arxiv: 2605.14430 · v1 · submitted 2026-05-14 · 🧮 math.OC

Recognition: no theorem link

OPTIMUS: Optimization Productivity Tool for Intelligent Management of Utilizable Space

Souvik Bhattacharyya , Nisha Singh , Salman Haider , Balaji Nagarajan , Ved Prakash Dwivedi , Nithin Surendran , Karthik Nair

Authors on Pith no claims yet

Pith reviewed 2026-05-15 01:53 UTC · model grok-4.3

classification 🧮 math.OC

keywords retail space optimizationknapsack modeldynamic programmingplanogram allocationmixed-integer programmingsales liftmargin optimization

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The pith

A linear binary knapsack model solved by dynamic programming optimizes retail bay space and delivers 11.8 percent average sales lift.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper formulates department-level retail space allocation as a linear binary knapsack problem in which planograms compete for limited bay capacity according to weighted contributions from sales, margin, units sold, and assortment similarity. Dynamic programming produces exact, reproducible assortment decisions in linear time, after which a second-stage mixed-integer program refines the bay layout. When the full pipeline runs end-to-end across ten optimization instances covering multiple departments and store clusters, it records an average sales increase of 11.8 percent and an average margin increase of 9.5 percent. A sympathetic reader would care because the method replaces heuristic variance with exact, interpretable decisions at enterprise scale.

Core claim

OPTIMUS treats potential SKUs as items in a linear binary knapsack whose value is a linear combination of sales, margin, units, and assortment similarity; dynamic programming solves the knapsack exactly in O(nc) time to select planograms, and these selections become input to a mixed-integer program that optimizes physical bay arrangement, producing the measured lifts when executed across the ten test runs.

What carries the argument

The linear binary knapsack model solved by dynamic programming to generate exact assortment decisions that feed a mixed-integer bay optimization program.

Load-bearing premise

A linear binary knapsack model with hand-chosen or fitted weights for sales, margin, units, and assortment similarity fully captures real business and operational constraints without significant omitted factors or non-linear effects.

What would settle it

Deploy the OPTIMUS recommendations in one or more live store clusters, measure actual sales and margin changes against a matched control set of stores, and check whether the observed lifts reach or exceed 11.8 percent sales and 9.5 percent margin.

read the original abstract

We study department-level retail space optimization, where limited bay capacity must be allocated among planograms (POGs) under business and operational constraints. The problem is formulated as a linear binary knapsack model, with potential SKUs treated as items characterized by space requirements and weighted value contributions from sales, margin, units, and assortment similarity. Dynamic Programming (DP) is employed to obtain exact and reproducible assortment decisions in O(nc) time, avoiding the variance inherent in heuristic approaches. These decisions are integrated with a second-stage bay optimization model formulated as a mixed-integer program. Evaluated end-to-end across ten optimization runs spanning multiple departments and store clusters, the OPTIMUS framework achieves an average sales lift of 11.8% and an average margin lift of 9.5%. Overall, OPTIMUS provides a scalable, interpretable, and profit-driven solution for enterprise-scale retail space management.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper packages standard binary knapsack plus dynamic programming into a two-stage retail space tool and reports 11.8% sales and 9.5% margin lifts from ten runs, but supplies no baselines or external checks.

read the letter

The main thing to know is that OPTIMUS formulates department-level planogram allocation as a linear binary knapsack problem solved exactly by dynamic programming in O(nc) time, then passes the decisions to a mixed-integer program for bay optimization. It reports average lifts of 11.8% in sales and 9.5% in margin across ten runs on multiple departments and store clusters, using weights on sales, margin, units, and assortment similarity. That is the core of the work. What it does well is the choice of exact DP for reproducibility instead of heuristics, plus the clean two-stage split that keeps the whole thing scalable and interpretable for an enterprise setting. If you need a packaged, profit-driven framework for retail space decisions, the structure is straightforward and the runtime claims are plausible on paper. The soft spots sit in the results section. The lifts come only from optimization runs with no reported comparison to current manual allocations, no alternative solvers as baselines, no statistical tests, and no hold-out periods or post-deployment sales data. The linear weighted objective also leaves room for omitted non-linear effects such as cross-elasticities or restocking costs that could shrink the real gains. With evaluation limited to ten runs and weights described as hand-chosen or fitted, it is difficult to judge robustness. This is aimed at retail operations teams or applied OR practitioners who want a ready tool rather than new theory. A reader working on similar allocation problems could pull useful implementation details from the DP and MIP integration. It deserves a serious referee because the methods are grounded and the claims are concrete enough to test, even if the current evidence needs strengthening. I would send it to peer review with a request for baseline comparisons and sensitivity checks on the weights.

Referee Report

2 major / 2 minor

Summary. The paper proposes the OPTIMUS framework for department-level retail space optimization. It formulates the allocation of limited bay capacity among planograms as a linear binary knapsack model, using dynamic programming to obtain exact assortment decisions in O(nc) time, followed by a mixed-integer program for bay optimization. Evaluated on ten optimization runs across departments and store clusters, it reports average sales lift of 11.8% and margin lift of 9.5%.

