed by taking all 2-way products as follows:(16)MNOQMNPR≡M2N2OPQR≡OPQR,MNOQMOPS≡M2NO2QPS≡NPQS,MNPRMOPS≡M2NOP2RS≡NORS.Table 5
Sixteen-run design and block generators form = 5, 6, 7, and 8 with 8 choices set of size 2.
mnbDesign generatorsBlock generatorsDefining contrastIBlock contrastFractionsAliasing effectsConfounding effectsNo. of runsClear effects513Q≡MNOb1=MPb2=NPb3=OPI≡MNOQMP,NP,OP,MN,MO,NO2IV5−1MN≡OQMO≡NQNO≡MQMP,NP,OP,MN,MO,NO,OQ,NQ,MQ16All 5 main effects623Q≡MNOR≡MNPb1=MOb2=NOb3=NPI≡MNOQ≡MNPR≡OPQRMO,NO,MP,MN,OP,MNOP2IV6−2MN≡OQ≡PRMO≡NQNO≡MQMP≡NRNP≡MROP≡QRPQ≡QRMO,NQ,NO,MQ,MP,NR,MN,OQ,PR,OP,QR,PQ,OR16All 6 main effects733Q≡MNOR≡MNPS≡MOPb1=MNb2=MOb3=MPI≡MNOQ≡MNPR≡MOPS≡OPQR≡NPQS≡NQRSMN,MO,MP,NO,NP,OP2IV7−3MN≡OQ≡PR,MO≡NQ≡PSNO≡MQ≡RS,NQ≡MOMP≡NR≡OS,NP≡MR≡QSOP≡MS≡QR,PQ≡OR≡NSNS≡PQ,RS≡NO,NQ≡PSMN,OQ,PR,MO,NQ,PS,NO,MQ,NP,MR,QS,OP,MS,QR16All 7 main effects843Q≡MNOR≡MNPS≡MOPT≡NOPb1=MNb2=MOb3=MPI≡MNOQ≡MNPR≡MOPS≡NOPT≡OPQR≡NPQS≡MPQT≡NORS≡MORT≡MNSTNN,MO,MP,NO,NP,OP2IV8−4MP≡NR≡OS≡QT,MN≡OQ≡PR≡ST,NP≡MR,OT≡QS,MO≡NQ≡PS≡RT , MS≡OP≡NT≡QR,NO≡MQ≡PT≡RS,PQ≡OR≡NS≡MTMN,MO,MP,NO,NP,OP16All 8 main effectsNote. The rest of the designs are illustrated in Table 5.That is, the treatment defining relation is given byI≡MNOQ≡MNPR≡MOPS≡OPQR≡NPQS≡NORS, and because the length of the shortest “word” is four, this design is resolution IV, denoted as 2IV7−3 fractional factorial design. The detailed results are contained in Table 5.


#### 3.3.2. Alias Structures

Considering the2IV7−3 design in Table 5 with design relation(17)I≡MNOQ≡MNPR≡MOPS≡OPQR≡NPQS≡NORS.The alias structure is obtained by multiplying the treatment effects by the design relation; hence, the main effects are aliased as follows:(18)M≡NOQ≡NPR≡OPS≡MOPQR≡MNPQS≡MNORS,N≡MOQ≡MPR≡MNOPS≡NOPQR≡PQS≡ORS,O≡MNQ≡MNOPR≡MPS≡PQR≡NOPQS≡NRS,P≡MNOPQ≡MNR≡MOS≡OQR≡NQS≡NOPRS,Q≡MNO≡MNPQR≡MOPQS≡OPR≡NPS≡NOQRS,R≡MNOQR≡MNP≡MOPRS≡OPQ≡NPQRS≡NOS,S≡MNOQS≡MNPRS≡MOP≡OPQRS≡NPQ≡NOR,PQ≡MNOP≡MNQR≡MOQS≡OR≡NS≡NOPQRS,NOP≡MPQ≡MOR≡MNS≡NQR≡OQS≡PRS,MQR≡MNOR≡MNPQ≡MOPQRS≡OP≡NPRS≡NOQRS,MRS≡NOQRS≡NPS≡OPR≡MOPQS≡MNPQR≡MNO,MQRS≡NORS≡NPQS≡OPQR≡MOPS≡MNPR≡MNOQ.


#### 3.3.3. Determining the Confounding of Effects

Consider the 8-run2IV7−3 design in four blocks in Table 5, with the treatment defining contrast subgroup(19)I≡MNOQ≡MNPR≡MOPS≡OPQR≡NPQS≡NORS.The generalized block defining contrast subgroup is formed by taking all 2-way products as follows:(20)MNMO≡M2NO≡NO,MNMP≡M2NP≡NP,MOMP≡M2OP≡OP.That is, the block defining contrast subgroup is given byMN, MO, MP, NO, NP, and OP.By multiplying each defining word byMN, MO, MP, NO, NP, and OP, we have(21)MNOQMN≡OQ,MNOQMO≡NQ,MNOQMP≡NOPQ,MNOQNO≡MQ,MNOQNP≡MOPQ,MNOQOP≡MNPQ,MNPRMN≡PR,MNPRMO≡NOPR,MNPRMP≡NR,MNPRNO≡MOPR,MNPRNP≡MR,MNPROP≡MNOR,MOPSMN≡NOPS,MOPSMO≡PS,MOPSMP≡OS,MOPSNO≡MNPS,MOPSNP≡MNOS,MOPSOP≡MS,OPQRMN≡MNOPQ,OPQRMO≡MPQR,OPQRMP≡MOQR,OPQRNO≡NPQR,OPQRNP≡NOQR,OPQROP≡QR,NPQSMN≡MPQS,NPQSMO≡MNOPQS,NPQSMP≡MNQS,NPQSNO≡OPQS,NPQSNP≡QS,NPQSOP≡NOQS,NORSMN≡MORS,NORSMO≡MNRS,NORSMP≡MNOPRS,NORSNO≡RS,NORSNP≡OPRS,NORSOP≡NPRS.Considering the block defining generators, it might be concluded that the block effectsb1, b2, and b3 are only confounded with MN, MO, MP, NO, NP, and OP. Because the treatment defining relation is also aliased with the effects illustrated previously, therefore, b1 and b2 are confounded with thirty-six effects:(22)OQ,NQ,NOPQ,MQ,MOPQ,MNPQ,PR,NOPR,NR,MOPR,MR,MNOR,NOPS,PS,OS,MNPS,MNOS,MNS,MNOPQR,MPQR,MOQR,NPQR,NOQR,QR,MPQS,MNOPQS,MNQS,OPQS,QS,NOQS,MORS,MNRS,MNOPRS,RS,OPRSandNPRS.


#### 3.3.4. Clear Effects

Hence, from the above illustrations, the remaining 13 (= 16 − 3) degrees of freedom that do not involve the block effects, each of the seven main effects is aliased with higher interactions, see the previous alias structure,M≡NOQ≡NPR≡OPS≡MOPQR≡MNPQS≡MNORS and so on. That is, the seven main effects are clear. The two-factor interactions are aliased with other two factor interaction or either confounded with a block effect. Therefore, the two-factor interactions are not eligible for estimation, see the previous alias structures and confounding effects, and MNPRMP≡NR, respectively, and so on.


#### 3.3.5. Blocked Fractional Factorial Design for 8 Choices Set of Size 2

The blocked fractional factorial design2IV7−3 is a sixteen-run design. The generators of this 2-level design are Q=MNO,R=MNP, and S=MOP. If y1,...,y7 represents the seven factors, then Q=MNO,R=MNP, and S=MOP are equivalent to y5 = y1+y2+y3 (mod 2), y6=y1+y2+y4 (mod 2), and y7=y1+y3+y4 (mod 2), respectively. The factor level for factor y5 is obtained by adding the corresponding treatment combinations of factor y1, y2, and y3; likewise, factor y6 is obtained by adding the corresponding treatment combinations of factor y1, y2, and y4, and y7 is obtained by adding the corresponding treatment combinations of factor y1, y3, and y4.To illustrate this for treatment combinationI, with the factor level for factor y1 = 0, y2=0, and y3 = 0, y5 = 0 + 0+0(mod 2) = 0, also with the factor level for factor y1 = 0, y2=0, and y4 = 0, y6 = 0 + 0+0(mod 2) = 0 and with the factor level for factor y1 = 0, y3=0, and y4 = 0, y7 = 0 + 0+0 (mod 2) = 0. Also, for treatment combination a, with the factor level for factor y1 = 1, y2=0, and y3 = 0, y5 = 1 + 0+0 (mod 2) = 1, also with the factor level for factor y1 = 1, y2=0, and y4 = 0, y6 = 1 + 0 + 0 (mod 2) = 1 and with the factor level for factor y1 = 1, y3=0, and y4 = 0, y7 = 1 + 0 + 0 (mod 2) = 1 and so on.To obtain 8 choice sets of size two we select three effects to be confounded with the block. Suppose we choosey1y2, y1y3, and y1y4 to be confounded with the blocks. These effects will have two block-defining contrasts, that is(23)b1=y1y2,b2=y1y3,b3=y1y4.All the treatment combinations are evaluated usingb1, b2, and b3. λ1 = λ2 = λ3 = λ4 = 1I: b1 = 1(0) + 1(0) + 0(0) + 0(0) = 0 (mod 2)b2 = 1(0) + 0(0) + 1(0) + 0(0) = 0 (mod 2)b3 = 1(0) +0(0) + 0(0) + 1(0) = 0 (mod 2)a: b1 = 1(1) + 1(0) + 0(0) + 0(0) = 1 (mod 2)b2 = 1(1) + 0(0) + 1(0) + 0(0) = 1 (mod 2)b3 = 1(1) + 0(0) + 0(0) + 1(0) = 1 (mod 2)b: b1 = 1(0) + 1(1) + 0(0) + 0(0) = 1 (mod 2)b2 = 1(0) + 0(1) + 1(0) + 0(0) = 0 (mod 2)b3 = 1(0) + 0(1) + 0(0) + 1(0) = 0 (mod 2).Similarly, we can obtainb1 (mod 2), b2 (mod 2) and b3 (mod 2) values for other treatments. Hence, from the above values, all treatment combinations with level 0 in b1, b2 and b3 form block 1 which is an inherent choice set one and also all treatment combinations with level 1 in b1, level 1 in b2 and level 1 in b3 form block 2, which also form choice set two. Also, all treatment combinations with level 1 in b1, level 0 in b2 and level 0 in b3 form block 3, which is an inherent choice set three, all treatment combinations with level 0 in b1, level 1 in b2 and level 1 in b3 form block 4, all treatment combinations with level 0 in b1, level 1 in b2 and level 0 in b3 form block 5, all treatment combinations with level 1 in b1, level 0 in b2 and level 1 in b3 form block 6, all treatment combinations with level 1 in b1, level 1 in b2 and level 0 in b3 form block 7 and finally, all treatment combinations with level 0 in b1, level 0 in b2 and level 1 in b3 form block 8, which also form choice set eight.Therefore, treatment combinationsI and abcd forms choice set 1, a and bcd forms choice set 2, b and acd form choice set 3 and ab and cd form choice set 4, c and abd forms choice set 5, ac and bd forms choice set 6, bc and ad form choice set 7, lastly, abc and d form choice set 8. Hence a choice set of 8 and size 2 is formed.


#### 3.3.6. Design Efficiency

The result can be seen in Table6, which shows 60.3–67.8 percent A-efficiency for all the designs.Table 6
Design efficiency design and block generators form = 5, 6, 7, and 8 with 8 choices set of size 2.
DesignsA-efficiency (%)2IV5−1−360.32IV6−2−366.72IV7−3−367.82IV8−4−365.5


### 3.4. Design and Block Generators form = 5, 6, 7, 8, 9 and 10 with 8 Choice Sets of Size 4

#### 3.4.1. Generators and Defining Relations

In Table7, consider a 25−0 fractional factorial design with no generator. Hence, there are no generalized interactions and also no treatment defining relation. The design is of resolution V, denoted as 2V5−0 fractional factorial design. The detailed results are contained in Table 7.Table 7
Sixteen-run design and block generators form = 5, 6, 7, and 8 with 8 choice sets of size 4.
mnbDesign generatorsBlock generatorsDefining contrastIBlock contrastFract- ionsAliasing effectsConfounding effectsNo. of runsClear effects503…b1=MOQb2=NOQb3=MNOP…MOQ,NOQ,MNOP,MN,NPQ,MPQ2V5…MN,OP,MOQ,NOQ,MNOP,NPQ,MPQ32All 5 main effects plus, all two factor interactions except MN andOP613R≡MNOPQb1=MOQb2=NOQb3=MPQI≡MNOPQMOQ,NOQ,PQ,MN,OP,MNOP2VI6−1M≡NOPQRMN≡OPQRMO≡NPQRMN,OP,QR,MOQ,NOQ,MPQ,MNOP32All 6 main effects plus, all two factor interactions except MN andOP and QR723R≡MNOS=MNPQb1=NOPb2=NOQb3=MOPQI≡MNOR≡MNPQS≡OPQRSNOP,NOQ,MOPQ,PQ,MNQ,MNP2IV7−2MN≡ORMO≡NRNO≡MRPS,QS,MNP,PQ,MNQ,NOP,NOQ,MOPQ,MO32All 7 main effects plus the following two factor interactionsMP,MQ,MS,NP,NQ,NS,OP,OQ,OS,PR,QR,RS833R≡MNOS≡MNPT≡MOPQb1=MOb2=NOb3=MPI≡MNOR≡MNPS≡MOPQT≡OPRS≡NPQRT≡NOQSTMO,NO,MP,MN,OP,MNOP2IV8−3MN≡OR≡PSMO≡NRNO≡MRMP≡NSNP≡MSOP≡RSPR≡OSNR,MR,NS,OR,PS,RS,PR,OS,MO,NO,MP,OP,MN,MNOP32All 8 main effects plus the following two factor interactionsMQ,NQ, OQ,PQ,QR,QS,MT,NT,OT,PT,QT,RT,ST943R≡MNOS≡MNPT≡MOPU≡NOPQb1=MNb2=MOb3=MPI≡MNOR≡MNPS≡MOPT≡NOPQU≡OPRS≡NPRT≡MPQRU≡NOST≡MQSU≡MNQTUMN,MO,MP,NO,NP,OP2IV9−4MN≡OR≡PS,MO≡NR≡PTMP≡NS≡OT,NO≡MR≡STNP≡MS≡RT,OP≡MT≡RSPR≡NT≡OSOR,PS,NR,PT,NS,OT,MR,ST,MS,MT,RS32All 9 main effects plus the following two factor interact- tionsMQ,NQ,OQ,PQ,QR,QT,MU,NU,QU,RU,SU,TU1053Q≡MNR≡MOS≡MPT≡NOPU≡NOPQb1=MNb2=MPb3=NQI≡MNQ≡MOR≡MPS≡NOPT≡NOPQU≡NOQR≡NPQS≡MOPQT≡MOPU≡OPRS≡MNPRT≡MNPQRUMN,MP,NQ,NP,NQ,MNPQ2III10−5MN≡Q,MQ≡NNQ≡M≡OR≡PSMO≡R,MR≡QMP≡S,MS≡PNO≡PT≡QR,NP≡OT≡QSOP≡NT≡RS,OQ≡NRPS≡OR,MO≡PUMP≡OUOP≡MU≡RSPR≡OSMN,MP,NQ,NP,MQ,MN,MNPQ32The following main and interac- tions effects are clearT,U,MT,QT,RT,ST,NU,QU,RU,SU,TU,MV,OV,PV,QV,RV,SV,TV,UVNote. The rest of the designs are illustrated in Table 7 above.


#### 3.4.2. Alias Structures

Considering the25−0 design in Table 7, which have no treatment defining relation, hence it does not have aliasing structure.


#### 3.4.3. Determing the Confounding of Effects

Consider the 32-run of25−0 design in eight blocks in Table 7, the generalized block-defining contrast subgroup is formed by taking all 2-way products as follows:(24)MOQNOQ≡MNO2Q2≡MN,MOQMNOP≡M2NO2PQ≡NPQ,NOQMNOP≡MN2O2PQ≡MPQ.Considering only the block-defining generators, it might be concluded that the block effectsb1, b2, and b3 are only confounded with MOQ, NOQ, MNOP, MN, NPQ, and MPQ, but OP is also confounded in the block effects. Therefore, b1, b2, and b3 are confounded with six effects: MOQ, NOQ, MNOP, MN, OP, NPQ, and MPQ.


#### 3.4.4. Clear Effects

Hence from the above illustrations, the remaining 29 (= 32 − 3) degrees of freedom that do not involve the block effects, all the 5 main effects and two factor effects are clear exceptMN and OP which are confounded in the block effects. Hence, M,N,O,P,Q,MO,NO,MP,NP,MQ,NQ, and OQ are clear effects.


#### 3.4.5. Blocked Fractional Factorial Design for 8 Choice Sets of Size 4

To obtain 8 choice sets of size four we select three effects to be confounded with the block. Suppose we choosey1y3y5, y2y3y5, and y1y2y3y4 to be confounded with the blocks. These effects will have three block-defining contrasts, that is b1 = y1y3y5, b2 = y2y3y5, and b3 = y1y2y3y4All the treatment combinations are evaluated usingb1, b2, and b3. λ1 = λ2 = λ3 = λ4 = λ5= 1(I):b1 = 1(0) + 0(0) + 1(0) + 0(0) + 1(0) = 0 (mod 2)b2 = 0(0) + 1(0) + 1(0) + 0(0) + 1(0) = 0 (mod 2)b3 = 1(0) + 1(0) + 1(0) + 1(0) + 0(0) = 0 (mod 2)a:b1 = 1(1) + 0(0) + 1(0) + 0(0) + 1(0) = 1 (mod 2)b2 = 0(1) + 1(0) + 1(0) + 0(0) + 1(0) = 0 (mod 2)b3 = 1(1) + 1(0) + 1(0) + 1(0) + 0(0) = 1 (mod 2).Similarly, we can obtainb1 (mod 2), b2 (mod 2), and b3 (mod 2) values for other treatments. Hence, from the above values, all treatment combinations with level 0 in b1, b2, and b3 form block 1 which is an inherent choice set one and also all treatment combinations with level 1 in b1, level 0 in b2, and level 1 in b3 form block 2, which also form choice set two. Also, all treatment combinations with level 0 in b1, level 1 in b2, and level 1 in b3 form block 3, which is an inherent choice set three, all treatment combinations with level 1 in b1, level 1 in b2, and level 0 in b3 form block 4, which is an inherent choice set four, all treatment combinations with level 1 in b1, level 1 in b2, and level 1 in b3 form block 5, which is an inherent choice set five, all treatment combinations with level 0 in b1, level 1 in b2, and level 0 in b3 form block 6, which is an inherent choice set six, all treatment combinations with level 1 in b1, level 0 in b2, and level 0 in b3 form block 7, which is an inherent choice set seven and finally, all treatment combinations with level 0 in b1, level 0 in b2, and level 1 in b3 form block 8, which is an inherent choice set eight.Therefore, treatment combinationsI,abcd,abe, and cde form choice set 1; a,bcd,be, and ade form choice set 2; b,acd,ae, and bcde form choice set 3; ab,cd,e, and abcde form choice set 4; c,abd,abce, and de form choice set 5; ac,bd,bce,ade form choice set 6; bc,ad,ace, and bde form choice set 7, and lastly, abc,d,ce, and abde form choice set 8. Hence, a choice set of 8 and size 4 is formed.


#### 3.4.6. Design Efficiency

The result can be seen in Table8, which shows 62.5–76.9 percent A-efficiency for all the designs.Table 8
Design efficiency design and block generators form = 5, 6, 7, 8, 9, and 10 with 8 choice sets of size 4.
DesignsA-efficiency (%)2V5−0−362.52VI6−1−366.72IV7−2−370.02IV8−3−372.02IV9−4−375.02III10−5−376.9


### 3.5. Application

The research makes use of service quality attributes and levels identified by [10] in their paper a discrete choice modeling of service quality in public transport. The paper used five symmetric attributes and designed twenty-eight choice sets from these attributes and levels. This research also used the same attributes and levels to illustrate an application of the proposed method. Table 9 shows the attributes and their levels.Table 9
Attributes and their levels included for the SCE.
AttributesAttribute levelsy1 = Walking distance to bus stopWithin 10 min (0)More than 10 min (1)y2 = Attitude of driver and mateVery polite (0)Very impolite (1)y3 = Transport fareNormal fare (0)15% more than normal fare (1)y4 = Bus stop facilityShelter, lighting, and seat available (0)No shelter, lighting, and seat (1)y5 = ReliabilityOn time (0)late (1)


#### 3.5.1. Stated Choice Experiment with Two-Level Attributes

Considering the five attributes with the two levels above, which requires25 = 32 treatment combinations of a complete factorial design. This can be precluded to run; therefore, we run one-half fraction of the complete factorial design. This research uses a 2v5−1−2 block fractional factorial design to construct 4 choices set of size 4. The generators, alias structure, confounding effects, and clear effects of the design are clearly illustrated and derived in Section 3.2.Lety1 = walking distance to bus stop, y2 = attitude of driver and mate, y3 = transport fare, y4 = bus stop facility, and y5 = reliability, with their levels coded as 0 and 1 as shown in Table 9. Hence, as shown in Section 3.2.5, all treatment combinations with level 0 in b1 and b2 form block 1 which is an inherent choice set one and also all treatment combinations with level 1 in b1 and level 1 in b2 form block 2, which also form choice set two. Also, all treatment combinations with level 1 in b1 and level 0 in b2 form block 3, which is an inherent choice set three, and finally, all treatment combinations with level 0 in b1 and level 1 in b2 form block 4, which also form choice set four.Therefore, from Table10, treatment combinations I,abc,d, and abcd form choice set 1; a,bc,ad, and bcd form choice set 2; b,ac,bd, and acd form choice set 3; ab,c,abd, and cd form choice set 4. Hence, a choice set of 4 and size 4 is formed. Table 11 shows choice set 1, and the remaining three choice sets are design similarly.Table 10
Design matrix for2V5−1−2.
y1y2y3y4y5 = y1y2y3y4b1 = y1y2b2 = y1y3I0000000a1000111b0100110ab1100001c0010101ac1010010bc0110011abc1110100d0001100ad1001011bd0101010abd1101101cd0011001acd1011110bcd0111111abcd1111000Table 11
A choice set one for2V5−1−2 design.
AttributesServiceAServiceBServiceCServiceDWalking distance to bus stopWithin 10 minsMore than 10 minsWithin 10 minsMore than 10 minsAttitude of driver and mateVery politeVery impoliteVery politeVery impoliteTransport fareNormal fare15% more than normal fareNormal fare15% more than normal fareBus stop facilityShelter, lighting and seat availableShelter, lighting and seat availableNo shelter, lighting and seat availableNo shelter, lighting and seat availableReliabilityOn timeLateLateOn timeWhich of these options do you prefer?( )( )( )( )


#### 3.5.2. Model Specification

The stated choice data are analysed based on the random utility model [11]. Stated choice experiment modeling is rooted in random utility theory (RUT). This supposes, given alternatives to select from, a respondent n (n=1…N) will select the one alternative that produces the higher utility amidst the choice group (i = 1, 2, 3, …) at the instant of choice. The utility of the respondent is defined by explainable component specified as a function of the attributes of the alternatives and characteristics of the respondent and an unexplainable component (random variation) εni.(25)Uni=Vni+εni,where Uni represents the utility of respondent, and Vni is the explainable component specified as a function of the attributes of the alternatives and characteristics of the respondent and an unexplainable component (random variation) εni following an independent identically distributed extreme value type 1 distribution. In equation (25), the probability of a respondent selecting a specified service type is modeled. The chance of a selecting a service type is resolute by the indirect utility function for the respondent n from selection i in choice set c, assuming this is linear and additive and of the following form:(26)vnic=xnicβ+εnic,where vnic stands for utility derived from a selection, xnicβ is the utility component, and εnic is the random component. The vector xnic is specified as follows, where β1−5 stand for design attributes of the choice experiment and β0 the constant.(27)xnicβj=β0+β1walkingdistancetobusstationi+β2attitudeofdriverandmatei+β3transportfaresi+β4Busstopfacilityi+β5reliabilityi.


## 3.1. Design and Block Generators form = 3, 4, and 5 with 4 Choice Sets of Size 2

### 3.1.1. Generators and Defining Relations

LetMNP and MNOQ be the generators of the 25−2 design, in Table 1. The generalized interactions are formed by taking all 2-way products as follows:(4)MNPMNOQ=M2N2OPQ=OPQ.Table 1
Eight-run design and block generators form = 3, 4, and 5 with 4 choices set of size 2.
mnbDesign generatorsBlock generatorsDefining contrastIBlock contrastsFractionsAliasing effectsConfounding effectsNo. of runsClear effects302…b1=MNb2=MOMN,MO,NO2III3…MN,MO, NO8All 3 main effects412P≡MNOb1=MNb2=MOI≡MNOPMN,MO,NO2IV4−1M≡OPM≡NPMO,NP,MN,OP,NO,MP8All 4 main effects522P≡MN, Q≡MOb1=MOb2=NOI≡MNP≡MNOQ≡OPQMO,NO,MN2III5−2N≡MPM≡NPP≡MN≡OQN≡MPQ≡OPO≡PQM≡MNN≡MOMO,NQ,NO,MQ,MN,P,OQ8NoneNote. The rest of the designs are illustrated in Table 1.That is, the treatment defining relation is given byI≡MNP≡MNOQ≡OPQ because the length of the shortest “word” is three, and this design is resolution III, denoted as 2III5−2 fractional factorial design. The detailed results are contained in Table 1.


