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Robust Discrete Pricing Optimization via Multiple-Choice Knapsack Reductions

Zi Yuan Eric Shao

Abstract

We study a discrete portfolio pricing problem that selects one price per product from a finite menu under margin and fairness constraints. To account for demand uncertainty, we incorporate a budgeted robust formulation that controls conservatism while remaining computationally tractable. By reducing the problem to a Multiple-Choice Knapsack Problem (MCKP), we identify structural properties of the LP relaxation, in particular upper-hull filtering and greedy filling over hull segments, that yield an exact solution method for the LP relaxation of the fixed-parameter subproblems. For the resulting fixed-parameter subproblems, we show that the integrality gap is bounded additively by a single-item hull jump, and that the corresponding relative gap decays as O(1/n) under standard boundedness and linear-growth assumptions. Numerical experiments on synthetic portfolios and a stylized retail case study with economically calibrated parameters are consistent with these bounds and indicate that robust margin protection can be achieved with less than 1 percent nominal revenue loss on the instances tested.

Robust Discrete Pricing Optimization via Multiple-Choice Knapsack Reductions

Abstract

We study a discrete portfolio pricing problem that selects one price per product from a finite menu under margin and fairness constraints. To account for demand uncertainty, we incorporate a budgeted robust formulation that controls conservatism while remaining computationally tractable. By reducing the problem to a Multiple-Choice Knapsack Problem (MCKP), we identify structural properties of the LP relaxation, in particular upper-hull filtering and greedy filling over hull segments, that yield an exact solution method for the LP relaxation of the fixed-parameter subproblems. For the resulting fixed-parameter subproblems, we show that the integrality gap is bounded additively by a single-item hull jump, and that the corresponding relative gap decays as O(1/n) under standard boundedness and linear-growth assumptions. Numerical experiments on synthetic portfolios and a stylized retail case study with economically calibrated parameters are consistent with these bounds and indicate that robust margin protection can be achieved with less than 1 percent nominal revenue loss on the instances tested.
Paper Structure (79 sections, 7 theorems, 73 equations, 4 figures, 1 table, 1 algorithm)

This paper contains 79 sections, 7 theorems, 73 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction and Reduction
  2. Contributions.
  3. Problem Formulation
  4. Reduction to MCKP
  5. Related Work
  6. MCKP foundations.
  7. Extensions and uncertainty.
  8. Gap addressed in this paper.
  9. Robust Pricing under Demand Uncertainty
  10. Uncertainty Model
  11. The Robust Counterpart
  12. Structural Analysis and Greedy Optimality
  13. Upper-hull geometry of the LP relaxation
  14. Robust Constraint Structure
  15. Box Uncertainty (Gamma=n): Separability
  16. ...and 64 more sections

Key Result

Theorem 1

Let $\Gamma \in \{0,1,\dots,n\}$ be an integer. Define where $|t|_{(1)}(\bm{x}) \ge \cdots \ge |t|_{(n)}(\bm{x})$ denotes the nonincreasing ordering of the numbers $\{|t_i(x_i)|\}_{i=1}^n$. We adopt the convention that $\beta(\bm{x},0)=0$. Then the robust optimization problem is equivalent to

Figures (4)

  • Figure 1: Integrality gap (\ref{['prop:igap_additive_theta_clean', 'cor:igap_relative_theta_clean']}). (a) Ratio of round-down loss to additive bound vs. $n$ for three $\Gamma$-regimes; all points lie well below the theoretical ceiling of $1$ (dashed), consistent with \ref{['prop:igap_additive_theta_clean']}. (b) Scaled relative gap $n\times\mathrm{Gap}_{\mathrm{LP}}$ vs. $n$; the product fluctuates around a stable median (dashed) with no upward trend, consistent with the $O(1/n)$ decay of \ref{['cor:igap_relative_theta_clean']}.
  • Figure 2: Scalability and hull compression. (a) Wall-clock runtime of \ref{['alg:robust_hull_greedy']} vs. $n$ for $m\in\{10,50\}$; reference lines at slopes $1$ and $2$ are overlaid. Both curves track close to slope $2$. (b) Box plot of per-item hull size $|\mathcal{H}_i^\theta|$ at $\theta=0$, grouped by raw menu size $m$, on the $n=500$ master instance; annotated values are medians. Dashed lines mark the raw menu size for reference; the hull retains roughly two-thirds of the menu across all $m$. (c) Stacked bar chart of runtime by algorithm phase ($m=50$); hull construction dominates at every $n$.
  • Figure 3: Revenue--risk frontier (\ref{['thm:robust']}, $n=200$). (a) Top: normalized revenue $N(\bm x)/N(\bm x^{\Gamma=0})$; bottom: violation probability $\hat{P}$ under adversarial shocks (solid) and i.i.d. shocks (dashed), for $\alpha\in\{0.10,0.20\}$. Revenue cost is below $1\%$ even at $\Gamma=n$; violations vanish by $\Gamma/n\approx 0.05$. (b) $5\%$-quantile of realized margin vs. $\Gamma/n$; higher $\alpha$ produces a larger margin buffer and higher tail quantile. (c) Heatmap of violation probability (adversarial, $\alpha=0.10$, log scale): below the diagonal $\Gamma_{\mathrm{attack}}=\Gamma$ (dashed) all cells are white (zero violations, \ref{['thm:robust']}); above the diagonal violations increase smoothly.
  • Figure 4: Retail category pricing ($n=300$, $m=20$, heterogeneous $\alpha_i$). (a) Top: normalized category revenue $N(\bm x)/N(\bm x^{\Gamma=0})$; bottom: violation probability $\hat{P}$ under adversarial shocks at $\Gamma_{\mathrm{attack}}=\lfloor1.5\Gamma\rfloor$ (solid) and i.i.d. shocks (dashed). Revenue cost is below $0.7\%$ even at $\Gamma=n$; i.i.d. violations vanish by $\Gamma/n\approx0.03$. (b) Sorted per-SKU margin contribution $s_i(x_i)$ for the nominal solution ($\Gamma=0$, gray) and the robust solution ($\Gamma=17\approx\lfloor\sqrt{n}\rfloor$, colored by segment). The robust solution compresses the right tail and spreads positive contributions across a broader SKU base.

Theorems & Definitions (17)

  • Definition 1: Uncertainty Set
  • Theorem 1: Robust Counterpart Formulation
  • proof
  • Remark 1
  • Remark 2: Relation to prior work
  • Theorem 2: Hull reduction and greedy optimality for the LP relaxation
  • proof
  • Theorem 3: Parametric decomposition of the coupled $\Gamma$-budget penalty
  • proof
  • Proposition 1: Finite candidate set for theta
  • ...and 7 more