An improved probability-based discrete particle swarm optimization algorithm for solving the product portfolio planning problem

Abstract

Product Portfolio Planning (PPP) is one of the most critical decisions for companies to gain an edge in the competitive market. It seeks for the optimal combination of products and attribute levels offered for customers in the target market, which is a NP-hard combinatorial optimization problem. In this paper, a Probability-based Discrete Particle Swarm Optimization (PDPSO) algorithm is proposed to solve the PPP problem. In PDPSO, the particle is encoded as discrete values, which can be straightforwardly used to represent the product portfolio with discrete attributes. PDPSO adopts a probability-based mechanism to update particles. Specifically, a probability vector is used to decide the probability of three search behaviors, i.e., learning from the personal best position, global best position, or random search. In experiments, the search performance of PDPSO has been compared with that of a Genetic Algorithm (GA) and a Simulated Annealing (SA) algorithm on generated PPP problem cases with different sizes. The results indicate that PDPSO obtains significantly better optimization results than GA and SA in most cases and obtains desirable/near-optimal solutions on various PPP problem cases. A case study of notebook computer portfolio planning is also presented to illustrate the efficiency and effectiveness of PDPSO.

Appendix A: Tuning the parameters for PDPSO

In this section, we verify different combinations of parameters for the proposed PDPSO on the PPP problem with different sizes and obtain a best compromise setting for the PDPSO parameters. The obtained PDPSO parameters have been used in the case study in Sect. 5 and further search performance analysis in Sect. 6. The tuning experiments are run on a PC with a 3.4 gigahertz CPU and a 16 gigabytes RAM with Matlab R2021a.

The parameters used in PDPSO that may affect its search performance are swarm size $N$, maximum number of iterations $T$, and parameter $p^{r}$ that controls the search behavior. In the tuning experiments, we set two candidate settings for $N$ and $T$. The first setting is $N = 100$ and $T = 200$. The second setting is $N = 50$ and $T = 400$. Both settings yield the same $20,000$ function evaluations. The parameter $p^{r}$ that controls the random search probability during the iterations is suggested to be a small value. In the tuning experiments, five values of $p^{r}$ (i.e., 0.001, 0.005, 0.01, 0.05, and 0.1) are examined. Therefore, a total number of $2 \times 5$ = 10 parameter combinations are yielded for $N$, $T$, and $p^{r}$.

A strategy like that in Sect. 6.2 is used to generate the PPP problem cases for parameter tuning. Problem parameters to generate the PPP problem cases are set to be $J \in \left\{ {5,8} \right\}$, $\left( {K + 1} \right) \in \left\{ {6,9,12} \right\}$, $L_{k} \in \left\{ {6,9,12} \right\}$, and $I \in \left\{ {3,4,5} \right\}$, which are the same as that used in Sect. 6.2. This results in $2 \times 3 \times 3 \times 3 = 54$ problem parameter combinations (please refer to Table 7 for the detailed description of each problem parameter combination). For each problem parameter combination, one problem case (which is smaller than that used in Sect. 6.2 considering limited computational resources) is generated and a total number of 54 problem cases (ID:1-54) are generated. Finally, the 54 problem cases and the 10 parameter combinations yield a total number of 540 tuning experiments.

For each of the 540 tuning experiments, the PDPSO algorithm is repeated 30 times to comprehensively evaluate the performance of PDPSO. Therefore, each of the tuning experiments would have 30 sets of the experimental results. The expected shared surplus ${\text{E}}\left[ V \right]$ as shown in Eq. (1) is the objective to be maximized by PDPSO and thus it is a good measure to evaluate the performance of PDPSO. Moreover, whether the PDPSO algorithm can obtain results with small standard deviation during the 30 runs is also a key measure to evaluate PDPSO since the standard deviation reflects the robustness of the search algorithm. Therefore, considering both the mean and standard deviation of the expected shared surplus ${\text{E}}\left[ V \right]$ over 30 runs, the signal-to-noise (S/N) ratio is used as the performance measure in order to find a desirable parameter setting for PDPSO as suggested in Sadeghi et al. (2011). The ${\text{E}}[V$] is larger-the-better and thus the corresponding loss function is defined as

$$ S/N_{ratio} = - 10{\text{lg}}\left( {\frac{1}{{N_{r} }}\mathop \sum \limits_{r = 1}^{{N_{r} }} \left( {\frac{1}{{E\left[ V \right]_{r} }}} \right)^{2} } \right), $$

(7)

where $N_{r}$ is the number of experimental runs (30 in these tuning experiments) and $E\left[ V \right]_{r}$ is the expected shared surplus obtained from the $r$ th run.

The S/N ratios obtained by the 10 PDPSO parameter settings on the generated 54 PPP problem cases are shown in Table

Table 11 The S/N ratios on the 54 PPP problem cases obtained by different parameter settings

11. The parameter setting that obtains the best S/N ratio on each problem case is highlighted in bold, and the #best row of the table denotes the number of cases that a parameter setting obtains the best (highest) S/N ratio compared with other parameter settings. According to the table, it can be found that the parameter setting $\left\{ {N = 50,\,{ }T = 400,{ }\,p^{r} = 0.01} \right\}$ obtains the highest S/N ratios in 33 of the 54 cases, which means that this parameter setting performs the best in most of the cases. According to the results, problem size seems not to be a key factor that affects the performance of the parameter setting $\left\{ {N = 50,{ }\,\,T = 400,{ }\,p^{r} = 0.01} \right\}$ since this setting can obtain the best S/N ratio results on different problem sizes. It can be also found that the parameter $p^{r}$ should be carefully set since a too high or a too low $p^{r}$ value can significantly affect the results. If only the five parameter settings where $N = 100$, $T = 200$ and $p^{r} \in \left\{ {0.001,\,\,{ }0.005,{ }\,\,0.01,\,\,{ }0.05,{ }\,\,{ }0.1} \right\}$ are considered, it can be found that the parameter setting $N = 100$, $T = 200$, and $p^{r} = 0.01$ obtains the best S/N ratios in 32 of the 54 cases. This denotes that no matter the {$N = 50, T = 400\}$ or $\left\{ {N = 100, T = 200} \right\}$ is used, $p^{r} = 0.01$ is a desirable setting for PDPSO. Overall, according to the S/N ratio results in Table 11, a desirable parameter setting (i.e., $N = 50$, ${ }T = 400$, and ${ }p^{r} = 0.01$) is obtained for PDPSO. This parameter setting is used in the case study and the search performance analysis sections.