### 3.1.2. Alias Structures

Considering25−2 design in Table 1, with design relation:(5)I≡MNP≡MNOQ≡OPQ.The aliased structure is determined by multiplying the effects with the design relation.Let consider the main effects,MMNOQ≡M2NOQ≡NOQ and MOPQ≡MOPQ. Similarly, we can obtain the other aliases effects as given as follows.Hence, the main effects and two-factor interactions are aliased as follows:(6)M≡NP≡NOQ≡MOPQ,Q≡MNPQ≡MNO≡OP,N≡MP≡MOQ≡NOPQ,O≡MNOP≡MNQ≡PQ,P≡MN≡MONPQ≡OQ,MNP≡OPQ≡MNOQ.


### 3.1.3. Determining the Confounding of Effects

Consider the 8-run of25−2 design in four blocks in Table 1, with the treatment defining contrast subgroup:(7)I≡MNP≡MNOQ≡OPQ.The generalized block defining contrast subgroup is formed by taking all 2-way products as follows:MONO≡MNO2≡MNThat is, the block-defining contrast subgroup is given byMO, NO, and MN.By multiplying each defining word byMO, NO, and MN, we have(8)MNPMO≡NOP,MNPNO≡MOP,MNPMN≡P,MNOQMO≡NQ,MNOQNO≡MQ,MNOQMN≡OQ,OPQMO≡MPQ,OPQNO≡NPQ,OPQMN≡MNOPQ.Considering only the block-defining generators, it might be concluded that the block effectsb1 and b2 are only confounded with MO, NO, and MN because the treatment-defining relation is also aliased with the effects NOP,NQ,MPQ,MOP,MQ,NPQ,P,OQ, and MNOPQ. Therefore, b1 and b2 are confounded with nine effects:(9)NOP,NQ,MPQ,MOP,MQ,NPQ,P,OQ,andMNOPQ.


### 3.1.4. Clear Effects

Hence, from the above illustrations, the remaining 6 (= 8 − 2) degrees of freedom that do not involve the block effects, each of them is aliased with a two-factor or higher interactions, detailed can be seen in alias structures previously, whereM≡NP≡NOQ≡MOPQ and so on. Hence, the five main effects are not clear. Among the two-factor interactions, NQ, MQ, OQ, NO, MN, and MO are confounded with the block effect b1 and b2 and are not eligible for estimation. Hence, none of the effects is clear.


### 3.1.5. Blocked Fractional Factorial Design for 4 Choice Sets of Size 2

The blocked fractional factorial design2III5−2−2 is an eight-run design. First, the design is constructed by choosing the column for P to be equal to column M plus column N (mod 2); that is, every entry in column P is sum of column M and column N (mod 2). Hence, the generators of this 2-level design are P=MN and Q=MO. If y1...y5 represents the five factors, then P=MN and Q=MO are equivalent to y4=y1+y2 (mod 2) and y5=y1+y3 (mod 2), respectively. The factor level for factor y4 is obtained by adding the corresponding treatment combinations of factor y1 and y2; likewise, factor y5 is obtained by adding the corresponding treatment combinations of factor y1 and y3.To illustrate this for treatment combination1, with the factor level for factor y1 = 0 and y2 = 0, factor y4 = 0 + 0 (mod 2) = 0 and y5 = 0 + 0 (mod 2) = 0. Also, for treatment combination a, with the factor level for factor y1 = 1 and y2 = 0, factor y4 = 1 + 0 (mod 2) = 1 and y5 = 1 + 0 (mod 2) = 1 and so on.To obtain 4 choice sets of size four, we select two effects to be confounded with the block. Suppose we choosey1y3 and y2y3 to be confounded with the blocks. These effects will have two block defining contrast, that is,(10)b1=y1y3,b2=y2y3.All the treatment combinations are evaluated usingb1 and b2. λ1=λ2=λ3=1.I: b1 = 1(0) + 0(0) + 1(0) = 0 (mod 2)b2 = 0(0) + 1(0) + 1(0) = 0 (mod 2)a: b1 = 1(1) + 0(0) + 1(0) = 1 (mod 2)b2 = 0(1) + 1(0) + 1(0) = 0 (mod 2)b: b1 = 1(0) + 0(1) + 1(0) = 0 (mod 2)b2 = 0(0) + 1(1) + 1(0) = 1 (mod 2)ab: b1 = 1(1) + 0(1) + 1(0) = 1 (mod 2)b2 = 0(1) + 1(1) + 1(0) = 1 (mod 2)c: b1 = 1(0) +0(0) + 1(1) = 1 (mod 2)b2 = 0(0) + 1(0) + 1(1) = 1 (mod 2)ac: b1 = 1(1) + 0(0) + 1(1) = 0 (mod 2)b2 = 0(1) + 1(0) + 1(1) = 1 (mod 2)bc: b1 = 1(0) + 0(1) + 1(1) = 1 (mod 2)b2 = 0(0) + 1(1) + 1(1) = 0 (mod 2)abc: b1 = 1(1) + 0(1) + 1(1) = 0 (mod 2)b2 = 0(1) + 1(1) + 1(1) = 0 (mod 2)Hence, from treatment combinations above, all treatment combinations with level 0 inb1 and b2 form block 1 which is an inherent choice set one, and also all treatment combinations with level 1 in b1 and level 0 in b2 form block 2, which also form choice set two. Also, all treatment combinations with level 0 in b1 and level 1 in b2 form block 3, which is an inherent choice set three, and finally, all treatment combinations with level 1 in b1 and level 1 in b2 form block 4, which also form choice set four.Therefore, from treatment combinationsI and abc forms choice set 1, a and bc form choice set 2, b and ac form choice set 3, and ab and c form choice set 4. Hence, a choice set of 4 and size 2 is formed.


### 3.1.6. Design Efficiency

The result can be seen in Table2, which shows 60.0–71.4% A-efficiency for all the designs.Table 2
Design efficiency form = 3, 4, and 5 with 4 choice sets of size 2.
DesignsA-efficiency (%)2III3−0−260.02IV4−1−266.72III5−2−271.4


## 3.1.1. Generators and Defining Relations

LetMNP and MNOQ be the generators of the 25−2 design, in Table 1. The generalized interactions are formed by taking all 2-way products as follows:(4)MNPMNOQ=M2N2OPQ=OPQ.Table 1
Eight-run design and block generators form = 3, 4, and 5 with 4 choices set of size 2.
mnbDesign generatorsBlock generatorsDefining contrastIBlock contrastsFractionsAliasing effectsConfounding effectsNo. of runsClear effects302…b1=MNb2=MOMN,MO,NO2III3…MN,MO, NO8All 3 main effects412P≡MNOb1=MNb2=MOI≡MNOPMN,MO,NO2IV4−1M≡OPM≡NPMO,NP,MN,OP,NO,MP8All 4 main effects522P≡MN, Q≡MOb1=MOb2=NOI≡MNP≡MNOQ≡OPQMO,NO,MN2III5−2N≡MPM≡NPP≡MN≡OQN≡MPQ≡OPO≡PQM≡MNN≡MOMO,NQ,NO,MQ,MN,P,OQ8NoneNote. The rest of the designs are illustrated in Table 1.That is, the treatment defining relation is given byI≡MNP≡MNOQ≡OPQ because the length of the shortest “word” is three, and this design is resolution III, denoted as 2III5−2 fractional factorial design. The detailed results are contained in Table 1.


## 3.1.2. Alias Structures

Considering25−2 design in Table 1, with design relation:(5)I≡MNP≡MNOQ≡OPQ.The aliased structure is determined by multiplying the effects with the design relation.Let consider the main effects,MMNOQ≡M2NOQ≡NOQ and MOPQ≡MOPQ. Similarly, we can obtain the other aliases effects as given as follows.Hence, the main effects and two-factor interactions are aliased as follows:(6)M≡NP≡NOQ≡MOPQ,Q≡MNPQ≡MNO≡OP,N≡MP≡MOQ≡NOPQ,O≡MNOP≡MNQ≡PQ,P≡MN≡MONPQ≡OQ,MNP≡OPQ≡MNOQ.


## 3.1.3. Determining the Confounding of Effects

Consider the 8-run of25−2 design in four blocks in Table 1, with the treatment defining contrast subgroup:(7)I≡MNP≡MNOQ≡OPQ.The generalized block defining contrast subgroup is formed by taking all 2-way products as follows:MONO≡MNO2≡MNThat is, the block-defining contrast subgroup is given byMO, NO, and MN.By multiplying each defining word byMO, NO, and MN, we have(8)MNPMO≡NOP,MNPNO≡MOP,MNPMN≡P,MNOQMO≡NQ,MNOQNO≡MQ,MNOQMN≡OQ,OPQMO≡MPQ,OPQNO≡NPQ,OPQMN≡MNOPQ.Considering only the block-defining generators, it might be concluded that the block effectsb1 and b2 are only confounded with MO, NO, and MN because the treatment-defining relation is also aliased with the effects NOP,NQ,MPQ,MOP,MQ,NPQ,P,OQ, and MNOPQ. Therefore, b1 and b2 are confounded with nine effects:(9)NOP,NQ,MPQ,MOP,MQ,NPQ,P,OQ,andMNOPQ.


## 3.1.4. Clear Effects

Hence, from the above illustrations, the remaining 6 (= 8 − 2) degrees of freedom that do not involve the block effects, each of them is aliased with a two-factor or higher interactions, detailed can be seen in alias structures previously, whereM≡NP≡NOQ≡MOPQ and so on. Hence, the five main effects are not clear. Among the two-factor interactions, NQ, MQ, OQ, NO, MN, and MO are confounded with the block effect b1 and b2 and are not eligible for estimation. Hence, none of the effects is clear.


## 3.1.5. Blocked Fractional Factorial Design for 4 Choice Sets of Size 2

The blocked fractional factorial design2III5−2−2 is an eight-run design. First, the design is constructed by choosing the column for P to be equal to column M plus column N (mod 2); that is, every entry in column P is sum of column M and column N (mod 2). Hence, the generators of this 2-level design are P=MN and Q=MO. If y1...y5 represents the five factors, then P=MN and Q=MO are equivalent to y4=y1+y2 (mod 2) and y5=y1+y3 (mod 2), respectively. The factor level for factor y4 is obtained by adding the corresponding treatment combinations of factor y1 and y2; likewise, factor y5 is obtained by adding the corresponding treatment combinations of factor y1 and y3.To illustrate this for treatment combination1, with the factor level for factor y1 = 0 and y2 = 0, factor y4 = 0 + 0 (mod 2) = 0 and y5 = 0 + 0 (mod 2) = 0. Also, for treatment combination a, with the factor level for factor y1 = 1 and y2 = 0, factor y4 = 1 + 0 (mod 2) = 1 and y5 = 1 + 0 (mod 2) = 1 and so on.To obtain 4 choice sets of size four, we select two effects to be confounded with the block. Suppose we choosey1y3 and y2y3 to be confounded with the blocks. These effects will have two block defining contrast, that is,(10)b1=y1y3,b2=y2y3.All the treatment combinations are evaluated usingb1 and b2. λ1=λ2=λ3=1.I: b1 = 1(0) + 0(0) + 1(0) = 0 (mod 2)b2 = 0(0) + 1(0) + 1(0) = 0 (mod 2)a: b1 = 1(1) + 0(0) + 1(0) = 1 (mod 2)b2 = 0(1) + 1(0) + 1(0) = 0 (mod 2)b: b1 = 1(0) + 0(1) + 1(0) = 0 (mod 2)b2 = 0(0) + 1(1) + 1(0) = 1 (mod 2)ab: b1 = 1(1) + 0(1) + 1(0) = 1 (mod 2)b2 = 0(1) + 1(1) + 1(0) = 1 (mod 2)c: b1 = 1(0) +0(0) + 1(1) = 1 (mod 2)b2 = 0(0) + 1(0) + 1(1) = 1 (mod 2)ac: b1 = 1(1) + 0(0) + 1(1) = 0 (mod 2)b2 = 0(1) + 1(0) + 1(1) = 1 (mod 2)bc: b1 = 1(0) + 0(1) + 1(1) = 1 (mod 2)b2 = 0(0) + 1(1) + 1(1) = 0 (mod 2)abc: b1 = 1(1) + 0(1) + 1(1) = 0 (mod 2)b2 = 0(1) + 1(1) + 1(1) = 0 (mod 2)Hence, from treatment combinations above, all treatment combinations with level 0 inb1 and b2 form block 1 which is an inherent choice set one, and also all treatment combinations with level 1 in b1 and level 0 in b2 form block 2, which also form choice set two. Also, all treatment combinations with level 0 in b1 and level 1 in b2 form block 3, which is an inherent choice set three, and finally, all treatment combinations with level 1 in b1 and level 1 in b2 form block 4, which also form choice set four.Therefore, from treatment combinationsI and abc forms choice set 1, a and bc form choice set 2, b and ac form choice set 3, and ab and c form choice set 4. Hence, a choice set of 4 and size 2 is formed.


