Dominance relationship analysis with budget constraints

Abstract

Creating a new product that dominates all its competitors is one of the main objectives in marketing. Nevertheless, this might not be feasible since in practice the development process is confined by some constraints, e.g., limited funding or low target selling price. We model these constraints by a constraint function, which determines the feasible characteristics of a new product. Given such a budget, our task is to decide the best possible features of the new product that maximize its profitability. In general, a product is marketable if it dominates a large set of existing products, while it is not dominated by many. Based on this, we define dominance relationship analysis and use it to measure the profitability of the new product. The decision problem is then modeled as a budget constrained optimization query (BOQ). Computing BOQ is challenging due to the exponential increase in the search space with dimensionality. We propose a divide-and-conquer based framework, which outperforms a baseline approach in terms of not only execution time but also space complexity. Based on the proposed framework, we further study an approximation solution, which provides a good trade-off between computation cost and quality of result.

Appendices

Appendix A: Compact MBR computation

Given an MBR $m$, a budget plane $\mathcal {BP}$, and an orthogonal hyperplane, we can partition $m$ into (at most) two compact MBRs as shown in Fig. 8a. In this section, we show how we can compute the boundaries of these new MBRs in $O(d^2)$ time. For the sake of presentation, we assume that the hyperplane splits $m$ into exactly two MBRs $m_1$ and $m_2$. Assume that $m$ is split by hyperplane $x[j]=v$ (i.e., parallel to dimension $j$ at value $v$). Let $m^l[i]$ ($m^u[i]$) be the lower (upper) bound of $m$ in dimension $i$. Then, two loose MBRs, whose union equals $m$, can be defined by the following equations.

$$\begin{aligned}&m^l_1:=m^l~~~~ m^u_1[i]:={\left\{ \begin{array}{ll} m^u[i] &{} \text { if } i\ne j \\ v &{} \text { if } i=j \end{array}\right. }\end{aligned}$$

(6)

$$\begin{aligned}&m^l_2[i]:={\left\{ \begin{array}{ll} m^l[i] &{} \text { if } i\ne j \\ v &{} \text { if } i=j \end{array}\right. }~~~ m^u_2:=m^u \end{aligned}$$

(7)

Note that the values of $m^u_1$ and $m^l_2$ are the tightest values already; in addition, $m^l_1[j]$ and $m^u_2[j]$ are also tightest. However, we can tighten the values ($m^l_1$ and $m^u_2$) in other dimensions, so that $m_1$ and $m_2$ tightly enclose ${\mathcal {BP}}$ using the following method. To find the maximum bound within an MBR in dimension $i$, we need to set the minimum values in all other dimensions and solve $C(x)=B$. For example, $m^l_1[i]$ can be computed by solving the equation $C(x)=B$, by setting $x[j]=m^u_1[j],\forall {j\ne i}$.

For instance, in Fig. 19, assume that we want to partition $m$ using hyperplane y$=0.5$. Initially, $m$ is split into two MBRs $m_1=\{m^u_1,m^l_1\}=\{[0,0.5],[1,1]\}$ and $m_2=\{m^u_2,m^l_2\}=\{[0,0],[1,0.5]\}$. Then, if $C=\frac{1}{xy}$ and $B=0.5$, we tighten $m^l_1[0]$ by solving $m^l_1[0]\cdot m^u_1[1] = B$, which gives $m^l_1[0]=0.5/0.5=0.25$. Therefore, the tightmost bound of $m_1$ becomes $\{[0,0.5],[0.25,1]\}$. In general, $d-1$ boundaries can be tightened for each new MBR and each tightening requires $O(d)$ computations, so the overall cost for tightening an MBR is $O(d^2)$.

Fig. 19

Compact MBR Computation

Appendix B: Solution of piecewise constraint functions

Even though our work does not address all non-monotonic constraint functions, our proposed techniques are able to compute GenBOQ if the constraint function $C(x)$ is a piecewise function, which is composed of a set of finite monotonic functions. For instance,

$$\begin{aligned} C(x) = \left\{ \begin{array}{l l} 1/x &{} \quad 0.5<x\le 1\\ 1-1/x &{} \quad 0\le x \le 0.5\\ \end{array} \right. \end{aligned}$$

(8)

The complete piecewise function may be non-monotonic. Figure 20 illustrates a non-monotonic piecewise function that composes of three finite monotonic functions. In case of such a function, we can partition the domain area into 9 subareas. To compute the result of GenBOQ, we first identify the relevant subareas of each finite monotonic function. For instance, the relevant areas of the red function are 2, 3, 4, 5, 6, 7, and 8 since there is no profitable region of the red function dominating or dominated by areas 1 and 9. Accordingly, we apply our algorithms to compute the GenBOQ result of the red constraint function using the data in the relevant areas. The final result can be straightforwardly produced by combining the result of each piecewise monotonic function.

Fig. 20

An example of piecewise functions