Redesigning product lines in a period of economic crisis: a hybrid simulated annealing algorithm with crossover

Abstract

The optimal product line design is an NP-hard optimization problem in marketing that involves a number of decisions, such as product line length and configuration. Simulated annealing constitutes the best performing approach so far, but with extremely large running times. In the current study simulated annealing is hybridized with an evolutionary algorithm to improve its search efficiency and alleviate its performance dependence on the selection of the parameters related to its cooling schedule. The presented approach outperforms genetic algorithms and classic simulated annealing, through the use of crossover as a neighborhood operator, along with the restricted tournament selection as the replacement strategy of the evolutionary algorithm’s population. Moreover, the paper describes the way that the proposed hybrid metaheuristic can be used for redesigning a firm’s product line. The issue of redesigning product lines becomes even more important in periods of economic crisis, as firms must adapt their offerings to new evolving patterns of consumer buying behavior and reduced levels of consumer’s purchasing power. The applicability of the proposed approach is illustrated through the case of the 2008 automotive industry crisis, by showing how the North American car manufacturers could have redesigned their lines on time, based on the configuration of the competitive products in the market as well as the new customer preferences emerged during the economic recession.

Appendices

Appendix 1: The optimal product line design problem

In the optimal product line design problem every product is represented as a bundle of features (attributes), with each attribute taking a number of specific levels from a predefined range. Using market research and conjoint analysis, consumer preferences can be measured. Through the evaluation of a small group of hypothetical product profiles, customers implicitly reveal their interests for each of the various levels of the different attributes. For example, a personal computer consists of monitor, processor, hard disc and memory (considering these as attributes), which may take the levels 17”, 21” or 23”, dual-core or quad-core, 500GB, 750GB, or 1T, and 4GB or 8GB respectively. Levels of attributes are being chosen by customers according to their needs and preferences. A computer engineer, for instance, probably prefers a fast processor, whereas an architect may prefer a large monitor. With the use of conjoint analysis a part-worth matrix is constructed for every consumer that contains the value associated with each level of each attribute. The sum of the corresponding partworths provides the utility of a product (profile). Using a choice model, product utilities for each customer are converted to choice probabilities for each product, the aggregation of which results in simulated (hypothetical) market shares.

For example, consider a firm that operates in the laptop industry and market research has shown that consumers consider four attributes (memory capacity, graphics card, screen size, hard disk capacity) as important when they decide which laptop to buy. All attributes take discrete values ranging from two to three different levels. Conjoint analysis involves the creation of a limited number of hypothetical laptop profiles (different combinations of attribute levels), which a sample of respondents rate or rank from the most to the least preferred. The method takes as input the product profiles evaluations and estimates for each respondent a partworth for each attribute level. An illustrative partworth matrix for two customers is provided in Table 3.

Table 3 Partworth matrix for laptop computers

It is assumed that the current laptop market consists of three products with the following configurations:

• Laptop $\mathrm{A}$: 4GB, GC2, 17.3, 500GB

• Laptop $\mathrm{B}$: 8GB, GC3, 15.6, 1TB

• Laptop C: 6GB, GC1, 15.6, 500GB

The total utility value $U_{ij}$ that customer i assigns to product j can be calculated as a simple weighted average of the corresponding partworths:

$$\begin{aligned} U_{ij}=\sum \nolimits _k w_{ik} \end{aligned}$$

where k the number of attributes that comprise the product.

According to this, the following product utility matrix arises (Table 4).

Table 4 Utility matrix for two customers

The next step is the application of a choice model. With the use of the probabilistic BTL choice model $\Big (P_{ij} =\frac{U_{ij}}{\sum \limits _{k\in C} {U_{ik}}}\Big )$ product choice probabilities are calculated for each customer $i\,(P_{iA},\, P_{iB},\, P_{iC})$. The aggregation of choice probabilities for each product j across the entire customer sample $(\sum _{i=1}^n {P_{ij}})$ provides the following simulated market shares: $MS_{A}=31.75\,\%,\,MS_{B}=32.5\,\%,\,MS_{C}=35.75\,\%$. In total, 36 different product profiles can be created (3x3x2x2 combinations of attribute levels). For a single product with the maximum market share, all the 36 profiles can be evaluated. For a line of two different products the entire solution space can still be enumerated since the number of different solutions is $36^{2}-36=1260$ (the 36 solutions with two identical products in the line are not considered).

However, in case of a product of 8 attributes with 5 levels each, the number of possible solutions is 390625, whereas for designing a line of three products the number of candidate solutions is more than $10^{16}$. As the problem belongs to the class of NP-hard, the provision of a good near-optimal solution requires the application of an optimization algorithm.

Appendix 2: Simulated annealing

Simulated annealing was first applied as an optimization method to the Travelling Salesman Problem (TSP) allowing uphill moves that enable the escaping from local optima. Its name originates from its analogy to the process of physical annealing with solids, in which a crystalline solid is heated and then allowed to cool very slowly until it achieves its most regular possible crystal lattice configuration (i.e. its minimum lattice energy state), and thus is free of crystal defects (Henderson et al. 2003). Kirkpatrick et al. (1983) formalized these ideas by mapping the elements of the physical process onto the elements of a combinatorial optimization problem as illustrated in Table 5 (Dowsland and Thompson 2013).

Table 5 Mapping of the physical process onto a combinatorial optimization problem

The SA algorithm simulates the changes that occur at the energy state of a system which is cooled until equilibrium is reached. In optimization problems this is represented by moving into the feasible search space and the algorithm stops when a feasible solution is found.

An important stage of SA is the choice of the function of the temperature reduction. This function describes the cooling program and determines when and how often temperature should be reduced. This procedure enables us to accept or not a solution and depicts the algorithm’s ability to find an optimal solution.

For the algorithm to converge, a number of termination criteria can be used such as:

• Temperature has reached a predefined sufficiently low level.

• A number of iterations with temperature reduction and no solution acceptance has been completed.

• The ratio of accepted moves has fallen below an acceptable value

• A maximum number of iterations have been reached.

A pseuso-code of the SA algorithm for minimization problems follows below: