Constrained optimisation: fitting assortment to budget and space

Retailers face a fundamental constraint problem: hundreds or thousands of products compete for limited shelf space, floor area and working capital. Modern assortment planning tools use linear programming and constraint solvers to find the combination of products that maximises profit or revenue whilst respecting hard limits on space, budget, minimum stock requirements and category breadth. The mathematics underlying these systems has roots in operations research developed decades ago, but software implementations have accelerated in fashion retail only in the past three years as computing power, data quality and integration have improved.

What is constrained optimisation in retail assortment planning?

At the selling department level, where profitability and product strategy converge, the challenge is to allocate limited bay space in a way that maximises returns whilst respecting category level and organisational constraints. Retailers aim to maximise sales under different constraints, such as limited purchasing budget, limited space of classes for displaying the products, and having at least a certain number of suppliers, developing an integrated mathematical model to optimise integrated assortment planning, space allocation and inventory control.

The objective is to obtain the list and quantity of articles that is to be kept in the store for a particular phase, such that overall revenue is maximised and the business constraints are satisfied. These constraints typically include total distinct articles per store, budget at sub class level, and minimum and maximum distinct articles at sub class level.

How does the knapsack problem apply to merchandise planning?

The problem of optimising the products offered for sale by a retailer is a generalisation of what is commonly referred to as the knapsack problem, which asks what is the most valuable combination of items that can be stored in a knapsack without breaking it, given a set of items with each item having two values and a constraint on one of the values. The knapsack problem and its variants have emerged as useful abstractions for modelling resource constrained allocation problems, including those found in retail.

Multiple knapsack problem optimisation presents significant challenges in retail resource allocation, particularly for product bundling scenarios with dual constraints, and evaluations compare seven distinct algorithms: base dynamic programming, Numba accelerated dynamic programming, parallel processing, parallel rolling, genetic algorithm, greedy algorithm and branch and bound. For resource constrained environments, branch and bound offers 90 per cent utilisation with minimal computational overhead, and statistical analysis confirms the superior performance of parallel variants compared to basic implementations.

What mathematical techniques solve assortment optimisation problems?

Research published in 2025 and 2026 describes multiple solution approaches. OPTIMUS, an enterprise scale framework for department level retail space optimisation, casts assortment construction as a linear binary knapsack problem and solves it via dynamic programming to produce deterministic, reproducible assortments under capacity constraints, with these outputs feeding a bay level mixed integer optimisation model that allocates space across planograms whilst enforcing operational constraints.

Academic work on assortment optimisation under general linear constraints, where customer choice behaviour is governed by the paired combinatorial logit model, develops four exact solution approaches to solve the problem with any linear constraints to optimality. A scenario based multi stage stochastic linear programming model is proposed for integrated sales planning, incorporating demand elasticity as a function of promotions and assortment size.

The assortment optimisation is formulated as integer linear programming with the objective to maximise profit subject to constraints. GAMS BARON solver is hired to solve the proposed model in small and medium scales, but because the problem is NP hard, an evolutionary genetic algorithm and an efficient local search vibration damping optimisation algorithm are proposed.

Which constraints matter most in fashion merchandising?

Fashion retailers face several categories of constraint. Retailers operate under increasing pressure to maximise profitability within constrained store footprints, and as customer demand for product variety grows, the tension between limited physical space and the need for broad assortments intensifies. The system's primary objective is to enhance the trading margin obtained from the sale whilst considering constraints related to commodity resources and shelf space.

• Space constraints: Physical shelf metres, fixture capacity, backroom storage and floor area impose hard limits on the number of facings and stock keeping units.
• Budget constraints: Open to buy allocations, working capital limits and supplier payment terms restrict the total cost value that can be purchased.
• Cardinality constraints: Minimum and maximum product counts per category, supplier diversity requirements and presentation standards govern assortment breadth.
• Service level constraints: Minimum in stock rates, availability targets and replenishment frequency commitments ensure customer satisfaction.

An optimisation method for commodities relies on mathematical modelling and the calculation of the consolidated profitability ratio, ensuring the necessary accuracy and providing assortment management within time and cost limits, without substantial investments in equipment and updating qualifications of employees.

How do solvers handle large scale retail problems?

Many existing formulations are nonlinear and computationally intensive, posing practical challenges for implementation in large scale retail environments. There is presently no known polynomial time algorithm for exactly solving this class of problems, so people often rely on approximations, and currently known approximation approaches, such as allowing fractional allocations, tend not to work well with very large scale data sets and are typically not useful for obtaining answers in real time.

Practical implementations decompose large problems into manageable subproblems. The multiple choice knapsack solver in one mode of operation may find an optimal solution to the problem of selecting one facing from each choice set within a global space constraint, selecting an optimal facing from each of the choice sets to fit within the total width of the shelves, and the shelf capacity controller supplies parameters such as the number of shelves, the width of each shelf and other relevant parameters to the shelf splitter which splits the solution into shelf specific components utilising the multiple knapsack problem solver.

The global assortment optimisation market was valued at 4.8 billion dollars in 2025 and is projected to reach 12.3 billion dollars by 2034, with the software component segment capturing 3.0 billion dollars in revenue in 2025, and software solutions encompassing core demand forecasting engines, inventory optimisation algorithms, product recommendation platforms and integrated dashboards, with enterprise grade software platforms from vendors including Oracle, SAP and Blue Yonder offering comprehensive assortment planning capabilities integrated with broader supply chain and retail management systems.

What does recent industry adoption look like?

Generative AI could significantly accelerate merchandising with applications ranging from automating assortment selection to detecting microtrends via social listening and translating them into actionable guidance for stock volumes. The report points to a renewed push to simplify assortments, perhaps best exemplified by Adidas' turnaround over the last few years, and reducing working capital tied up in stock, with companies beginning to adopt more disciplined approaches to forecasting and product creation, supported by digital tools that help them understand how and where value leaks.

The research outcomes are applicable to apparel retail, the practical outcome is maximising the return on investment for goods sold per day, and the algorithm's benefits and effectiveness were calculated based on real data after implementation. Major trends in the forecast period include demand forecast driven assortment optimisation, store level gap detection using real time sales signals, localised assortment planning for micro markets, automated planogram and shelf space recommendations, and inventory allocation optimisation with constraint models.

Despite the mathematical sophistication available, practical barriers remain. A McKinsey survey found 71 per cent of merchants say AI merchandising tools have had limited to no effect so far, and the gap traces to sales history quality and attribute consistency, not the algorithms. Implementation success depends on clean data, consistent product attributes and integration with existing enterprise resource planning and point of sale systems, not just on algorithm choice.