Lecture VI - Minimizing Split Orders

Applied Optimization with Julia

Author

Affiliation

Dr. Tobias Vlćek

University of Hamburg - Fall 2025

Introduction

E-Commerce Trends

Question: What are current trends in e-commerce?

E-Commerce Sales

E-Commerce sales are growing fast:
- Products are no longer bound between borders
- Product variety is rising
- Consumer shopping patterns are shifting
- Brick-and-mortar stores loose customers to the internet
- Covid-19 accelerated this trend even more

Parcels Worldwide

The number of parcels is rising:
- 2014: 44 billion parcels (Pitney Bowes Inc. 2017)
- 2019: 103 billion parcels (Pitney Bowes Inc. 2019)
- 2026: 220 – 262 billion¹ (Pitney Bowes Inc. 2020)

Pitney Bowes Inc. 2017. “Pitney Bowes Parcel Shipping Index Reveals 48 Percent Growth in Parcel Volume since 2014.” 2017. https://www.businesswire.com/news/home/20170830005628/en/Pitney-Bowes-Parcel-Shipping-Index-Reveals-48.

———. 2019. “Pitney Bowes Parcel Shipping Index Reports Continued Growth Bolstered by China and Emerging Markets.” 2019. https://www.businesswire.com/news/home/20191010005148/en/.

¹ Forecast, not actual number

———. 2020. “Pitney Bowes Parcel Shipping Index Reports Continued Growth as Global Parcel Volume Exceeds 100 billion for First Time Ever.” 2020. https://www.businesswire.com/news/home/20201012005150/en/.

Pressure on infrastructure

Consumers nowadays expect free, fast deliveries and returns
Existing warehouses have to store an increasing range of products
Better customer service requires faster deliveries
Incurred fulfillment costs depend on the number of parcels

Pressure on the environment

Each parcel packaging consumes resources during production
Every dispatched parcel to the customer causes CO₂ emissions
In case of returns, more parcels cause more emissions

Problem Structure

Split Order

Question: What is a split order?

. . .

No Split Order

Reason for Split Orders

Question: Why might they occur?

. . .

Stock availability: Some products are out of stock at a warehouse and need to be fulfilled from another warehouse
Capacity constraints: Some products are stored at different warehouses and need to be shipped from elsewhere

Impact of Split Orders

Question: What are the consequences?

. . .

Higher shipping costs
Increased packaging material
More CO₂ emissions
Higher operational complexity
Lower customer satisfaction

Mitigations?

Question: What are possible mitigations?

. . .

Consolidation: Ship to a central warehouse before dispatch
Cross-docking: Ship directly from supplier to customer
Transshipment: Ship between warehouses before delivery
Co-allocation: Predict co-appearance of products and allocate them to the same warehouse

Case Study

Key information about the case:

a large European e-commerce retailer
the retailer has two warehouses
product range cannot be stored in either warehouse
product deliveries can be made to both warehouses
products do not have to be stored exclusively

Problem Structure - Version 1

Optimizing Co-allocation

Question: What could be our objective?

We aim to improve the SKU²-warehouse allocation to minimize the number of split parcels resulting from SKUs being stored in different warehouses.

² SKU: Stock Keeping Unit

Available Sets

Question: What could be the sets here?

$\mathcal{I}$ - Set of products indexed by $i \in \{1,2,...,|\mathcal{I}|\}$
$\mathcal{K}$ - Set of warehouses indexed by $k \in \{1,\dots,|\mathcal{K}|\}$
$\mathcal{M}$ - Set of customer orders $m \in \{1,2,...,|\mathcal{M}|\}$

Available Parameters

Question: What are possible parameters?

$c_k$ - Storage space of warehouse $k \in \{1,\dots,|\mathcal{K}|\}$
$\boldsymbol{T}= (t_{m,i})$ - Past customer orders for SKUs

. . .

Question: What could the transactional data look like?

Transactional Data

Example of $\boldsymbol{T}$
$t_{m,i}$	A	B	C	D
1	1	1	1	0
2	1	1	1	0
3	1	1	0	0
4	1	0	0	1
5	1	0	0	1
6	1	0	0	1
7	1	0	0	1
8	0	0	1	1

Past vs. Future

The transactional data $\boldsymbol{T}$ is based on past orders
It is a binary matrix of customer orders and SKUs
We use this data to assume future co-occurrence
- Past co-occurrence predicts future co-occurrence

. . .