Significance. If the empirical results hold under proper validation, the framework could offer a scalable, interpretable, and exact (via DP) alternative to heuristic methods for enterprise retail space management. The reproducibility of the DP stage and the two-stage integration of knapsack and MIP are strengths that could support profit-driven decisions at scale.

major comments (2)

[Evaluation section] Evaluation section (results paragraph): the central claim of 11.8% average sales lift and 9.5% margin lift across ten optimization runs supplies no baselines, statistical tests, data sources, exclusion rules, hold-out validation, or post-deployment comparisons, so the improvement cannot be assessed.
[Model formulation] Knapsack model formulation: the linear weighted objective (sales + margin + units + assortment similarity) plus MIP stage is presented as directly capturing the problem, but the hand-chosen or fitted weights and strict linearity omit potential non-linear effects such as cross-elasticities and restocking costs; this assumption is load-bearing for the reported lifts yet lacks sensitivity analysis or real-data grounding.

minor comments (2)

[Abstract and DP description] The O(nc) time complexity is stated without defining n and c, reducing clarity for readers.
[Model section] Notation for the weighted value contributions in the knapsack objective should be formalized with an explicit equation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments. We address each major point below and indicate the revisions we will make to strengthen the manuscript.

read point-by-point responses

Referee: [Evaluation section] Evaluation section (results paragraph): the central claim of 11.8% average sales lift and 9.5% margin lift across ten optimization runs supplies no baselines, statistical tests, data sources, exclusion rules, hold-out validation, or post-deployment comparisons, so the improvement cannot be assessed.

Authors: We agree that the evaluation section requires substantially more detail to allow proper assessment of the reported lifts. The 11.8% sales and 9.5% margin improvements are computed by comparing the OPTIMUS allocations against the retailer’s existing planogram configurations using historical transaction data. The ten runs cover distinct department–store-cluster pairs drawn from the same internal dataset. In the revision we will add: (i) explicit description of the data source and time window, (ii) the precise baseline (current allocation), (iii) any exclusion rules applied to SKUs, (iv) the exact formula used to calculate the lifts, and (v) a limitations paragraph noting the absence of hold-out validation and post-deployment results. Because the study is retrospective, statistical hypothesis tests were not performed; we will discuss this limitation and, if the data permit, include basic confidence intervals. revision: yes
Referee: [Model formulation] Knapsack model formulation: the linear weighted objective (sales + margin + units + assortment similarity) plus MIP stage is presented as directly capturing the problem, but the hand-chosen or fitted weights and strict linearity omit potential non-linear effects such as cross-elasticities and restocking costs; this assumption is load-bearing for the reported lifts yet lacks sensitivity analysis or real-data grounding.

Authors: The linear weighted objective was deliberately chosen to admit an exact dynamic-programming solution in O(nc) time, which is central to the claimed scalability and reproducibility. The weights were set through iterative consultation with category managers to reflect business priorities. We acknowledge that the linearity assumption excludes cross-elasticities and restocking costs. In the revised manuscript we will (a) add a dedicated sensitivity-analysis subsection that varies the four weights over plausible ranges and reports the resulting changes in selected assortments and objective values, and (b) expand the discussion of modeling assumptions to explicitly note the omitted non-linear effects and their potential impact on the reported lifts. The real-data grounding remains the end-to-end evaluation on retailer data; we will clarify how the weights were elicited from that data. revision: partial

Circularity Check

0 steps flagged

No circularity: lifts are empirical outcomes of applying the stated linear model

full rationale

The paper formulates the problem directly as a linear binary knapsack with hand-chosen or fitted weights on sales, margin, units, and assortment similarity, solves it exactly via DP plus a second-stage MIP, and reports average lifts from ten optimization runs on real data. No equation or derivation reduces the reported 11.8%/9.5% lifts to quantities defined by the same weights; the lifts are measured post-optimization results rather than algebraic identities or fitted predictions. No self-citations, uniqueness theorems, or ansatzes are invoked in the abstract or described chain. This is the normal non-circular case for an applied optimization paper whose central claim is empirical performance of an explicitly stated model.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The framework rests on the modeling choice that retail constraints are linear and binary and that value weights can be supplied externally; no new entities are postulated.

free parameters (1)

value weights
Weights combining sales, margin, units, and assortment similarity into each item's value are not derived from first principles and must be supplied or fitted.

axioms (1)

domain assumption Retail space allocation can be represented exactly as a linear binary knapsack problem under the stated business constraints.
The paper assumes linearity and binary decisions suffice without demonstrating why non-linear effects or additional constraints are negligible.

pith-pipeline@v0.9.0 · 5473 in / 1318 out tokens · 38622 ms · 2026-05-15T01:53:45.481270+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

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