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Exact Logit-Based Product Design

İrem Akchen, Velibor V. Mišić

TL;DR

This work studies designing a single product under a logit-based share-of-choice objective, proving the problem is NP-hard and APX-hard in broad settings. It advances exact solvability by proposing three mixed-integer exponential-cone formulations (RA, P, P-RPT) that leverage modern solvers to obtain provable optimality on large synthetic and real conjoint data. The paper also develops extensions including profit objectives, product-line design, and robust optimization, and compares against several heuristics, showing substantial performance gains in practice. Empirically, the approach solves problems up to 70 attributes and 30 customer types to within small optimality gaps, with real-data experiments achieving provable optimality and outpacing heuristics, highlighting strong practical impact for data-driven product design under uncertainty.

Abstract

The share-of-choice product design (SOCPD) problem is to find the product, as defined by its attributes, that maximizes market share arising from a collection of customer types or segments. When customers follow a logit model of choice, the market share is given by a weighted sum of logistic probabilities, leading to the logit-based share-of-choice product design problem. In this paper, we develop a methodology for solving this problem to provable optimality. We first analyze the complexity of this problem, and show that this problem is theoretically intractable: it is NP-Hard to solve exactly, even when there are only two customer types, and it is furthermore NP-Hard to approximate to within a non-trivial factor. Motivated by the difficulty of this problem, we propose three different mixed-integer exponential cone programs of increasing strength for solving the problem exactly, which allow us to leverage modern integer conic program solvers such as Mosek. Using both synthetic problem instances and instances derived from real conjoint data sets, we show that our methodology can solve large instances to provable optimality or near optimality in operationally feasible time frames and yields solutions that generally achieve higher market share than previously proposed heuristics.

Exact Logit-Based Product Design

TL;DR

This work studies designing a single product under a logit-based share-of-choice objective, proving the problem is NP-hard and APX-hard in broad settings. It advances exact solvability by proposing three mixed-integer exponential-cone formulations (RA, P, P-RPT) that leverage modern solvers to obtain provable optimality on large synthetic and real conjoint data. The paper also develops extensions including profit objectives, product-line design, and robust optimization, and compares against several heuristics, showing substantial performance gains in practice. Empirically, the approach solves problems up to 70 attributes and 30 customer types to within small optimality gaps, with real-data experiments achieving provable optimality and outpacing heuristics, highlighting strong practical impact for data-driven product design under uncertainty.

Abstract

The share-of-choice product design (SOCPD) problem is to find the product, as defined by its attributes, that maximizes market share arising from a collection of customer types or segments. When customers follow a logit model of choice, the market share is given by a weighted sum of logistic probabilities, leading to the logit-based share-of-choice product design problem. In this paper, we develop a methodology for solving this problem to provable optimality. We first analyze the complexity of this problem, and show that this problem is theoretically intractable: it is NP-Hard to solve exactly, even when there are only two customer types, and it is furthermore NP-Hard to approximate to within a non-trivial factor. Motivated by the difficulty of this problem, we propose three different mixed-integer exponential cone programs of increasing strength for solving the problem exactly, which allow us to leverage modern integer conic program solvers such as Mosek. Using both synthetic problem instances and instances derived from real conjoint data sets, we show that our methodology can solve large instances to provable optimality or near optimality in operationally feasible time frames and yields solutions that generally achieve higher market share than previously proposed heuristics.

Paper Structure

This paper contains 58 sections, 15 theorems, 200 equations, 4 figures, 24 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Single product design:
  4. Product line design/assortment optimization:
  5. Representative agent model:
  6. Optimization literature:
  7. Model
  8. Problem definition
  9. Complexity
  10. Mixed-Integer Convex Programming Formulations
  11. Formulation RA
  12. Formulation P
  13. Formulation P-RPT
  14. Extensions and other formulations
  15. Numerical Experiments
  16. ...and 43 more sections

Key Result

Theorem 1

The logit-based SOCPD problem prob:SOCPD_abstract with $\mathcal{A} = \{0,1\}^n$ and with $K = 2$ customer types is NP-Hard.

Figures (4)

  • Figure EC.1: Visualization of rotated second order cone constraint for $n = 5$ example in constraints \ref{['eq:SOC_example_a1']} - \ref{['eq:SOC_example_xk0_r_rotatedcone']}.
  • Figure EC.2: Plots of partworth distributions for LC-MNL models estimated from real data sets in Section \ref{['subsec:numerical_experiments_real']}.
  • Figure EC.3: Plots of partworth distributions for HB models estimated from real data sets in Section \ref{['subsec:numerical_experiments_real']}.
  • Figure EC.4: Plot of the approximation factor $\Gamma$ as a function of the ratio $U/L$, for different values of $K$. Note that $\boldsymbol \lambda$ is assumed to be the uniform distribution, i.e., $\boldsymbol \lambda = (1/K, \dots, 1/K)$.

Theorems & Definitions (15)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition EC.1
  • Theorem EC.1
  • Theorem EC.2
  • Lemma EC.1
  • ...and 5 more