Question: What is your opinion on the assumption?

Split-Order Minimization

Question: What could be our decision variable/s?

. . .

We have the following sets:

$\mathcal{I}$ - Set of products indexed by $i \in \{1,2,...,|\mathcal{I}|\}$
$\mathcal{K}$ - Set of warehouses indexed by $k \in \{1,\dots,|\mathcal{K}|\}$
$\mathcal{M}$ - Set of customer orders $m \in \{1,2,...,|\mathcal{M}|\}$

. . .

$X_{i,k}$ - 1, if $i\in\mathcal{I}$ is stored in $k\in\mathcal{K}$, 0 otherwise
$Y_{m,i,k}$ - 1, if SKU $i\in\mathcal{I}$ is shipped from warehouse $k\in\mathcal{K}$ for customer order $m\in\mathcal{M}$, 0 otherwise

Integer Programming Model

Catalán and Fisher (2012) created an integer model
Number of SKUs of E-Commerce retailers can easily be between 10,000 - 100,000
Number of customer orders necessary for “stable” results have to be higher in the order of 100,000 - 10,000,000

. . .

Question: Anybody an idea what this could mean?

Implementation Challenges

Small instance with 10 SKUs and 1000 customer orders
CPLEX 20.1.0 needs 3100 seconds to solve the problem
Computation times scales exponentially
$\rightarrow$ Not applicable in real world applications!

Any idea what

could be done?

Problem Structure - Version 2

Heuristic Approach

Heuristic: Fast, but not necessarily optimal
Approximation: Not guaranteed to be optimal, but close
Computational Effort: Reasonable even for large instances

. . .

Different view on the problem

Focus on the warehouses and the co-appearance of SKUs! Discard the exact information about the customer orders.

Objective

Question: What could be the objective?

Maximize the coappearance of products that are often part of the same customer orders.

Transaction Matrix

T = [
    1 1 1 0;
    1 1 1 0;
    1 1 0 0;
    1 0 0 1;
    1 0 0 1;
    1 0 0 1;
    1 0 0 1;
    0 0 1 1
]

# Create the coappearance matrix
Q = T' * T
println("Coappearance matrix Q:")
display(Q)

Coappearance matrix Q:

4×4 Matrix{Int64}:
 7  3  2  4
 3  3  2  0
 2  2  3  1
 4  0  1  5

Coappearance Matrix

$\boldsymbol{Q}$ is a symmetric matrix
Proposed by Catalán and Fisher (2012)
$\boldsymbol{Q} = (\boldsymbol{T}^T \cdot \boldsymbol{T})$ where $\boldsymbol{Q} = (q_{ij})_{i \in \{1,\dots,\mathcal{I}\},j \in \{1,\dots,\mathcal{I}\}}$
$q_{ij}$ shows how often $i$ and $j$ appear in the same order

. . .

Question: What do the principal diagonal values tell us?

. . .

How often each SKU appeared over all orders (binary!)

How to approach the problem?

Greedy Heuristic³: Allocation based on matrix
Mathematical Model⁴: Maximizes coappearance
GRASP⁵: Good on small instances
New: Max. coappearance with non-linear solver
New: Heuristic based on Chi-Square Tests

³ Simple and very fast, Catalán and Fisher (2012)

Catalán, Andrés, and Marshall Fisher. 2012. “Assortment Allocation to Distribution Centers to Minimize Split Customer Orders.” SSRN Electronic Journal. https://doi.org/10.2139/ssrn.2166687.

⁴ Computationally intensive with CPLEX, Zhu et al. (2021)

⁵ Greedy Randomized Adaptive Search Procedure, Zhu et al. (2021)

Zhu, Shan, Xiangpei Hu, Kai Huang, and Yufei Yuan. 2021. “Optimization of Product Category Allocation in Multiple Warehouses to Minimize Splitting of Online Supermarket Customer Orders.” European Journal of Operational Research 290 (2): 556–71. https://doi.org/10.1016/j.ejor.2020.08.024.