## 3.1.6. Design Efficiency

The result can be seen in Table2, which shows 60.0–71.4% A-efficiency for all the designs.Table 2
Design efficiency form = 3, 4, and 5 with 4 choice sets of size 2.
DesignsA-efficiency (%)2III3−0−260.02IV4−1−266.72III5−2−271.4


## 3.2. Design and Block Generators for m = 4, 5, 6, 7, 8, and 9 with 4 Choice Sets of Size 4

### 3.2.1. Generators and Defining Relations

LetMNOPQ be the generator of the 25−1 design in Table 3. The treatment defining relation is given by I≡MNOPQ, and because the length of the shortest “word” is five, this design is resolution V, denoted as 2V5−1 fractional factorial design. The detailed results are contained in Table 3.Table 3
Sixteen-run design and block generators form = 4, 5, 6, 7, 8, and 9 with 4 choice sets of size 4.
mnbDesign generatorsBlock generatorsDefining contrastIBlock contrastFractionsAliasing effectsConfounding effectsNo. of runsClear effects402…b1=MOPMO,NOP,2IV4…MOP,16All 4 main effects plus all 2 factor interactions exceptMN which is confounded in the block effect.b2=NOPMNPNOP,MN512Q≡MNOPb1=MNI≡MNOPQMN,MO,2V5−1MN≡OPQMN,MO,16All 5 main effects plus all 2 factor interactions exceptMN, MO, and NO.MO≡NPQNO,OPQ,b2=MONOM≡NOPQNPQ,MPQP≡MNOQ622Q≡MNOb1=MOPI≡MNOQ≡MNPR≡OPQRMOP,2IV6−2MN≡OQ≡PRMN,16All 6 main effectsR≡MNPb2=NOPNOP,MNMO≡NQOQ,PRNO≡MQMP≡NRNP≡MROP≡QRPQ≡QR732Q≡MNOb1=MN, b2=MOI≡MNOQ≡MNPR≡MOPS≡OPQR≡NPQS≡NORSMN,MO,NO2IV7−3MN≡OQ≡PRMN,OQ,16All 7 main effectsMO≡NQ≡PSPR,MO,NO≡MQ≡RSNQ,PS,R≡MNPMP≡NR≡OSNO,MQ,NP≡MR≡QSS≡MOPOP≡MS≡QRRS.OR≡NS≡PQ842Q≡MNOb1=MNI≡MNOQ≡MNPR≡MOPS≡NOPT≡OPQR≡NPQS≡MPQT≡NORS≡MORT≡MNSTMN,MO,NO2IV8−4MN≡ST≡OQ≡PRMN,ST,16All 8 main effectsR≡MNPMO≡RT≡NQ≡PSOQ,PR,NO≡RS≡MQ≡PTMO,RT,S≡MOPb2=MOMP≡QT≡NR≡OSNQ,PS,NP≡QS≡MR≡OTNO,RS,T≡NOPOP≡QR≡MS≡NTMQ,PTPQ≡QR≡NS≡MT952Q≡MNb1=NOb2=NPI≡MNQ≡MOR≡MPS≡NOPT≡NOPQU≡NOQR≡NPQS≡MOPQT≡MOPU≡OPRS≡MNPRT≡MNOST≡MNPQRU≡QTU≡MNOQSUNO,NP,2III9−5MN≡QNO,QR,16NoneR≡MOMO≡PU≡RPT,NP,MP≡OU≡SS≡MPMQ≡NQS,OT,OPNQ≡OR≡PS≡MT≡NOPM≡OROP,RS,MS≡PU≡NOPQQT≡UNTQU≡TNote. The rest of the designs are illustrated in Table 3.


### 3.2.2. Alias Structures

Considering25−1 design in Table 3 with design relation:(11)I≡MNOPQ.The alias structure is obtained similarly by multiplying the treatment effects by the design relation. Hence, the main effects are aliased as follows:(12)M≡NOPQ,N≡MOPQ,O≡MNPQ,P≡MNOQ,Q≡MNOP,OP≡MNQ,NQ≡MOP,MQ≡NOP,OQ≡MNP,NO≡MPQ,PQ≡MNO,MP≡NOQQ,NP≡MOQ.


### 3.2.3. Determining the Confounding of Effects

Consider the 16-run of25−1 design in four blocks in Table 3, with the treatment defining contrast subgroup(13)I≡MNOPQ.The generalized block defining contrast subgroup is formed by taking all 2-way products as follows:MNMO≡M2NO≡NOThat is, the block defining contrast subgroup is given byMN, MO, and NO.By multiplying the defining word byMN, MO, and NO, we have(14)MNOPQMN≡M2N2OPQ≡OPQ,MNOPQMO≡M2NO2PQ≡NPQ,MNOPQNO≡MN2O2PQ≡MPQ.Considering the block defining generators, it might be concluded that the block effectsb1 and b2 are only confounded with MN, MO, and NObecause of the treatment defining relation is also aliased with the effects OPQ, NPQ, and MPQ. Therefore, b1 and b2 are confounded with three effects: OPQ, NPQ, and MPQ.


### 3.2.4. Clear Effects

Hence, from the above illustrations, the remaining 14 (= 16 − 2) degrees of freedom do not involve the block effects, each of the 5 main effects is aliased with 4-factor interactions, and the 7 two-factor interactions effects are aliased with three-factor interaction, refer to alias structures above, whereM≡NOPQ,OP≡MNQ,NQ≡MOP, and so on. The 5-factor interaction is chosen as a generator. Hence, all 5 main effects and all two main effects are clear, except MN, MO, and NO which are confounded with the block effect b1 and b2 and are not eligible for estimation.


### 3.2.5. Blocked Fractional Factorial Design for 4 Choices Set of Size 4

The blocked fractional factorial design2V5−1−2 is a sixteen-run design. Hence, the generator of this 2-level design is =NMNOP. If y1,...,y5 represents the five factors, then Q=NMNOP is equivalent to y5 = y1+y2+y3+y4 (mod 2). The factor level for factor y5 is obtained by adding the corresponding treatment combinations of factor y1, y2, y3, and y4.To illustrate this for treatment combinationI, suppose the level for factors y1 = 0, y2 = 0, y3 = 0, and y4 = 0, and y5 = 0 + 0+0 + 0 = 0 (mod 2). Also, for treatment combination a, with the level for factors y1 = 1, y2 = 0, y3 = 0, y4 = 0, y5 = 1 + 0+0 + 0 = 1 (mod 2), and so on.To obtain 4 choice sets of size four, we select two effects to be confounded with the block. Suppose we choosey1y2 and y1y3 to be confounded with the blocks. These effects will have two block defining contrast, that is,(15)b1=y1y2,b2=y1y3.All the treatment combinations are evaluated usingb1 and b2. λ1 = λ2 = λ3 = 1I: b1 = 1(0) + 1(0) + 0(0) = 0 (mod 2)b2 = 1(0) + 0(0) + 1(0) = 0 (mod 2)a: b1 = 1(1) + 1(0) +0(0) = 1 (mod 2)b2 = 1(1) + 0(0) + 1(0) = 1 (mod 2).Similarly, we can obtainb1 (mod 2) and b2 (mod 2) values for other treatments. Hence, from the values above, all treatment combinations with level 0 in b1 and b2 form block 1 which is an inherent choice set one, and also all treatment combinations with level 1 in b1 and level 1 in b2 form block 2, which also form choice set two. Also, all treatment combinations with level 1 in b1 and level 0 in b2 form block 3, which is an inherent choice set three, and finally, all treatment combinations with level 0 in b1 and level 1 in b2 form block 4, which also form choice set four.


### 3.2.6. Design Efficiency

The result can be seen in Table4, which shows 66.7–81.8 percent A-efficiency for all the designs.Table 4
Design efficiencies’ design and block generators form = 4, 5, 6, 7, 8, and 9 with 4 choice sets of size 4.
DesignsA-efficiency (%)2IV4−0−266.72V5−1−271.42IV6−2−275.02IV7−3−277.72IV8−4−280.02III9−5−281.8


## 3.2.1. Generators and Defining Relations

LetMNOPQ be the generator of the 25−1 design in Table 3. The treatment defining relation is given by I≡MNOPQ, and because the length of the shortest “word” is five, this design is resolution V, denoted as 2V5−1 fractional factorial design. The detailed results are contained in Table 3.Table 3
Sixteen-run design and block generators form = 4, 5, 6, 7, 8, and 9 with 4 choice sets of size 4.
mnbDesign generatorsBlock generatorsDefining contrastIBlock contrastFractionsAliasing effectsConfounding effectsNo. of runsClear effects402…b1=MOPMO,NOP,2IV4…MOP,16All 4 main effects plus all 2 factor interactions exceptMN which is confounded in the block effect.b2=NOPMNPNOP,MN512Q≡MNOPb1=MNI≡MNOPQMN,MO,2V5−1MN≡OPQMN,MO,16All 5 main effects plus all 2 factor interactions exceptMN, MO, and NO.MO≡NPQNO,OPQ,b2=MONOM≡NOPQNPQ,MPQP≡MNOQ622Q≡MNOb1=MOPI≡MNOQ≡MNPR≡OPQRMOP,2IV6−2MN≡OQ≡PRMN,16All 6 main effectsR≡MNPb2=NOPNOP,MNMO≡NQOQ,PRNO≡MQMP≡NRNP≡MROP≡QRPQ≡QR732Q≡MNOb1=MN, b2=MOI≡MNOQ≡MNPR≡MOPS≡OPQR≡NPQS≡NORSMN,MO,NO2IV7−3MN≡OQ≡PRMN,OQ,16All 7 main effectsMO≡NQ≡PSPR,MO,NO≡MQ≡RSNQ,PS,R≡MNPMP≡NR≡OSNO,MQ,NP≡MR≡QSS≡MOPOP≡MS≡QRRS.OR≡NS≡PQ842Q≡MNOb1=MNI≡MNOQ≡MNPR≡MOPS≡NOPT≡OPQR≡NPQS≡MPQT≡NORS≡MORT≡MNSTMN,MO,NO2IV8−4MN≡ST≡OQ≡PRMN,ST,16All 8 main effectsR≡MNPMO≡RT≡NQ≡PSOQ,PR,NO≡RS≡MQ≡PTMO,RT,S≡MOPb2=MOMP≡QT≡NR≡OSNQ,PS,NP≡QS≡MR≡OTNO,RS,T≡NOPOP≡QR≡MS≡NTMQ,PTPQ≡QR≡NS≡MT952Q≡MNb1=NOb2=NPI≡MNQ≡MOR≡MPS≡NOPT≡NOPQU≡NOQR≡NPQS≡MOPQT≡MOPU≡OPRS≡MNPRT≡MNOST≡MNPQRU≡QTU≡MNOQSUNO,NP,2III9−5MN≡QNO,QR,16NoneR≡MOMO≡PU≡RPT,NP,MP≡OU≡SS≡MPMQ≡NQS,OT,OPNQ≡OR≡PS≡MT≡NOPM≡OROP,RS,MS≡PU≡NOPQQT≡UNTQU≡TNote. The rest of the designs are illustrated in Table 3.


## 3.2.2. Alias Structures

Considering25−1 design in Table 3 with design relation:(11)I≡MNOPQ.The alias structure is obtained similarly by multiplying the treatment effects by the design relation. Hence, the main effects are aliased as follows:(12)M≡NOPQ,N≡MOPQ,O≡MNPQ,P≡MNOQ,Q≡MNOP,OP≡MNQ,NQ≡MOP,MQ≡NOP,OQ≡MNP,NO≡MPQ,PQ≡MNO,MP≡NOQQ,NP≡MOQ.


## 3.2.3. Determining the Confounding of Effects

Consider the 16-run of25−1 design in four blocks in Table 3, with the treatment defining contrast subgroup(13)I≡MNOPQ.The generalized block defining contrast subgroup is formed by taking all 2-way products as follows:MNMO≡M2NO≡NOThat is, the block defining contrast subgroup is given byMN, MO, and NO.By multiplying the defining word byMN, MO, and NO, we have(14)MNOPQMN≡M2N2OPQ≡OPQ,MNOPQMO≡M2NO2PQ≡NPQ,MNOPQNO≡MN2O2PQ≡MPQ.Considering the block defining generators, it might be concluded that the block effectsb1 and b2 are only confounded with MN, MO, and NObecause of the treatment defining relation is also aliased with the effects OPQ, NPQ, and MPQ. Therefore, b1 and b2 are confounded with three effects: OPQ, NPQ, and MPQ.


## 3.2.4. Clear Effects

Hence, from the above illustrations, the remaining 14 (= 16 − 2) degrees of freedom do not involve the block effects, each of the 5 main effects is aliased with 4-factor interactions, and the 7 two-factor interactions effects are aliased with three-factor interaction, refer to alias structures above, whereM≡NOPQ,OP≡MNQ,NQ≡MOP, and so on. The 5-factor interaction is chosen as a generator. Hence, all 5 main effects and all two main effects are clear, except MN, MO, and NO which are confounded with the block effect b1 and b2 and are not eligible for estimation.


