Interaction effects in pairwise ordering model

Abstract In an order-of-addition (OofA) experiment, the response is a function of the addition order of components. The key objective of the OofA experiments is to find the optimal order of addition. The most popularly used model for OofA experiments is perhaps the pairwise ordering (PWO) model, which assumes that the response can be fully accounted by the pairwise ordering of components. Recently, the PWO model has been extended by adding the interactions of PWO factors, to account for variations caused by the ordering of sets of three or more components, where the interaction term is defined by the multiplication of two PWO factors. This paper introduces a novel class of conditional PWO effect to study the interaction effect between PWO factors. The advantages of the proposed interaction terms are studied. Based on these conditional effects, a new model is proposed. The optimal order of addition can be straightforwardly obtained via the proposed model.


Introduction
In the order-of-addition (OofA) experiment, the response is a function of the order of which components are added. The goal of OofA experiments is to reveal how the order of the components affects the response, and eventually find the optimal order of addition to optimize the response.
The PWO model assumes that the response can be fully accounted by the pairwise ordering of components. In some cases, however, these pairwise ordering of components may not be sufficient, and the higher order effects need to be considered. Mee (2020) extended the PWO model to triplets model by including interactions of the PWO factors. The interaction terms there is defined by directly multiplying two PWO factors together. Under such a triplets model, the optimal order can only be found by evaluating the predicted responses for all m! orders.
Here, we introduce a new class of conditional effect to study the interaction effect between two PWO factors called the conditional PWO effect, which can capture the conditional effect of a PWO factor at a fixed level of another PWO factor. The theoretical dependencies among these effects are studied. Based on these conditional effects, a new model is proposed. An illustrative example and a simulation study are provided to show the analysis produce for the proposed model. With the proposed model, the optimal order can be found straightforwardly. Most importantly, the proposed interaction terms can capture the conditional effect of a PWO factor at a fixed level of another PWO factor, and thus can be appropriate tools to study the interaction effects between two PWO factors.
The rest of the paper is organized as follows. Section 2 introduces the new interaction effects and their properties. A model is proposed for OofA experiments by including these effects to the PWO model. An illustrative example and a simulation study based on job scheduling are provided in Sections 3 and 4, respectively. Concluding remarks are provided in Section 5.

Preliminary
The PWO factor between components i and j, denoted as I ij , is defined to be 1 if component i is added before component j; and À1 if component j is added before component i. The PWO model is defined to be where b ¼ ðb 0 , b 12 , :::, b ðmÀ1Þm Þ is the parameter vector and $ Nð0, r 2 Þ is random error.
Note that the interaction of two distinct PWO factors I ST and I PQ involve with four or less distinct components. Specifically, they involve with four components S, T, P and Q. Of course, they could involve with three distinct components when for example S ¼ P. Mee (2020) extended the PWO model to include special interactions that involve only three distinct components (I ij and I ik ). Their triplets model is In particular, under the condition i < j < k, I ij I ik À I ij I jk þ I ik I jk ¼ 1: Model [1] becomes In the triplets model, the interaction effect between two PWO factors is defined by elementwise multiplication of these two columns. In this paper, we study the interaction effects between two PWO factors by introducing a class of conditional PWO effects, which can capture the conditional effect of a PWO factor at a fixed level of another PWO factor.

Pwo model with interaction effects
Conditional main effect was proposed by Su and Wu (2017) to de-alias the entangled effects in two-level designs. In this paper, we apply its basic idea to the OofA experiments and define interaction effects between two PWO factors.
Definition 1. For an OofA experiment with m components, the conditional effect of PWO factor I PQ given I ST at level þ, denoted as I PQ jI STþ , is defined as The conditional effect I PQ jI STÀ can be defined in a similar manner. It is clear that I PQ jI STþ can capture the effect of I PQ with I ST at level þ; in other words, it measures the effect of the order of components P and Q on the response when component S is added before There are four interaction terms I PQ jI STþ , I PQ jI STÀ , I ST jI PQþ and I ST jI PQÀ , for I PQ and I ST . For runs with I ST at level þ, the entries of I PQ I ST are the same as the entries of I PQ ; whereas, for runs with I ST at level -, the entries of I PQ I ST have the opposite sign of the entries of I PQ . In this way, we have the following algebraic relationships.
This implies that I PQ jI STþ þ I PQ jI STÀ ¼ I PQ , I ST jI PQþ þ I ST jI PQÀ ¼ I ST and I PQ jI STþ À I PQ jI STÀ ¼ I ST jI PQþ À I ST jI PQÀ : Therefore, any three of I PQ jI STþ , I PQ jI STÀ , I ST jI PQþ and I ST jI PQÀ can be linearly represented by the remaining one. Without loss of generality, we select the term I PQ jI STþ where P S, and if P ¼ S, Q < T.
Consider interactions involved with three distinct components. That is, for any three distinct components i < j < k, we consider three interaction terms I ij jI ikþ , I ij jI jkþ and I ik jI jkþ : From Theorem 1, we have the following result.
2ðI ij jI ikþ À I ij jI jkþ þ I ik jI jkþ Þ À I ik ¼ 1, for any i < j < k: From Theorem 2, given I ik , any one of I ij jI ikþ , I ij jI jkþ and I ik jI jkþ can be linearly represented by the remaining two. Without loss of generality, we select the terms I ij jI ikþ and I ij jI jkþ : Next we propose the following PWO model with these proposed interaction effects (called the interaction PWO model): There are mðm À 1Þ=2 PWO effects, and mðm À 1Þ ðm À 2Þ=3 conditional effects. Compared to the triplets model in [2], the term I ij jI ikþ , instead of I ij I ik , is employed as the interaction effect between PWO factors I ij and I ik . Therein I ij jI ikþ captures the conditional effect of I ij with I ik at level þ.

An illustrative example
To illustrate the proposed model with a fair comparison, the OofA experiment with m ¼ 4 components in Mee (2020)  , where the standard deviation of each coefficient is listed below its estimated value. Its AIC and BIC are À1500.80 and À1491.37 respectively. A comparison with the conventional PWO and triplets model, is displayed in Table 1. It is clear that the proposed interaction model has a smaller AIC and BIC than PWO model. Compared to the triplets model, the proposed interaction model is comparable in terms of AIC and BIC criteria. The interactions in our model can capture the conditional effect of a PWO factor at a fixed level of another PWO factor, and thus are interpretable and meaningful. For example, b 13j14þ is significant, indicating that the order of 1 and 3 affects the response when 1 precedes 4. Using the proposed model, the optimal order can be obtained as follows. Table 2 shows all significant parameters b 13 , b 23 , b 13j14þ , b 14 , b 23j24þ , b 24 , sorted by descending absolute value of estimated effect. We next find the optimal order by accommodating these significant effects sequentially. That is, [1] b 13 is þ, thus 1 precedes 3; [2] b 23 is -, thus 3 precedes 2; [3] b 13j14þ is -, as I 13 has been set to be þ, so I 14 is -, thus 4 precedes 1. The optimal order is then 4 ! 1 ! 3 ! 2: Figure 1 displays the directed graph of these significant parameters, where i precedes j indicates a directional edge from node i to j. The triplets model can identify the same optimal order, but it was obtained by evaluating all m! ¼ 24 predicted responses. Note that, this is infeasible for large m (e.g., when m ¼ 10, m! ¼ 10! % 3:6 millions).

A simulation study based on job scheduling
Job scheduling plays a crucial decision-making role in manufacturing and service industries. In the current competitive environment, effective job scheduling has become a necessity in the marketplace. In general, the objective of a job scheduling problem is to find the optimal order of jobs, such that the penalty function (a.k.a. cost) is optimized. See, for example, Pinedo (2016) and Wei (2019).
Suppose there are m jobs to be processed sequentially. For any job order s ¼ s 1 Á Á Á s m , the completion time of job s i is c i ¼ P i j¼1 t j , where t j is the processing time of job j for j ¼ 1, :::, m: The quadratic penalty function is then defined as P ¼ P m i¼1 w i c 2 i , where w i is the weight of job i for i ¼ 1, :::, m: Consider a 6-job scheduling problem as an illustrative example. The processing time t i 's and weights w i 's  Table 2. The significant parameters.