Basic Setting

Available Data (Version 2)

Question: What could be the sets?

. . .

$\mathcal{I}$ - Set of products indexed by $i \in \{1,2,...,|\mathcal{I}|\}$
$\mathcal{K}$ - Set of warehouses indexed by $k \in \{1,\dots,|\mathcal{K}|\}$

. . .

No customer order information is needed!

We can focus on the SKUs and the warehouses, making the problem much smaller!

Available Parameters

Question: What are possible parameters?

$c_k$ - Storage space of warehouse $k \in \{1,\dots,|\mathcal{K}|\}$
$\boldsymbol{Q}= (q_{ij})_{i \in \{1,\dots,\mathcal{I}\},j \in \{1,\dots,\mathcal{I}\}}$ - Coappearance matrix

. . .

Transactional Data replaced

Instead of the transactional data, we just use the coappearance matrix in our model!

Model Formulation

Decision Variables?

We have the following sets:

$\mathcal{I}$ - Set of products indexed by $i \in \{1,2,...,|\mathcal{I}|\}$
$\mathcal{K}$ - Set of warehouses indexed by $k \in \{1,\dots,|\mathcal{K}|\}$

. . .

Our objective is to:

Maximize the coappearance of products that are often part of the same customer orders. In more mathematical terms: Maximize the sum of all unique pair-wise values $q_{i,j}$ of all SKUs stored in the same warehouse.

. . .

Question: What could be our decision variable/s?

Decision Variables

$X_{i,k}$ - 1, if SKU $i\in\mathcal{I}$ is stored in $k\in\mathcal{K}$, 0 otherwise

. . .

Only one variable per SKU and warehouse!

As we don’t need the customer order information, we only need to make a decision for each SKU and warehouse pair!

Decision Variable in Julia

Question: How could we formulate the variable in Julia?

import Pkg; Pkg.add("SCIP")
using JuMP, SCIP # SCIP is a non-commercial MIQCP solver

warehouses = ["Hamburg", "Berlin"] # Add warehouses as a vector
skus = ["Smartphone", "Socks", "Charger"] # Add SKUs as a vector

warehouse_model = Model(SCIP.Optimizer)

. . .

@variable(warehouse_model, X[i in skus, k in warehouses], Bin)

2-dimensional DenseAxisArray{VariableRef,2,...} with index sets:
    Dimension 1, ["Smartphone", "Socks", "Charger"]
    Dimension 2, ["Hamburg", "Berlin"]
And data, a 3×2 Matrix{VariableRef}:
 X[Smartphone,Hamburg]  X[Smartphone,Berlin]
 X[Socks,Hamburg]       X[Socks,Berlin]
 X[Charger,Hamburg]     X[Charger,Berlin]

Objective Function

We need the following:

$X_{i,k}$ - 1, if SKU $i\in\mathcal{I}$ is stored in $k\in\mathcal{K}$, 0 otherwise
$q_{ij}$ - Coappearance of SKU $i\in\mathcal{I}$ and $j\in\mathcal{I}$

Our objective is to:

Maximize the sum of all unique pair-wise values $q_{i,j}$ of all SKUs stored in the same warehouse. Note, that this is a quadratic objective function!

. . .

Question: What could the objective function look like?

. . .

Quadratic Objective Function

\[\text{maximize} \quad \sum_{i=2}^{\mathcal{I}} \sum_{j=1}^{i-1} \sum_{k \in \mathcal{K}} X_{ik}\times X_{jk} \times q_{ij}\]

. . .

This is a quadratic objective function!

The quadratic terms are $X_{ik}\times X_{jk}$. This objective function is based on the Quadratic Multiple Knapsack Problem (QMKP), formulated by Hiley and Julstrom (2006).

Hiley, Amanda, and Bryant A. Julstrom. 2006. “The Quadratic Multiple Knapsack Problem and Three Heuristic Approaches to It.” In Proceedings of the 8th Annual Conference on Genetic and Evolutionary Computation, edited by M. Keijzer, 547–52. New York, NY: Association for Computing Machinery. https://doi.org/10.1145/1143997.1144096.

Objective Function in Julia

Question: How could we formulate this in Julia?