## 3.2.5. Blocked Fractional Factorial Design for 4 Choices Set of Size 4

The blocked fractional factorial design2V5−1−2 is a sixteen-run design. Hence, the generator of this 2-level design is =NMNOP. If y1,...,y5 represents the five factors, then Q=NMNOP is equivalent to y5 = y1+y2+y3+y4 (mod 2). The factor level for factor y5 is obtained by adding the corresponding treatment combinations of factor y1, y2, y3, and y4.To illustrate this for treatment combinationI, suppose the level for factors y1 = 0, y2 = 0, y3 = 0, and y4 = 0, and y5 = 0 + 0+0 + 0 = 0 (mod 2). Also, for treatment combination a, with the level for factors y1 = 1, y2 = 0, y3 = 0, y4 = 0, y5 = 1 + 0+0 + 0 = 1 (mod 2), and so on.To obtain 4 choice sets of size four, we select two effects to be confounded with the block. Suppose we choosey1y2 and y1y3 to be confounded with the blocks. These effects will have two block defining contrast, that is,(15)b1=y1y2,b2=y1y3.All the treatment combinations are evaluated usingb1 and b2. λ1 = λ2 = λ3 = 1I: b1 = 1(0) + 1(0) + 0(0) = 0 (mod 2)b2 = 1(0) + 0(0) + 1(0) = 0 (mod 2)a: b1 = 1(1) + 1(0) +0(0) = 1 (mod 2)b2 = 1(1) + 0(0) + 1(0) = 1 (mod 2).Similarly, we can obtainb1 (mod 2) and b2 (mod 2) values for other treatments. Hence, from the values above, all treatment combinations with level 0 in b1 and b2 form block 1 which is an inherent choice set one, and also all treatment combinations with level 1 in b1 and level 1 in b2 form block 2, which also form choice set two. Also, all treatment combinations with level 1 in b1 and level 0 in b2 form block 3, which is an inherent choice set three, and finally, all treatment combinations with level 0 in b1 and level 1 in b2 form block 4, which also form choice set four.


## 3.2.6. Design Efficiency

The result can be seen in Table4, which shows 66.7–81.8 percent A-efficiency for all the designs.Table 4
Design efficiencies’ design and block generators form = 4, 5, 6, 7, 8, and 9 with 4 choice sets of size 4.
DesignsA-efficiency (%)2IV4−0−266.72V5−1−271.42IV6−2−275.02IV7−3−277.72IV8−4−280.02III9−5−281.8


## 3.3. Design and Block Generators form = 5, 6, 7, and 8 with 8 Choices Set of Size 2

### 3.3.1. Generators and Defining Relations

In Table5, let MNOQ, MNPR, and MOPS be the generators of the 2IV7−3 design. The generalized interactions are formed by taking all 2-way products as follows:(16)MNOQMNPR≡M2N2OPQR≡OPQR,MNOQMOPS≡M2NO2QPS≡NPQS,MNPRMOPS≡M2NOP2RS≡NORS.Table 5
Sixteen-run design and block generators form = 5, 6, 7, and 8 with 8 choices set of size 2.
mnbDesign generatorsBlock generatorsDefining contrastIBlock contrastFractionsAliasing effectsConfounding effectsNo. of runsClear effects513Q≡MNOb1=MPb2=NPb3=OPI≡MNOQMP,NP,OP,MN,MO,NO2IV5−1MN≡OQMO≡NQNO≡MQMP,NP,OP,MN,MO,NO,OQ,NQ,MQ16All 5 main effects623Q≡MNOR≡MNPb1=MOb2=NOb3=NPI≡MNOQ≡MNPR≡OPQRMO,NO,MP,MN,OP,MNOP2IV6−2MN≡OQ≡PRMO≡NQNO≡MQMP≡NRNP≡MROP≡QRPQ≡QRMO,NQ,NO,MQ,MP,NR,MN,OQ,PR,OP,QR,PQ,OR16All 6 main effects733Q≡MNOR≡MNPS≡MOPb1=MNb2=MOb3=MPI≡MNOQ≡MNPR≡MOPS≡OPQR≡NPQS≡NQRSMN,MO,MP,NO,NP,OP2IV7−3MN≡OQ≡PR,MO≡NQ≡PSNO≡MQ≡RS,NQ≡MOMP≡NR≡OS,NP≡MR≡QSOP≡MS≡QR,PQ≡OR≡NSNS≡PQ,RS≡NO,NQ≡PSMN,OQ,PR,MO,NQ,PS,NO,MQ,NP,MR,QS,OP,MS,QR16All 7 main effects843Q≡MNOR≡MNPS≡MOPT≡NOPb1=MNb2=MOb3=MPI≡MNOQ≡MNPR≡MOPS≡NOPT≡OPQR≡NPQS≡MPQT≡NORS≡MORT≡MNSTNN,MO,MP,NO,NP,OP2IV8−4MP≡NR≡OS≡QT,MN≡OQ≡PR≡ST,NP≡MR,OT≡QS,MO≡NQ≡PS≡RT , MS≡OP≡NT≡QR,NO≡MQ≡PT≡RS,PQ≡OR≡NS≡MTMN,MO,MP,NO,NP,OP16All 8 main effectsNote. The rest of the designs are illustrated in Table 5.That is, the treatment defining relation is given byI≡MNOQ≡MNPR≡MOPS≡OPQR≡NPQS≡NORS, and because the length of the shortest “word” is four, this design is resolution IV, denoted as 2IV7−3 fractional factorial design. The detailed results are contained in Table 5.


### 3.3.2. Alias Structures

Considering the2IV7−3 design in Table 5 with design relation(17)I≡MNOQ≡MNPR≡MOPS≡OPQR≡NPQS≡NORS.The alias structure is obtained by multiplying the treatment effects by the design relation; hence, the main effects are aliased as follows:(18)M≡NOQ≡NPR≡OPS≡MOPQR≡MNPQS≡MNORS,N≡MOQ≡MPR≡MNOPS≡NOPQR≡PQS≡ORS,O≡MNQ≡MNOPR≡MPS≡PQR≡NOPQS≡NRS,P≡MNOPQ≡MNR≡MOS≡OQR≡NQS≡NOPRS,Q≡MNO≡MNPQR≡MOPQS≡OPR≡NPS≡NOQRS,R≡MNOQR≡MNP≡MOPRS≡OPQ≡NPQRS≡NOS,S≡MNOQS≡MNPRS≡MOP≡OPQRS≡NPQ≡NOR,PQ≡MNOP≡MNQR≡MOQS≡OR≡NS≡NOPQRS,NOP≡MPQ≡MOR≡MNS≡NQR≡OQS≡PRS,MQR≡MNOR≡MNPQ≡MOPQRS≡OP≡NPRS≡NOQRS,MRS≡NOQRS≡NPS≡OPR≡MOPQS≡MNPQR≡MNO,MQRS≡NORS≡NPQS≡OPQR≡MOPS≡MNPR≡MNOQ.


### 3.3.3. Determining the Confounding of Effects

Consider the 8-run2IV7−3 design in four blocks in Table 5, with the treatment defining contrast subgroup(19)I≡MNOQ≡MNPR≡MOPS≡OPQR≡NPQS≡NORS.The generalized block defining contrast subgroup is formed by taking all 2-way products as follows:(20)MNMO≡M2NO≡NO,MNMP≡M2NP≡NP,MOMP≡M2OP≡OP.That is, the block defining contrast subgroup is given byMN, MO, MP, NO, NP, and OP.By multiplying each defining word byMN, MO, MP, NO, NP, and OP, we have(21)MNOQMN≡OQ,MNOQMO≡NQ,MNOQMP≡NOPQ,MNOQNO≡MQ,MNOQNP≡MOPQ,MNOQOP≡MNPQ,MNPRMN≡PR,MNPRMO≡NOPR,MNPRMP≡NR,MNPRNO≡MOPR,MNPRNP≡MR,MNPROP≡MNOR,MOPSMN≡NOPS,MOPSMO≡PS,MOPSMP≡OS,MOPSNO≡MNPS,MOPSNP≡MNOS,MOPSOP≡MS,OPQRMN≡MNOPQ,OPQRMO≡MPQR,OPQRMP≡MOQR,OPQRNO≡NPQR,OPQRNP≡NOQR,OPQROP≡QR,NPQSMN≡MPQS,NPQSMO≡MNOPQS,NPQSMP≡MNQS,NPQSNO≡OPQS,NPQSNP≡QS,NPQSOP≡NOQS,NORSMN≡MORS,NORSMO≡MNRS,NORSMP≡MNOPRS,NORSNO≡RS,NORSNP≡OPRS,NORSOP≡NPRS.Considering the block defining generators, it might be concluded that the block effectsb1, b2, and b3 are only confounded with MN, MO, MP, NO, NP, and OP. Because the treatment defining relation is also aliased with the effects illustrated previously, therefore, b1 and b2 are confounded with thirty-six effects:(22)OQ,NQ,NOPQ,MQ,MOPQ,MNPQ,PR,NOPR,NR,MOPR,MR,MNOR,NOPS,PS,OS,MNPS,MNOS,MNS,MNOPQR,MPQR,MOQR,NPQR,NOQR,QR,MPQS,MNOPQS,MNQS,OPQS,QS,NOQS,MORS,MNRS,MNOPRS,RS,OPRSandNPRS.


### 3.3.4. Clear Effects

Hence, from the above illustrations, the remaining 13 (= 16 − 3) degrees of freedom that do not involve the block effects, each of the seven main effects is aliased with higher interactions, see the previous alias structure,M≡NOQ≡NPR≡OPS≡MOPQR≡MNPQS≡MNORS and so on. That is, the seven main effects are clear. The two-factor interactions are aliased with other two factor interaction or either confounded with a block effect. Therefore, the two-factor interactions are not eligible for estimation, see the previous alias structures and confounding effects, and MNPRMP≡NR, respectively, and so on.


### 3.3.5. Blocked Fractional Factorial Design for 8 Choices Set of Size 2

The blocked fractional factorial design2IV7−3 is a sixteen-run design. The generators of this 2-level design are Q=MNO,R=MNP, and S=MOP. If y1,...,y7 represents the seven factors, then Q=MNO,R=MNP, and S=MOP are equivalent to y5 = y1+y2+y3 (mod 2), y6=y1+y2+y4 (mod 2), and y7=y1+y3+y4 (mod 2), respectively. The factor level for factor y5 is obtained by adding the corresponding treatment combinations of factor y1, y2, and y3; likewise, factor y6 is obtained by adding the corresponding treatment combinations of factor y1, y2, and y4, and y7 is obtained by adding the corresponding treatment combinations of factor y1, y3, and y4.To illustrate this for treatment combinationI, with the factor level for factor y1 = 0, y2=0, and y3 = 0, y5 = 0 + 0+0(mod 2) = 0, also with the factor level for factor y1 = 0, y2=0, and y4 = 0, y6 = 0 + 0+0(mod 2) = 0 and with the factor level for factor y1 = 0, y3=0, and y4 = 0, y7 = 0 + 0+0 (mod 2) = 0. Also, for treatment combination a, with the factor level for factor y1 = 1, y2=0, and y3 = 0, y5 = 1 + 0+0 (mod 2) = 1, also with the factor level for factor y1 = 1, y2=0, and y4 = 0, y6 = 1 + 0 + 0 (mod 2) = 1 and with the factor level for factor y1 = 1, y3=0, and y4 = 0, y7 = 1 + 0 + 0 (mod 2) = 1 and so on.To obtain 8 choice sets of size two we select three effects to be confounded with the block. Suppose we choosey1y2, y1y3, and y1y4 to be confounded with the blocks. These effects will have two block-defining contrasts, that is(23)b1=y1y2,b2=y1y3,b3=y1y4.All the treatment combinations are evaluated usingb1, b2, and b3. λ1 = λ2 = λ3 = λ4 = 1I: b1 = 1(0) + 1(0) + 0(0) + 0(0) = 0 (mod 2)b2 = 1(0) + 0(0) + 1(0) + 0(0) = 0 (mod 2)b3 = 1(0) +0(0) + 0(0) + 1(0) = 0 (mod 2)a: b1 = 1(1) + 1(0) + 0(0) + 0(0) = 1 (mod 2)b2 = 1(1) + 0(0) + 1(0) + 0(0) = 1 (mod 2)b3 = 1(1) + 0(0) + 0(0) + 1(0) = 1 (mod 2)b: b1 = 1(0) + 1(1) + 0(0) + 0(0) = 1 (mod 2)b2 = 1(0) + 0(1) + 1(0) + 0(0) = 0 (mod 2)b3 = 1(0) + 0(1) + 0(0) + 1(0) = 0 (mod 2).Similarly, we can obtainb1 (mod 2), b2 (mod 2) and b3 (mod 2) values for other treatments. Hence, from the above values, all treatment combinations with level 0 in b1, b2 and b3 form block 1 which is an inherent choice set one and also all treatment combinations with level 1 in b1, level 1 in b2 and level 1 in b3 form block 2, which also form choice set two. Also, all treatment combinations with level 1 in b1, level 0 in b2 and level 0 in b3 form block 3, which is an inherent choice set three, all treatment combinations with level 0 in b1, level 1 in b2 and level 1 in b3 form block 4, all treatment combinations with level 0 in b1, level 1 in b2 and level 0 in b3 form block 5, all treatment combinations with level 1 in b1, level 0 in b2 and level 1 in b3 form block 6, all treatment combinations with level 1 in b1, level 1 in b2 and level 0 in b3 form block 7 and finally, all treatment combinations with level 0 in b1, level 0 in b2 and level 1 in b3 form block 8, which also form choice set eight.Therefore, treatment combinationsI and abcd forms choice set 1, a and bcd forms choice set 2, b and acd form choice set 3 and ab and cd form choice set 4, c and abd forms choice set 5, ac and bd forms choice set 6, bc and ad form choice set 7, lastly, abc and d form choice set 8. Hence a choice set of 8 and size 2 is formed.


### 3.3.6. Design Efficiency

The result can be seen in Table6, which shows 60.3–67.8 percent A-efficiency for all the designs.Table 6
Design efficiency design and block generators form = 5, 6, 7, and 8 with 8 choices set of size 2.
DesignsA-efficiency (%)2IV5−1−360.32IV6−2−366.72IV7−3−367.82IV8−4−365.5


## 3.3.1. Generators and Defining Relations

In Table5, let MNOQ, MNPR, and MOPS be the generators of the 2IV7−3 design. The generalized interactions are formed by taking all 2-way products as follows:(16)MNOQMNPR≡M2N2OPQR≡OPQR,MNOQMOPS≡M2NO2QPS≡NPQS,MNPRMOPS≡M2NOP2RS≡NORS.Table 5
Sixteen-run design and block generators form = 5, 6, 7, and 8 with 8 choices set of size 2.
mnbDesign generatorsBlock generatorsDefining contrastIBlock contrastFractionsAliasing effectsConfounding effectsNo. of runsClear effects513Q≡MNOb1=MPb2=NPb3=OPI≡MNOQMP,NP,OP,MN,MO,NO2IV5−1MN≡OQMO≡NQNO≡MQMP,NP,OP,MN,MO,NO,OQ,NQ,MQ16All 5 main effects623Q≡MNOR≡MNPb1=MOb2=NOb3=NPI≡MNOQ≡MNPR≡OPQRMO,NO,MP,MN,OP,MNOP2IV6−2MN≡OQ≡PRMO≡NQNO≡MQMP≡NRNP≡MROP≡QRPQ≡QRMO,NQ,NO,MQ,MP,NR,MN,OQ,PR,OP,QR,PQ,OR16All 6 main effects733Q≡MNOR≡MNPS≡MOPb1=MNb2=MOb3=MPI≡MNOQ≡MNPR≡MOPS≡OPQR≡NPQS≡NQRSMN,MO,MP,NO,NP,OP2IV7−3MN≡OQ≡PR,MO≡NQ≡PSNO≡MQ≡RS,NQ≡MOMP≡NR≡OS,NP≡MR≡QSOP≡MS≡QR,PQ≡OR≡NSNS≡PQ,RS≡NO,NQ≡PSMN,OQ,PR,MO,NQ,PS,NO,MQ,NP,MR,QS,OP,MS,QR16All 7 main effects843Q≡MNOR≡MNPS≡MOPT≡NOPb1=MNb2=MOb3=MPI≡MNOQ≡MNPR≡MOPS≡NOPT≡OPQR≡NPQS≡MPQT≡NORS≡MORT≡MNSTNN,MO,MP,NO,NP,OP2IV8−4MP≡NR≡OS≡QT,MN≡OQ≡PR≡ST,NP≡MR,OT≡QS,MO≡NQ≡PS≡RT , MS≡OP≡NT≡QR,NO≡MQ≡PT≡RS,PQ≡OR≡NS≡MTMN,MO,MP,NO,NP,OP16All 8 main effectsNote. The rest of the designs are illustrated in Table 5.That is, the treatment defining relation is given byI≡MNOQ≡MNPR≡MOPS≡OPQR≡NPQS≡NORS, and because the length of the shortest “word” is four, this design is resolution IV, denoted as 2IV7−3 fractional factorial design. The detailed results are contained in Table 5.


## 3.3.2. Alias Structures

Considering the2IV7−3 design in Table 5 with design relation(17)I≡MNOQ≡MNPR≡MOPS≡OPQR≡NPQS≡NORS.The alias structure is obtained by multiplying the treatment effects by the design relation; hence, the main effects are aliased as follows:(18)M≡NOQ≡NPR≡OPS≡MOPQR≡MNPQS≡MNORS,N≡MOQ≡MPR≡MNOPS≡NOPQR≡PQS≡ORS,O≡MNQ≡MNOPR≡MPS≡PQR≡NOPQS≡NRS,P≡MNOPQ≡MNR≡MOS≡OQR≡NQS≡NOPRS,Q≡MNO≡MNPQR≡MOPQS≡OPR≡NPS≡NOQRS,R≡MNOQR≡MNP≡MOPRS≡OPQ≡NPQRS≡NOS,S≡MNOQS≡MNPRS≡MOP≡OPQRS≡NPQ≡NOR,PQ≡MNOP≡MNQR≡MOQS≡OR≡NS≡NOPQRS,NOP≡MPQ≡MOR≡MNS≡NQR≡OQS≡PRS,MQR≡MNOR≡MNPQ≡MOPQRS≡OP≡NPRS≡NOQRS,MRS≡NOQRS≡NPS≡OPR≡MOPQS≡MNPQR≡MNO,MQRS≡NORS≡NPQS≡OPQR≡MOPS≡MNPR≡MNOQ.


## 3.3.3. Determining the Confounding of Effects

Consider the 8-run2IV7−3 design in four blocks in Table 5, with the treatment defining contrast subgroup(19)I≡MNOQ≡MNPR≡MOPS≡OPQR≡NPQS≡NORS.The generalized block defining contrast subgroup is formed by taking all 2-way products as follows:(20)MNMO≡M2NO≡NO,MNMP≡M2NP≡NP,MOMP≡M2OP≡OP.That is, the block defining contrast subgroup is given byMN, MO, MP, NO, NP, and OP.By multiplying each defining word byMN, MO, MP, NO, NP, and OP, we have(21)MNOQMN≡OQ,MNOQMO≡NQ,MNOQMP≡NOPQ,MNOQNO≡MQ,MNOQNP≡MOPQ,MNOQOP≡MNPQ,MNPRMN≡PR,MNPRMO≡NOPR,MNPRMP≡NR,MNPRNO≡MOPR,MNPRNP≡MR,MNPROP≡MNOR,MOPSMN≡NOPS,MOPSMO≡PS,MOPSMP≡OS,MOPSNO≡MNPS,MOPSNP≡MNOS,MOPSOP≡MS,OPQRMN≡MNOPQ,OPQRMO≡MPQR,OPQRMP≡MOQR,OPQRNO≡NPQR,OPQRNP≡NOQR,OPQROP≡QR,NPQSMN≡MPQS,NPQSMO≡MNOPQS,NPQSMP≡MNQS,NPQSNO≡OPQS,NPQSNP≡QS,NPQSOP≡NOQS,NORSMN≡MORS,NORSMO≡MNRS,NORSMP≡MNOPRS,NORSNO≡RS,NORSNP≡OPRS,NORSOP≡NPRS.Considering the block defining generators, it might be concluded that the block effectsb1, b2, and b3 are only confounded with MN, MO, MP, NO, NP, and OP. Because the treatment defining relation is also aliased with the effects illustrated previously, therefore, b1 and b2 are confounded with thirty-six effects:(22)OQ,NQ,NOPQ,MQ,MOPQ,MNPQ,PR,NOPR,NR,MOPR,MR,MNOR,NOPS,PS,OS,MNPS,MNOS,MNS,MNOPQR,MPQR,MOQR,NPQR,NOQR,QR,MPQS,MNOPQS,MNQS,OPQS,QS,NOQS,MORS,MNRS,MNOPRS,RS,OPRSandNPRS.


## 3.3.4. Clear Effects

Hence, from the above illustrations, the remaining 13 (= 16 − 3) degrees of freedom that do not involve the block effects, each of the seven main effects is aliased with higher interactions, see the previous alias structure,M≡NOQ≡NPR≡OPS≡MOPQR≡MNPQS≡MNORS and so on. That is, the seven main effects are clear. The two-factor interactions are aliased with other two factor interaction or either confounded with a block effect. Therefore, the two-factor interactions are not eligible for estimation, see the previous alias structures and confounding effects, and MNPRMP≡NR, respectively, and so on.


## 3.3.5. Blocked Fractional Factorial Design for 8 Choices Set of Size 2

The blocked fractional factorial design2IV7−3 is a sixteen-run design. The generators of this 2-level design are Q=MNO,R=MNP, and S=MOP. If y1,...,y7 represents the seven factors, then Q=MNO,R=MNP, and S=MOP are equivalent to y5 = y1+y2+y3 (mod 2), y6=y1+y2+y4 (mod 2), and y7=y1+y3+y4 (mod 2), respectively. The factor level for factor y5 is obtained by adding the corresponding treatment combinations of factor y1, y2, and y3; likewise, factor y6 is obtained by adding the corresponding treatment combinations of factor y1, y2, and y4, and y7 is obtained by adding the corresponding treatment combinations of factor y1, y3, and y4.To illustrate this for treatment combinationI, with the factor level for factor y1 = 0, y2=0, and y3 = 0, y5 = 0 + 0+0(mod 2) = 0, also with the factor level for factor y1 = 0, y2=0, and y4 = 0, y6 = 0 + 0+0(mod 2) = 0 and with the factor level for factor y1 = 0, y3=0, and y4 = 0, y7 = 0 + 0+0 (mod 2) = 0. Also, for treatment combination a, with the factor level for factor y1 = 1, y2=0, and y3 = 0, y5 = 1 + 0+0 (mod 2) = 1, also with the factor level for factor y1 = 1, y2=0, and y4 = 0, y6 = 1 + 0 + 0 (mod 2) = 1 and with the factor level for factor y1 = 1, y3=0, and y4 = 0, y7 = 1 + 0 + 0 (mod 2) = 1 and so on.To obtain 8 choice sets of size two we select three effects to be confounded with the block. Suppose we choosey1y2, y1y3, and y1y4 to be confounded with the blocks. These effects will have two block-defining contrasts, that is(23)b1=y1y2,b2=y1y3,b3=y1y4.All the treatment combinations are evaluated usingb1, b2, and b3. λ1 = λ2 = λ3 = λ4 = 1I: b1 = 1(0) + 1(0) + 0(0) + 0(0) = 0 (mod 2)b2 = 1(0) + 0(0) + 1(0) + 0(0) = 0 (mod 2)b3 = 1(0) +0(0) + 0(0) + 1(0) = 0 (mod 2)a: b1 = 1(1) + 1(0) + 0(0) + 0(0) = 1 (mod 2)b2 = 1(1) + 0(0) + 1(0) + 0(0) = 1 (mod 2)b3 = 1(1) + 0(0) + 0(0) + 1(0) = 1 (mod 2)b: b1 = 1(0) + 1(1) + 0(0) + 0(0) = 1 (mod 2)b2 = 1(0) + 0(1) + 1(0) + 0(0) = 0 (mod 2)b3 = 1(0) + 0(1) + 0(0) + 1(0) = 0 (mod 2).Similarly, we can obtainb1 (mod 2), b2 (mod 2) and b3 (mod 2) values for other treatments. Hence, from the above values, all treatment combinations with level 0 in b1, b2 and b3 form block 1 which is an inherent choice set one and also all treatment combinations with level 1 in b1, level 1 in b2 and level 1 in b3 form block 2, which also form choice set two. Also, all treatment combinations with level 1 in b1, level 0 in b2 and level 0 in b3 form block 3, which is an inherent choice set three, all treatment combinations with level 0 in b1, level 1 in b2 and level 1 in b3 form block 4, all treatment combinations with level 0 in b1, level 1 in b2 and level 0 in b3 form block 5, all treatment combinations with level 1 in b1, level 0 in b2 and level 1 in b3 form block 6, all treatment combinations with level 1 in b1, level 1 in b2 and level 0 in b3 form block 7 and finally, all treatment combinations with level 0 in b1, level 0 in b2 and level 1 in b3 form block 8, which also form choice set eight.Therefore, treatment combinationsI and abcd forms choice set 1, a and bcd forms choice set 2, b and acd form choice set 3 and ab and cd form choice set 4, c and abd forms choice set 5, ac and bd forms choice set 6, bc and ad form choice set 7, lastly, abc and d form choice set 8. Hence a choice set of 8 and size 2 is formed.