Significant parameters
Estimator of the effects Arranged order b 13 15.00 1 precedes 3 b 23 -7.50 3 precedes 2 b 13j14þ -7.50 (1 precedes 4) and (3 precedes 1) b 14 -6.25 4 precedes 1 b 23j24þ 3.75 (2 precedes 4) and (2 precedes 3) b 24 3.13 2 precedes 4 Figure 1. Pairwise orders determined by the significant parameters in Table 2. are (randomly) generated from a Chi-square distribution with one degree of freedom (to ensure they are positive): t ¼ ð0:01, 0:13, 0:01, 0:63, 0:91, 0:16Þ and w ¼ ð1:05, 0:15, 0:04, 3:14, 2:69, 1:35Þ: Take n ¼ 60 as an example, we randomly generate 60 sequences. Thus, there are 60 order-sequences and their corresponding costs. Fit model (3) to the data and obtain the estimators of parameters. Next, we use the halfnormal probability plot to identify significant effects. From Figure 2, it is clearly seen that I 45 is the most significant. It is followed by b 15 , b 14 , b 56 , b 14j15þ , b 24 , b 45j56þ and b 25 : The signs of these estimators indicate the desirable order of the corresponding jobs, as summarized in the last column of Table 3. Figure 3 displays the directed graph of these significant parameters, where i precedes j indicates a directional edge from node i to j. Using a topological sorting algorithm (Knuth and Szwarcfiter,1974), all possible paths in Figure 3 can be obtained. Table 4 lists all these orders. Their corresponding costs were  then evaluated and displayed at the last column. Note that the orders in Table 4 are obtained based on the model with significant effects listed in Table 3. Thus, all those orders generated from the model are considered to be insignificant difference in costs. All these orders can be considered as the optimal orders. If only one order is to be recommended, the order 1 ! 6 ! 4 ! 5 ! 3 ! 2 is preferred. We also fit the PWO and triplets model to the same data, respectively. Under these two models, however, the optimal order can only be found by evaluating all possible 720 order-sequences. The optimal order identified by the PWO model is found to be 1 ! 6 ! 4 ! 5 ! 2 ! 3, with the cost 10.559; while the optimal order identified by the triplets model is 1 ! 6 ! 4 ! 5 ! 3 ! 2, with the cost 10.546. Note that both orders are listed in Table 4, i.e. they can be identified by our model, without evaluating all the possible 720 order-sequences.
In summary, the newly proposed model can be used to find the optimal order straightforwardly. It is particularly useful when the the number of components m is large (when evaluating all m! predicted responses is not feasible). Moreover, the interaction effect of our model captures the conditional effect of a PWO factor at a fixed level of another PWO factor, and thus can be appropriate tools to study the interaction effects between two PWO factors.

Conclusion
The order-of-addition (OofA) experiments have recently received a great deal of attention in many fields. The PWO model is popularly used for the analysis of OofA experiments. Mee (2020) extended the PWO model to the triplets model by adding the interactions of PWO factors, where the interaction is defined by multiplying two PWO factors directly. Under the triplets model, one needs to evaluate all m! predicted responses for finding the optimal order. This paper studies the interaction effect between two PWO factors by introducing conditional PWO effect, which can capture the conditional effect of a PWO factor at a fixed level of another PWO factor. Based on the proposed interactions, a PWO model with interaction effects is provided. Compared to the existing model, the new model enjoys two advantages: (i) the interactions terms can capture the conditional effect of a PWO factor at a fixed level of another PWO factor, and thus can be appropriate tools to study the interaction effects between two PWO factors; and (ii) the proposed model can be used to find the optimal order efficiently (without evaluating all m! orders).

About the authors
Dr. Wang is a Lecturer of Center for Applied Statistics and School of Statistics, Renmin University of China. Her email address is chunyanwang@ruc.edu.cn.
Dr. Lin is a Distinguished Professor and Head in the Department of Statistics at Purdue University. He is a fellow of ASQ (also fellows of IMS, RSS, ISI, and ASA). His email address is dkjlin@purdue.edu.

Disclosure statement
No potential conflict of interest was reported by the authors.  Table 3.