. . .

Q = [2 1 2; 1 2 1; 2 1 2]

@objective(warehouse_model,
    Max,
    sum(
        X[skus[i], warehouses[k]] * X[skus[j], warehouses[k]] * Q[i,j]
        for i in 2:length(skus)
        for j in 1:i-1
        for k in 1:length(warehouses)
    )
)

$ X_{Socks,Hamburg}X_{Smartphone,Hamburg} + X_{Socks,Berlin}X_{Smartphone,Berlin} + 2 X_{Charger,Hamburg}X_{Smartphone,Hamburg} + 2 X_{Charger,Berlin}X_{Smartphone,Berlin} + X_{Charger,Hamburg}X_{Socks,Hamburg} + X_{Charger,Berlin}X_{Socks,Berlin} $

Constraints

What constraints?

Question: What constraints?

Allocate each SKU at least once
Warehouses have a finite capacity
Capacity is not exceeded

Single Allocation Constraint?

The goal of this constraint is to:

Ensure that each SKU is allocated at least once.

. . .

We need the following variable:

$X_{i,k}$ - 1, if SKU $i\in\mathcal{I}$ is stored in $k\in\mathcal{K}$, 0 otherwise

. . .

Question: What could the constraint look like?

Single Allocation Constraint

\[\sum_{k \in \mathcal{K}} X_{ik} \geq 1 \quad \forall i \in \mathcal{I}\]

. . .

Remember, this is the variable:

$X_{i,k}$ - 1, if SKU $i\in\mathcal{I}$ is stored in $k\in\mathcal{K}$, 0 otherwise

. . .

Question: How could we change the constraint to ensure that each SKU is allocated only once?

. . .

Question: How could we add the constraint in Julia?

Single Allocation in Julia

@constraint(warehouse_model, single_allocation[i in skus],
    sum(X[i, k] for k in warehouses) >= 1
)

1-dimensional DenseAxisArray{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.GreaterThan{Float64}}, ScalarShape},1,...} with index sets:
    Dimension 1, ["Smartphone", "Socks", "Charger"]
And data, a 3-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.GreaterThan{Float64}}, ScalarShape}}:
 single_allocation[Smartphone] : X[Smartphone,Hamburg] + X[Smartphone,Berlin] ≥ 1
 single_allocation[Socks] : X[Socks,Hamburg] + X[Socks,Berlin] ≥ 1
 single_allocation[Charger] : X[Charger,Hamburg] + X[Charger,Berlin] ≥ 1

Capacity Constraints?

The goal of these constraints is to:

Ensure that the capacity of each warehouse is not exceeded.

. . .

We need the following variables and parameters:

$X_{i,k}$ - 1, if SKU $i\in\mathcal{I}$ is stored in $k\in\mathcal{K}$, 0 otherwise
$c_k$ - Storage space of warehouse $k\in\mathcal{K}$

. . .

Question: What could the second constraint be?

Capacity Constraints

\[\sum_{i \in \mathcal{I}} X_{ik} \leq c_k \quad \forall k \in \mathcal{K}\]

. . .

And that’s basically it!

. . .

Question: How could we add the second constraint in Julia?

Capacity Constraints in Julia

capacities = Dict("Hamburg" => 2, "Berlin" => 1) # Add capacities

@constraint(warehouse_model, capacity[k in warehouses],
    sum(X[i, k] for i in skus) <= capacities[k]
)

1-dimensional DenseAxisArray{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape},1,...} with index sets:
    Dimension 1, ["Hamburg", "Berlin"]
And data, a 2-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape}}:
 capacity[Hamburg] : X[Smartphone,Hamburg] + X[Socks,Hamburg] + X[Charger,Hamburg] ≤ 2
 capacity[Berlin] : X[Smartphone,Berlin] + X[Socks,Berlin] + X[Charger,Berlin] ≤ 1

