Kraus and Yano (2003) established the share-of-surplus product line optimisation model and developed a heuristic procedure for this nonlinear mixed-integer optimisation model. In their model, price of a product is defined as a continuous decision variable. However, because product line optimisation is a planning process in the early stage of product development, pricing decisions usually are not very precise. In this research, a nonlinear integer programming share-of-surplus product line optimization model that allows the selection of candidate price levels for products is established. The model is further transformed into an equivalent linear mixed-integer optimisation model by applying linearisation techniques. Experimental results in different market scenarios show that the computation time of the transformed model is much less than that of the original model.
Today, many firms adopt the strategy of product line optimisation to satisfy diverse customer requirements and gain competitive advantages. There are already many product line optimisation models proposed by scholars, for example, buyers’ welfare [
Kraus and Yano [
In the Kraus and Yano [
In this research, the Kraus and Yano [
The rest of this paper is organised as follows: in Section
The optimisation problem is described as follows. A firm is going to develop a product line. There are
Assumptions of the optimisation problem include [
Let
The objective function of the model is to maximise the total profit (revenue minus cost) of the product line in all segments. The expression
Consider the following linear mixed-integer programming model (Model B):
Model B has the same optimisation result as that of Model A.
Proof of Theorem
Compared with Model A, Model B is a mixed-linear model and thus can be solved by many well-developed algorithms integrating mature linear programming methods (e.g., simplex, barrier). Most of the commercial optimisation software packages, such as IBM ILOG and LINGO, provide modules for solving this type of problem.
The cost of the transformation lies in the extra intermediate variables and constraints appended to the model. Model B has
All the experiments in this section were run on a personal computer (4 GB RAM, 3.30 GHz CPU, Windows 7). The IBM ILOG software package is version 12.4. The LINGO software package is version 11.0. All the generated cases, modelling files, and computational results can be found at:
Three types of product line design problems, that is, “random,” “rich-poor,” and “quality”, applied in Kraus and Yano [
For the first scenario “random,” as the name implies, all the parameters of the model, including
For the second scenario “rich-poor,” it shows the situation where there are a few “rich” consumers who prefer high-price products and a lot of “poor” consumers who prefer low-price products. The formula
For the third scenario “quality,” it assumes that the cost is the most important influential factor for product quality and higher costs ensure better quality. According to the formula
Table
Parameter settings for case generation.
Type | Parameter generation |
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Random |
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Rich-poor |
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Quality |
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Note:
Because all parameters of the cases are randomly generated, for a specific type of product line design problem at given size of market segment (
Average computation time (second) of Models A and B.
Random | Rich-poor | Quality | ||||
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Model A | Model B | Model A | Model B | Model A | Model B | |
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4.51 | 2.96 | 5.09 | 2.87 | 9.50 | 3.23 |
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5.19 | 3.19 | 5.91 | 3.09 | 13.20 | 3.26 |
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6.68 | 3.21 | 7.39 | 3.28 | 15.34 | 3.42 |
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7.87 | 3.64 | 8.60 | 3.66 | 17.62 | 3.47 |
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8.60 | 3.73 | 9.65 | 3.79 | 19.84 | 3.78 |
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10.31 | 3.92 | 14.05 | 4.34 | 24.45 | 3.81 |
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4.51 | 2.96 | 5.09 | 2.87 | 9.50 | 3.23 |
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33.78 | 4.25 | 49.93 | 7.26 | 63.47 | 3.74 |
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173.91 | 9.72 | 282.09 | 16.22 | 340.62 | 7.37 |
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838.31 | 19.06 | N/A | 50.32 | N/A | 14.97 |
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N/A | 77.33 | N/A | 215.81 | N/A | 35.42 |
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N/A | 291.77 | N/A | 951.90 | N/A | 132 |
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4.51 | 2.96 | 5.09 | 2.87 | 9.50 | 3.23 |
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13.86 | 3.38 | 17.37 | 3.83 | 28.52 | 4.43 |
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35.14 | 5.35 | 43.29 | 7.10 | 66.62 | 5.62 |
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81.09 | 6.97 | 76.66 | 9.39 | 129.07 | 7.13 |
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146.08 | 13.86 | 125.8 | 11.64 | 226.33 | 9.73 |
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244.59 | 20.61 | 189.3 | 16.69 | 354.34 | 14.85 |
In the first experiment, Models A and B under the three scenarios were solved when the size of market segment (
Comparison of Models A and B when
Random
Rich-poor
Quality
In the second experiment, the average computation times of Models A and B under the three scenarios were compared when the number of products (
Comparison of Models A and B when
Random
Rich-poor
quality
In the third experiment, Models A and B under the three scenarios are solved when the number of price levels
Comparison of Models A and B when
Random
Rich-poor
Quality
The value of
The above-mentioned three experiments were also performed by LINGO. The comparison between computation time results of Models A and B has similar trends and conclusions as those by IBM ILOG, except that the computation time of LINGO is much longer than IBM ILOG. The computation results using LINGO can be found at the URL mentioned in the first paragraph of this section.
In this study, the share-of-surplus product line optimisation model by Kraus and Yano [
The transformed linear model adds a number of extra auxiliary continuous variables and linear constraints to the original model. However, the numeric experiments show that the computation time of the transformed model is much less than that of the original one in different market scenarios, and the efficiency of the transformed model increases with the scale of the problem.
It must be noted that for large-scale problems, the computation time of the transformed linear model is still too long due to the integer variables, and a metaheuristic algorithm such as evolutionary computation is probably a better choice. However, exact approaches can achieve the global optimal solutions and thus are attractive to small-scale practical problems (e.g., service products, some products of SME). In addition, global optimal solutions can also be used to evaluate the performance of heuristic or metaheuristic algorithms because they can provide a base line for comparison.
Let
In Model B, there are three constraints, (
Similarly,
Note that
For Model B, if
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research is financially supported by the National Science Foundation of China (NSFC) (Projects nos. 71171039, 61273204, and 71021061) and the Fundamental Research Funds for Central Universities (Project no. N110204005). The work described in this paper was supported by a grant from The Hong Kong Polytechnic University (Project no. G-YK06). The authors would like to thank the two anonymous reviewers for their valuable comments and suggestions.