## 3.3.6. Design Efficiency

The result can be seen in Table6, which shows 60.3–67.8 percent A-efficiency for all the designs.Table 6
Design efficiency design and block generators form = 5, 6, 7, and 8 with 8 choices set of size 2.
DesignsA-efficiency (%)2IV5−1−360.32IV6−2−366.72IV7−3−367.82IV8−4−365.5


## 3.4. Design and Block Generators form = 5, 6, 7, 8, 9 and 10 with 8 Choice Sets of Size 4

### 3.4.1. Generators and Defining Relations

In Table7, consider a 25−0 fractional factorial design with no generator. Hence, there are no generalized interactions and also no treatment defining relation. The design is of resolution V, denoted as 2V5−0 fractional factorial design. The detailed results are contained in Table 7.Table 7
Sixteen-run design and block generators form = 5, 6, 7, and 8 with 8 choice sets of size 4.
mnbDesign generatorsBlock generatorsDefining contrastIBlock contrastFract- ionsAliasing effectsConfounding effectsNo. of runsClear effects503…b1=MOQb2=NOQb3=MNOP…MOQ,NOQ,MNOP,MN,NPQ,MPQ2V5…MN,OP,MOQ,NOQ,MNOP,NPQ,MPQ32All 5 main effects plus, all two factor interactions except MN andOP613R≡MNOPQb1=MOQb2=NOQb3=MPQI≡MNOPQMOQ,NOQ,PQ,MN,OP,MNOP2VI6−1M≡NOPQRMN≡OPQRMO≡NPQRMN,OP,QR,MOQ,NOQ,MPQ,MNOP32All 6 main effects plus, all two factor interactions except MN andOP and QR723R≡MNOS=MNPQb1=NOPb2=NOQb3=MOPQI≡MNOR≡MNPQS≡OPQRSNOP,NOQ,MOPQ,PQ,MNQ,MNP2IV7−2MN≡ORMO≡NRNO≡MRPS,QS,MNP,PQ,MNQ,NOP,NOQ,MOPQ,MO32All 7 main effects plus the following two factor interactionsMP,MQ,MS,NP,NQ,NS,OP,OQ,OS,PR,QR,RS833R≡MNOS≡MNPT≡MOPQb1=MOb2=NOb3=MPI≡MNOR≡MNPS≡MOPQT≡OPRS≡NPQRT≡NOQSTMO,NO,MP,MN,OP,MNOP2IV8−3MN≡OR≡PSMO≡NRNO≡MRMP≡NSNP≡MSOP≡RSPR≡OSNR,MR,NS,OR,PS,RS,PR,OS,MO,NO,MP,OP,MN,MNOP32All 8 main effects plus the following two factor interactionsMQ,NQ, OQ,PQ,QR,QS,MT,NT,OT,PT,QT,RT,ST943R≡MNOS≡MNPT≡MOPU≡NOPQb1=MNb2=MOb3=MPI≡MNOR≡MNPS≡MOPT≡NOPQU≡OPRS≡NPRT≡MPQRU≡NOST≡MQSU≡MNQTUMN,MO,MP,NO,NP,OP2IV9−4MN≡OR≡PS,MO≡NR≡PTMP≡NS≡OT,NO≡MR≡STNP≡MS≡RT,OP≡MT≡RSPR≡NT≡OSOR,PS,NR,PT,NS,OT,MR,ST,MS,MT,RS32All 9 main effects plus the following two factor interact- tionsMQ,NQ,OQ,PQ,QR,QT,MU,NU,QU,RU,SU,TU1053Q≡MNR≡MOS≡MPT≡NOPU≡NOPQb1=MNb2=MPb3=NQI≡MNQ≡MOR≡MPS≡NOPT≡NOPQU≡NOQR≡NPQS≡MOPQT≡MOPU≡OPRS≡MNPRT≡MNPQRUMN,MP,NQ,NP,NQ,MNPQ2III10−5MN≡Q,MQ≡NNQ≡M≡OR≡PSMO≡R,MR≡QMP≡S,MS≡PNO≡PT≡QR,NP≡OT≡QSOP≡NT≡RS,OQ≡NRPS≡OR,MO≡PUMP≡OUOP≡MU≡RSPR≡OSMN,MP,NQ,NP,MQ,MN,MNPQ32The following main and interac- tions effects are clearT,U,MT,QT,RT,ST,NU,QU,RU,SU,TU,MV,OV,PV,QV,RV,SV,TV,UVNote. The rest of the designs are illustrated in Table 7 above.


### 3.4.2. Alias Structures

Considering the25−0 design in Table 7, which have no treatment defining relation, hence it does not have aliasing structure.


### 3.4.3. Determing the Confounding of Effects

Consider the 32-run of25−0 design in eight blocks in Table 7, the generalized block-defining contrast subgroup is formed by taking all 2-way products as follows:(24)MOQNOQ≡MNO2Q2≡MN,MOQMNOP≡M2NO2PQ≡NPQ,NOQMNOP≡MN2O2PQ≡MPQ.Considering only the block-defining generators, it might be concluded that the block effectsb1, b2, and b3 are only confounded with MOQ, NOQ, MNOP, MN, NPQ, and MPQ, but OP is also confounded in the block effects. Therefore, b1, b2, and b3 are confounded with six effects: MOQ, NOQ, MNOP, MN, OP, NPQ, and MPQ.


### 3.4.4. Clear Effects

Hence from the above illustrations, the remaining 29 (= 32 − 3) degrees of freedom that do not involve the block effects, all the 5 main effects and two factor effects are clear exceptMN and OP which are confounded in the block effects. Hence, M,N,O,P,Q,MO,NO,MP,NP,MQ,NQ, and OQ are clear effects.


### 3.4.5. Blocked Fractional Factorial Design for 8 Choice Sets of Size 4

To obtain 8 choice sets of size four we select three effects to be confounded with the block. Suppose we choosey1y3y5, y2y3y5, and y1y2y3y4 to be confounded with the blocks. These effects will have three block-defining contrasts, that is b1 = y1y3y5, b2 = y2y3y5, and b3 = y1y2y3y4All the treatment combinations are evaluated usingb1, b2, and b3. λ1 = λ2 = λ3 = λ4 = λ5= 1(I):b1 = 1(0) + 0(0) + 1(0) + 0(0) + 1(0) = 0 (mod 2)b2 = 0(0) + 1(0) + 1(0) + 0(0) + 1(0) = 0 (mod 2)b3 = 1(0) + 1(0) + 1(0) + 1(0) + 0(0) = 0 (mod 2)a:b1 = 1(1) + 0(0) + 1(0) + 0(0) + 1(0) = 1 (mod 2)b2 = 0(1) + 1(0) + 1(0) + 0(0) + 1(0) = 0 (mod 2)b3 = 1(1) + 1(0) + 1(0) + 1(0) + 0(0) = 1 (mod 2).Similarly, we can obtainb1 (mod 2), b2 (mod 2), and b3 (mod 2) values for other treatments. Hence, from the above values, all treatment combinations with level 0 in b1, b2, and b3 form block 1 which is an inherent choice set one and also all treatment combinations with level 1 in b1, level 0 in b2, and level 1 in b3 form block 2, which also form choice set two. Also, all treatment combinations with level 0 in b1, level 1 in b2, and level 1 in b3 form block 3, which is an inherent choice set three, all treatment combinations with level 1 in b1, level 1 in b2, and level 0 in b3 form block 4, which is an inherent choice set four, all treatment combinations with level 1 in b1, level 1 in b2, and level 1 in b3 form block 5, which is an inherent choice set five, all treatment combinations with level 0 in b1, level 1 in b2, and level 0 in b3 form block 6, which is an inherent choice set six, all treatment combinations with level 1 in b1, level 0 in b2, and level 0 in b3 form block 7, which is an inherent choice set seven and finally, all treatment combinations with level 0 in b1, level 0 in b2, and level 1 in b3 form block 8, which is an inherent choice set eight.Therefore, treatment combinationsI,abcd,abe, and cde form choice set 1; a,bcd,be, and ade form choice set 2; b,acd,ae, and bcde form choice set 3; ab,cd,e, and abcde form choice set 4; c,abd,abce, and de form choice set 5; ac,bd,bce,ade form choice set 6; bc,ad,ace, and bde form choice set 7, and lastly, abc,d,ce, and abde form choice set 8. Hence, a choice set of 8 and size 4 is formed.


### 3.4.6. Design Efficiency

The result can be seen in Table8, which shows 62.5–76.9 percent A-efficiency for all the designs.Table 8
Design efficiency design and block generators form = 5, 6, 7, 8, 9, and 10 with 8 choice sets of size 4.
DesignsA-efficiency (%)2V5−0−362.52VI6−1−366.72IV7−2−370.02IV8−3−372.02IV9−4−375.02III10−5−376.9


## 3.4.1. Generators and Defining Relations

In Table7, consider a 25−0 fractional factorial design with no generator. Hence, there are no generalized interactions and also no treatment defining relation. The design is of resolution V, denoted as 2V5−0 fractional factorial design. The detailed results are contained in Table 7.Table 7
Sixteen-run design and block generators form = 5, 6, 7, and 8 with 8 choice sets of size 4.
mnbDesign generatorsBlock generatorsDefining contrastIBlock contrastFract- ionsAliasing effectsConfounding effectsNo. of runsClear effects503…b1=MOQb2=NOQb3=MNOP…MOQ,NOQ,MNOP,MN,NPQ,MPQ2V5…MN,OP,MOQ,NOQ,MNOP,NPQ,MPQ32All 5 main effects plus, all two factor interactions except MN andOP613R≡MNOPQb1=MOQb2=NOQb3=MPQI≡MNOPQMOQ,NOQ,PQ,MN,OP,MNOP2VI6−1M≡NOPQRMN≡OPQRMO≡NPQRMN,OP,QR,MOQ,NOQ,MPQ,MNOP32All 6 main effects plus, all two factor interactions except MN andOP and QR723R≡MNOS=MNPQb1=NOPb2=NOQb3=MOPQI≡MNOR≡MNPQS≡OPQRSNOP,NOQ,MOPQ,PQ,MNQ,MNP2IV7−2MN≡ORMO≡NRNO≡MRPS,QS,MNP,PQ,MNQ,NOP,NOQ,MOPQ,MO32All 7 main effects plus the following two factor interactionsMP,MQ,MS,NP,NQ,NS,OP,OQ,OS,PR,QR,RS833R≡MNOS≡MNPT≡MOPQb1=MOb2=NOb3=MPI≡MNOR≡MNPS≡MOPQT≡OPRS≡NPQRT≡NOQSTMO,NO,MP,MN,OP,MNOP2IV8−3MN≡OR≡PSMO≡NRNO≡MRMP≡NSNP≡MSOP≡RSPR≡OSNR,MR,NS,OR,PS,RS,PR,OS,MO,NO,MP,OP,MN,MNOP32All 8 main effects plus the following two factor interactionsMQ,NQ, OQ,PQ,QR,QS,MT,NT,OT,PT,QT,RT,ST943R≡MNOS≡MNPT≡MOPU≡NOPQb1=MNb2=MOb3=MPI≡MNOR≡MNPS≡MOPT≡NOPQU≡OPRS≡NPRT≡MPQRU≡NOST≡MQSU≡MNQTUMN,MO,MP,NO,NP,OP2IV9−4MN≡OR≡PS,MO≡NR≡PTMP≡NS≡OT,NO≡MR≡STNP≡MS≡RT,OP≡MT≡RSPR≡NT≡OSOR,PS,NR,PT,NS,OT,MR,ST,MS,MT,RS32All 9 main effects plus the following two factor interact- tionsMQ,NQ,OQ,PQ,QR,QT,MU,NU,QU,RU,SU,TU1053Q≡MNR≡MOS≡MPT≡NOPU≡NOPQb1=MNb2=MPb3=NQI≡MNQ≡MOR≡MPS≡NOPT≡NOPQU≡NOQR≡NPQS≡MOPQT≡MOPU≡OPRS≡MNPRT≡MNPQRUMN,MP,NQ,NP,NQ,MNPQ2III10−5MN≡Q,MQ≡NNQ≡M≡OR≡PSMO≡R,MR≡QMP≡S,MS≡PNO≡PT≡QR,NP≡OT≡QSOP≡NT≡RS,OQ≡NRPS≡OR,MO≡PUMP≡OUOP≡MU≡RSPR≡OSMN,MP,NQ,NP,MQ,MN,MNPQ32The following main and interac- tions effects are clearT,U,MT,QT,RT,ST,NU,QU,RU,SU,TU,MV,OV,PV,QV,RV,SV,TV,UVNote. The rest of the designs are illustrated in Table 7 above.


## 3.4.2. Alias Structures

Considering the25−0 design in Table 7, which have no treatment defining relation, hence it does not have aliasing structure.