QMK Model

\[\text{maximize} \quad \sum_{i=2}^{\mathcal{I}} \sum_{j=1}^{i-1} \sum_{k \in \mathcal{K}} X_{ik}\times X_{jk} \times q_{ij}\]

subject to:

\[ \begin{align*} & \sum_{k \in \mathcal{K}} X_{ik} \geq 1 && \forall i \in \mathcal{I}\\ & \sum_{i \in \mathcal{I}} X_{ik} \leq c_{k} && \forall k \in \mathcal{K}\\ & X_{ik} \in \{0,1\} && \forall i \in \mathcal{I}, \forall k \in \mathcal{K} \end{align*} \]

QMK Model in Julia

set_attribute(warehouse_model, "display/verblevel", 0) # Hide solver output
optimize!(warehouse_model)

println("The optimal objective value is: ", objective_value(warehouse_model))
println("The optimal solution is: ", value.(X))

The optimal objective value is: 2.0
The optimal solution is: 2-dimensional DenseAxisArray{Float64,2,...} with index sets:
    Dimension 1, ["Smartphone", "Socks", "Charger"]
    Dimension 2, ["Hamburg", "Berlin"]
And data, a 3×2 Matrix{Float64}:
  1.0  0.0
 -0.0  1.0
  1.0  0.0

. . .

Question: What does this value tell us?

Model Characteristics

Characteristics

Is the model formulation linear/ non-linear?
What kind of variable domain do we have?
Do we know the split-orders based on the objective value?
Why couldn’t we use HiGHS as solver?

Choosing a solver

Identify problem structure, e.g. LP, MIP, NLP, QCP, MIQCP, …
What is the size of the problem?
Is a commercial solver needed?

. . .

Commercial Solvers

Commercial solvers are faster and more robust as open source solvers but also more expensive. During your studies, you can use most of them for free though! Nonetheless, we will only use open source solvers in this course.

Global vs Local Optimality

Local vs Global Optimum by Christoph Roser

Solver Comparison⁶

SKUs	SCIP	HiGHS	Gurobi
100	118s	X	~400s
1,000	1,011s (18%)	X	~200s (2%)
10,000	X	X	X

. . .

SCIP: Best open-source for MIQCP, but limited scalability
Commercial solvers: Better but still fail on 10,000+ SKUs
Conclusion: Heuristics necessary for realistic problems!

⁶ Based on numerical experiments (QMK). The percentage in the table shows instances that could be solved within one hour and their average time (if solved).

Model Assumptions

Questions: On model assumptions

What assumptions have we made?
Problem with allocating SKUs to multiple warehouses?
What else might pose a problem in the real world?

Model Limitations

Stock-outs: Model assumes perfect availability
- Real stockouts create splits despite optimal allocation
Storage costs: No differentiation between costs
Demand dynamics: Weather, promotions, seasonality
SKU sizes: Uniform capacity assumption unrealistic
Shipping costs: No consideration of distance or weight

Practical Limitations

Allocation changes:

Frequent reallocation disrupts operations
Physical inventory movements costly
Need for change minimization

Workload imbalance:

Optimal allocation may overload some warehouses
High-frequency SKUs cluster together
Trade-off: splits vs. balance

Impact

Can this be

applied?

Problem Size is Crucial

Up to 1,000 SKUs → commercial solvers
More than 1,000 SKUs → heuristics
For example, the CHI heuristic

. . .

CHI-Heuristic

Detect dependencies between products and allocate them accordingly, as products within orders can have dependencies and products are bought with different frequencies!

CHI Heuristic

Why CHI?

Question: What’s wrong with QMK?

. . .

QMK Problems:

Slow for large instances
May allocate SKUs to multiple warehouses
Doesn’t distinguish dependencies

CHI Solution:

Fast: $O(|\mathcal{K}| \times |\mathcal{I}|^2)$
Uses statistical tests
50,000 SKUs in 10 minutes

Statistical Independence

Question: When are two SKUs independent?

. . .

If purchasing one doesn’t affect purchasing the other:

\[P(A \cap B) = P(A) \times P(B)\]

. . .

Example: If $A$ appears in 10% of orders and $B$ in 20%:

Independent: Appear together in $0.10 \times 0.20 = 2\%$ of orders
Dependent: Appear together in 5% of orders → customers buy them together!

Chi-Square Test Intuition

Use chi-square tests to detect if SKUs are dependent:

Calculate expected coappearances under independence
Compare with actual coappearances
If difference is significant → SKUs are dependent
Allocate dependent SKUs to same warehouse!

. . .

Two-Phase Approach

Phase 1: Build allocation using dependencies
Phase 2: Refine with local search

CHI Algorithm Overview

Identify dependencies: Chi-square tests on all SKU pairs
Separate matrices:
- $\boldsymbol{E}$: Independent coappearances
- $\boldsymbol{D}$: “Dependency premium”
Allocate strategically: Cluster dependent SKUs together
Refine: Local search for improvements

. . .

Result: Better than QMK on large instances!

Practical Implementation

Real-World Challenges

Question: What challenges might arise in practice?

New products: Allocate SKUs without historical data?
Dynamic inventory: SKUs appear and disappear
Different sizes: SKUs consume varying storage space
Computational limits: Problems with 50,000+ SKUs

Rolling Horizon Approach

Use 5-week training window for allocation decisions
Update allocations weekly based on recent patterns
Balance stability vs. responsiveness

. . .

Grouped SKUs by category-brand combinations to reduce problem size while maintaining allocation quality. New SKUs inherit allocation from their cluster!

Case Study

More than 100,000 SKUs and several millions of orders
Comparison of different heuristics⁷
- CHI: based on Chi-Square tests Vlćek and Voigt (2024)
- GP, GO, GS, BS: based on greedy algorithms (Catalán and Fisher 2012)
- RA: Random allocation of SKUs to warehouses

⁷ QMKP is not applicable for instance in case study

Vlćek, Tobias, and Guido Voigt. 2024. “Optimizing SKU-Warehouse Allocations to Minimize Split Parcels in E-Commerce Environments.” To Be Submitted Soon.

Real Data Set

Implementation Results

Theoretical improvements:

CHI: 82.25% reduction
BS: 62.63% reduction
GS: 59.16% reduction
vs. retailer: 6.95% split ratio

Conclusion

Splits are of no benefit, except faster customer deliveries
Increase workload, packaging and shipping costs
Mathematical Optimisation of “full” problem not solvable
CHI Heuristic close to mathematical optimisation

. . .

And that’s it for todays lecture!

We now have covered the Quadratic Multiple Knapsack Problem and are ready to start solving some tasks in the upcoming tutorial.

Questions?

Literature

Literature I

For more interesting literature to learn more about Julia, take a look at the literature list of this course.

Example of \(\boldsymbol{T}\)
\(t_{m,i}\)	A	B	C	D
1	1	1	1	0
2	1	1	1	0
3	1	1	0	0
4	1	0	0	1
5	1	0	0	1
6	1	0	0	1
7	1	0	0	1
8	0	0	1	1

Introduction

E-Commerce Trends

E-Commerce Sales

Parcels Worldwide

Pressure on infrastructure

Pressure on the environment

Problem Structure

Split Order

No Split Order

Reason for Split Orders

Impact of Split Orders

Mitigations?

Case Study

Problem Structure - Version 1

Optimizing Co-allocation

Available Sets

Available Parameters

Transactional Data

Past vs. Future

Split-Order Minimization

Integer Programming Model

Implementation Challenges

Problem Structure - Version 2

Heuristic Approach

Objective

Transaction Matrix

Coappearance Matrix

How to approach the problem?

Basic Setting

Available Data (Version 2)

Available Parameters

Model Formulation

Decision Variables?

Decision Variables

Decision Variable in Julia

Objective Function

Quadratic Objective Function

Objective Function in Julia

Constraints

What constraints?

Single Allocation Constraint?

Single Allocation Constraint

Single Allocation in Julia

Capacity Constraints?

Capacity Constraints

Capacity Constraints in Julia

QMK Model

QMK Model in Julia

Model Characteristics

Characteristics

Choosing a solver

Global vs Local Optimality

Solver Comparison6

Model Assumptions

Model Limitations

Practical Limitations

Impact

Problem Size is Crucial

CHI Heuristic

Why CHI?

Statistical Independence

Chi-Square Test Intuition

CHI Algorithm Overview

Practical Implementation

Real-World Challenges

Rolling Horizon Approach

Case Study

Real Data Set

Implementation Results

Conclusion

Literature

Literature I

Solver Comparison⁶