## 3.4.3. Determing the Confounding of Effects

Consider the 32-run of25−0 design in eight blocks in Table 7, the generalized block-defining contrast subgroup is formed by taking all 2-way products as follows:(24)MOQNOQ≡MNO2Q2≡MN,MOQMNOP≡M2NO2PQ≡NPQ,NOQMNOP≡MN2O2PQ≡MPQ.Considering only the block-defining generators, it might be concluded that the block effectsb1, b2, and b3 are only confounded with MOQ, NOQ, MNOP, MN, NPQ, and MPQ, but OP is also confounded in the block effects. Therefore, b1, b2, and b3 are confounded with six effects: MOQ, NOQ, MNOP, MN, OP, NPQ, and MPQ.


## 3.4.4. Clear Effects

Hence from the above illustrations, the remaining 29 (= 32 − 3) degrees of freedom that do not involve the block effects, all the 5 main effects and two factor effects are clear exceptMN and OP which are confounded in the block effects. Hence, M,N,O,P,Q,MO,NO,MP,NP,MQ,NQ, and OQ are clear effects.


## 3.4.5. Blocked Fractional Factorial Design for 8 Choice Sets of Size 4

To obtain 8 choice sets of size four we select three effects to be confounded with the block. Suppose we choosey1y3y5, y2y3y5, and y1y2y3y4 to be confounded with the blocks. These effects will have three block-defining contrasts, that is b1 = y1y3y5, b2 = y2y3y5, and b3 = y1y2y3y4All the treatment combinations are evaluated usingb1, b2, and b3. λ1 = λ2 = λ3 = λ4 = λ5= 1(I):b1 = 1(0) + 0(0) + 1(0) + 0(0) + 1(0) = 0 (mod 2)b2 = 0(0) + 1(0) + 1(0) + 0(0) + 1(0) = 0 (mod 2)b3 = 1(0) + 1(0) + 1(0) + 1(0) + 0(0) = 0 (mod 2)a:b1 = 1(1) + 0(0) + 1(0) + 0(0) + 1(0) = 1 (mod 2)b2 = 0(1) + 1(0) + 1(0) + 0(0) + 1(0) = 0 (mod 2)b3 = 1(1) + 1(0) + 1(0) + 1(0) + 0(0) = 1 (mod 2).Similarly, we can obtainb1 (mod 2), b2 (mod 2), and b3 (mod 2) values for other treatments. Hence, from the above values, all treatment combinations with level 0 in b1, b2, and b3 form block 1 which is an inherent choice set one and also all treatment combinations with level 1 in b1, level 0 in b2, and level 1 in b3 form block 2, which also form choice set two. Also, all treatment combinations with level 0 in b1, level 1 in b2, and level 1 in b3 form block 3, which is an inherent choice set three, all treatment combinations with level 1 in b1, level 1 in b2, and level 0 in b3 form block 4, which is an inherent choice set four, all treatment combinations with level 1 in b1, level 1 in b2, and level 1 in b3 form block 5, which is an inherent choice set five, all treatment combinations with level 0 in b1, level 1 in b2, and level 0 in b3 form block 6, which is an inherent choice set six, all treatment combinations with level 1 in b1, level 0 in b2, and level 0 in b3 form block 7, which is an inherent choice set seven and finally, all treatment combinations with level 0 in b1, level 0 in b2, and level 1 in b3 form block 8, which is an inherent choice set eight.Therefore, treatment combinationsI,abcd,abe, and cde form choice set 1; a,bcd,be, and ade form choice set 2; b,acd,ae, and bcde form choice set 3; ab,cd,e, and abcde form choice set 4; c,abd,abce, and de form choice set 5; ac,bd,bce,ade form choice set 6; bc,ad,ace, and bde form choice set 7, and lastly, abc,d,ce, and abde form choice set 8. Hence, a choice set of 8 and size 4 is formed.


## 3.4.6. Design Efficiency

The result can be seen in Table8, which shows 62.5–76.9 percent A-efficiency for all the designs.Table 8
Design efficiency design and block generators form = 5, 6, 7, 8, 9, and 10 with 8 choice sets of size 4.
DesignsA-efficiency (%)2V5−0−362.52VI6−1−366.72IV7−2−370.02IV8−3−372.02IV9−4−375.02III10−5−376.9


## 3.5. Application

The research makes use of service quality attributes and levels identified by [10] in their paper a discrete choice modeling of service quality in public transport. The paper used five symmetric attributes and designed twenty-eight choice sets from these attributes and levels. This research also used the same attributes and levels to illustrate an application of the proposed method. Table 9 shows the attributes and their levels.Table 9
Attributes and their levels included for the SCE.
AttributesAttribute levelsy1 = Walking distance to bus stopWithin 10 min (0)More than 10 min (1)y2 = Attitude of driver and mateVery polite (0)Very impolite (1)y3 = Transport fareNormal fare (0)15% more than normal fare (1)y4 = Bus stop facilityShelter, lighting, and seat available (0)No shelter, lighting, and seat (1)y5 = ReliabilityOn time (0)late (1)


### 3.5.1. Stated Choice Experiment with Two-Level Attributes

Considering the five attributes with the two levels above, which requires25 = 32 treatment combinations of a complete factorial design. This can be precluded to run; therefore, we run one-half fraction of the complete factorial design. This research uses a 2v5−1−2 block fractional factorial design to construct 4 choices set of size 4. The generators, alias structure, confounding effects, and clear effects of the design are clearly illustrated and derived in Section 3.2.Lety1 = walking distance to bus stop, y2 = attitude of driver and mate, y3 = transport fare, y4 = bus stop facility, and y5 = reliability, with their levels coded as 0 and 1 as shown in Table 9. Hence, as shown in Section 3.2.5, all treatment combinations with level 0 in b1 and b2 form block 1 which is an inherent choice set one and also all treatment combinations with level 1 in b1 and level 1 in b2 form block 2, which also form choice set two. Also, all treatment combinations with level 1 in b1 and level 0 in b2 form block 3, which is an inherent choice set three, and finally, all treatment combinations with level 0 in b1 and level 1 in b2 form block 4, which also form choice set four.Therefore, from Table10, treatment combinations I,abc,d, and abcd form choice set 1; a,bc,ad, and bcd form choice set 2; b,ac,bd, and acd form choice set 3; ab,c,abd, and cd form choice set 4. Hence, a choice set of 4 and size 4 is formed. Table 11 shows choice set 1, and the remaining three choice sets are design similarly.Table 10
Design matrix for2V5−1−2.
y1y2y3y4y5 = y1y2y3y4b1 = y1y2b2 = y1y3I0000000a1000111b0100110ab1100001c0010101ac1010010bc0110011abc1110100d0001100ad1001011bd0101010abd1101101cd0011001acd1011110bcd0111111abcd1111000Table 11
A choice set one for2V5−1−2 design.
AttributesServiceAServiceBServiceCServiceDWalking distance to bus stopWithin 10 minsMore than 10 minsWithin 10 minsMore than 10 minsAttitude of driver and mateVery politeVery impoliteVery politeVery impoliteTransport fareNormal fare15% more than normal fareNormal fare15% more than normal fareBus stop facilityShelter, lighting and seat availableShelter, lighting and seat availableNo shelter, lighting and seat availableNo shelter, lighting and seat availableReliabilityOn timeLateLateOn timeWhich of these options do you prefer?( )( )( )( )


### 3.5.2. Model Specification

The stated choice data are analysed based on the random utility model [11]. Stated choice experiment modeling is rooted in random utility theory (RUT). This supposes, given alternatives to select from, a respondent n (n=1…N) will select the one alternative that produces the higher utility amidst the choice group (i = 1, 2, 3, …) at the instant of choice. The utility of the respondent is defined by explainable component specified as a function of the attributes of the alternatives and characteristics of the respondent and an unexplainable component (random variation) εni.(25)Uni=Vni+εni,where Uni represents the utility of respondent, and Vni is the explainable component specified as a function of the attributes of the alternatives and characteristics of the respondent and an unexplainable component (random variation) εni following an independent identically distributed extreme value type 1 distribution. In equation (25), the probability of a respondent selecting a specified service type is modeled. The chance of a selecting a service type is resolute by the indirect utility function for the respondent n from selection i in choice set c, assuming this is linear and additive and of the following form:(26)vnic=xnicβ+εnic,where vnic stands for utility derived from a selection, xnicβ is the utility component, and εnic is the random component. The vector xnic is specified as follows, where β1−5 stand for design attributes of the choice experiment and β0 the constant.(27)xnicβj=β0+β1walkingdistancetobusstationi+β2attitudeofdriverandmatei+β3transportfaresi+β4Busstopfacilityi+β5reliabilityi.


## 3.5.1. Stated Choice Experiment with Two-Level Attributes

Considering the five attributes with the two levels above, which requires25 = 32 treatment combinations of a complete factorial design. This can be precluded to run; therefore, we run one-half fraction of the complete factorial design. This research uses a 2v5−1−2 block fractional factorial design to construct 4 choices set of size 4. The generators, alias structure, confounding effects, and clear effects of the design are clearly illustrated and derived in Section 3.2.Lety1 = walking distance to bus stop, y2 = attitude of driver and mate, y3 = transport fare, y4 = bus stop facility, and y5 = reliability, with their levels coded as 0 and 1 as shown in Table 9. Hence, as shown in Section 3.2.5, all treatment combinations with level 0 in b1 and b2 form block 1 which is an inherent choice set one and also all treatment combinations with level 1 in b1 and level 1 in b2 form block 2, which also form choice set two. Also, all treatment combinations with level 1 in b1 and level 0 in b2 form block 3, which is an inherent choice set three, and finally, all treatment combinations with level 0 in b1 and level 1 in b2 form block 4, which also form choice set four.Therefore, from Table10, treatment combinations I,abc,d, and abcd form choice set 1; a,bc,ad, and bcd form choice set 2; b,ac,bd, and acd form choice set 3; ab,c,abd, and cd form choice set 4. Hence, a choice set of 4 and size 4 is formed. Table 11 shows choice set 1, and the remaining three choice sets are design similarly.Table 10
Design matrix for2V5−1−2.
y1y2y3y4y5 = y1y2y3y4b1 = y1y2b2 = y1y3I0000000a1000111b0100110ab1100001c0010101ac1010010bc0110011abc1110100d0001100ad1001011bd0101010abd1101101cd0011001acd1011110bcd0111111abcd1111000Table 11
A choice set one for2V5−1−2 design.
AttributesServiceAServiceBServiceCServiceDWalking distance to bus stopWithin 10 minsMore than 10 minsWithin 10 minsMore than 10 minsAttitude of driver and mateVery politeVery impoliteVery politeVery impoliteTransport fareNormal fare15% more than normal fareNormal fare15% more than normal fareBus stop facilityShelter, lighting and seat availableShelter, lighting and seat availableNo shelter, lighting and seat availableNo shelter, lighting and seat availableReliabilityOn timeLateLateOn timeWhich of these options do you prefer?( )( )( )( )


## 3.5.2. Model Specification

The stated choice data are analysed based on the random utility model [11]. Stated choice experiment modeling is rooted in random utility theory (RUT). This supposes, given alternatives to select from, a respondent n (n=1…N) will select the one alternative that produces the higher utility amidst the choice group (i = 1, 2, 3, …) at the instant of choice. The utility of the respondent is defined by explainable component specified as a function of the attributes of the alternatives and characteristics of the respondent and an unexplainable component (random variation) εni.(25)Uni=Vni+εni,where Uni represents the utility of respondent, and Vni is the explainable component specified as a function of the attributes of the alternatives and characteristics of the respondent and an unexplainable component (random variation) εni following an independent identically distributed extreme value type 1 distribution. In equation (25), the probability of a respondent selecting a specified service type is modeled. The chance of a selecting a service type is resolute by the indirect utility function for the respondent n from selection i in choice set c, assuming this is linear and additive and of the following form:(26)vnic=xnicβ+εnic,where vnic stands for utility derived from a selection, xnicβ is the utility component, and εnic is the random component. The vector xnic is specified as follows, where β1−5 stand for design attributes of the choice experiment and β0 the constant.(27)xnicβj=β0+β1walkingdistancetobusstationi+β2attitudeofdriverandmatei+β3transportfaresi+β4Busstopfacilityi+β5reliabilityi.


## 4. Conclusions

This paper proposed design and block generators for stated choice set of size 2 and 4 with resolution three, four, and five. This method tries to eliminate the influence of extraneous factors in an experiment. This was shown from the derived alias structure and confounding effects as well as the clear effects. The paper also used the2V5−1−2 design to demonstrate a practical application of the proposed method by designing 4 choice sets of sizes 4 using service quality in public transport. These attributes and levels were also used by [10] to design 28 choice sets of size 2. Hence, this method not only tries to eliminate the influence of extraneous factors but also reduces the number of choice sets presented to respondents, thereby reducing the work load on respondents. The efficiencies of the proposed designs computed lies between 60%–81.8%. These proposed designs have been demonstrated and are applicable to data sets and can easily be replicated by different researchers.



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