Exoticproducepackingoptimization

Overview

ExoticProducePackingOptimization is a MATLAB-based modeling and simulation project where I explore how to optimally pack and route exotic produce in a warehouse or distribution grid. I treat the warehouse as a 2D lattice, build spatial interaction weights with Gaussian kernels, and explore combinatorial packing patterns for different product categories. The focus is on translating an abstract logistics problem into concrete mathematical objects that I can simulate, analyze, and iterate on.

Role & Context

I built this project as an individual contributor in the context of a modeling and simulation coursework assignment (“ModSim Project 2”). My goals were:

• To frame a realistic logistics/packing problem as a mathematical model.
• To implement the core computations in MATLAB using vectorized, numerically stable methods.
• To experiment with how different neighborhood structures and product mixes affect feasible packing configurations.

Tech Stack

• MATLAB
• MATLAB Project (.prj) structure
• MATLAB sparse matrix utilities

Problem

The high-level problem I wanted to study was:

Given a grid-like storage or packing area and multiple categories of exotic produce, how can we:

• Represent spatial interactions between locations (e.g., cooling influence, handling proximity, cross-contamination risk)?
• Enumerate or sample viable packing patterns under categorical constraints?
• Prepare a structure that could be used for further optimization (e.g., minimizing damage risk, travel time, or temperature variance)?

In particular, I needed:

• A robust way to build spatial weight matrices that decay smoothly with distance and can be truncated beyond a practical radius.
• A way to generate all possible category assignments for a fixed number of slots, given fixed counts per category (e.g., equal counts of three produce types).

Approach / Architecture

I decomposed the problem into two core modeling components:

1. Spatial interaction modeling

   • Represent the warehouse (or packing surface) as an Nside × Nside grid.
   • Construct a Gaussian-based weight matrix W defining how strongly each cell interacts with its neighbors, with distance-based decay and radius truncation.
   • Normalize the weights row-wise so they can represent probabilities or influence coefficients.

2. Combinatorial packing patterns

   • Model a packing scenario with three categories (e.g., three types of produce) and n items of each category.
   • Encode each possible packing configuration as a vector of labels of length 3n.
   • Generate the full set of patterns using combinatorics (nchoosek) such that each pattern respects the required counts for each category.

The project is wrapped in a MATLAB Project (.prj), which keeps the environment, paths, and metadata organized for reproducible runs and easy extension into more complex simulations (e.g., objective functions, optimization loops, or Monte Carlo experiments).

Key Features

• Gaussian-based spatial weight matrix generator with distance truncation and normalization.
• Efficient use of sparse matrices for large grid-based interaction networks.
• Exhaustive pattern generator for equal-count tri-category packing ((3n)! / (n!)^3 configurations).
• Clear separation between spatial modeling (layout) and combinatorial modeling (category assignments).
• MATLAB Project structure for reproducible, IDE-integrated workflows.

Technical Details

Spatial Weight Matrix: makeW_gaussian.m

The makeW_gaussian function constructs a normalized, sparse interaction matrix W over a 2D grid:

function W = makeW_gaussian(Nside, r, sigma)
    N = Nside*Nside;
    [JJ,II] = meshgrid(1:Nside,1:Nside);
    xy = [II(:), JJ(:)];
    D = pdist2(xy,xy,'euclidean');

    A = exp(-(D./sigma).^2); 
    A(D==0) = 0;             % remove self-interaction
    A(D>r)  = 0;             % truncate beyond radius
    A = sparse(A);

    rowSum = sum(A,2);
    W = spdiags(1./max(rowSum,1),0,N,N)*A;
end

Key design choices:

• Grid encoding: I flatten the 2D grid into N = Nside^2 nodes, while still storing their (i, j) coordinates in xy. This allows me to compute pairwise Euclidean distances using pdist2 once, then reuse them for building interaction kernels.
• Gaussian kernel: A_ij = exp(-(d_ij / sigma)^2) yields smooth decay with distance. sigma controls how quickly influence drops off.
• Radius truncation: For computational efficiency and realism, I set entries where D > r to zero. This gives a banded, local-interaction structure matching physical intuition (cells far apart do not meaningfully interact).
• Sparsity: After truncation, A is sparse. Converting to sparse(A) is crucial when Nside grows, dramatically reducing memory footprint and improving performance for matrix operations.
• Row normalization: I normalize each row so that sum_j W_ij = 1, whenever there are neighbors. This supports multiple interpretations (e.g., as a stochastic matrix, influence weights, or convex combination coefficients), and I guard against empty rows via max(rowSum,1) to avoid division by zero.

Pattern Enumeration: patterns3.m

The patterns3 function enumerates all possible assignments of three labels with equal counts:

function P = patterns3(n)
% P is (3n) x K where K = (3n)!/(n!)^3
m = 3*n;
K = factorial(m)/(factorial(n)^3);
P = zeros(m, K, 'uint8');

col = 1;
idx1 = nchoosek(1:m, n);           % choose positions for label 1
for i = 1:size(idx1,1)
    rest = setdiff(1:m, idx1(i,:));    % remaining positions
    idx2 = nchoosek(rest, n);          % choose positions for label 2
    for j = 1:size(idx2,1)
        v = uint8(3*ones(m,1));        % default label 3
        v(idx1(i,:)) = 1;
        v(idx2(j,:)) = 2;
        P(:,col) = v;
        col = col + 1;
    end
end
end

Key aspects:

• Combinatorial basis: I construct all patterns by:
  • First choosing which positions get label 1 (idx1).
  • Then, for each such choice, choosing which of the remaining positions get label 2 (idx2).
  • All remaining positions default to label 3.
• Memory awareness: The combinatorial explosion is explicit in K = (3n)! / (n!)^3. This makes the function very useful for small n (e.g., to validate optimization heuristics or constraints) while also highlighting when exhaustive enumeration becomes infeasible.
• Compact storage: Using uint8 for labels keeps memory usage down and is sufficient for categorical IDs.

Conceptually, each column of P is a full packing configuration across 3n slots, where the counts of each type are fixed to n. This structure is flexible: I can later map labels {1,2,3} to specific produce types, handling constraints like “do not place type 1 adjacent to type 2 under a given W.”

MATLAB Project Structure

The .prj file and associated resources/project/*.xml files:

• Capture project metadata (name, root, and categories such as design, derived, and artifact).
• Allow MATLAB to restore the project environment, making it easier to extend the work into:
  • Scripted simulations over different Nside, r, and sigma.
  • Integration with optimization routines (e.g., fmincon, custom heuristics, or metaheuristics).

Results

This project produced:

• A reusable, parameterized spatial interaction generator that can be applied to:
  • Packing and layout problems.
  • Diffusion-like simulations or Markov processes on grids.
• A combinatorial pattern generator that enumerates feasible tri-category layouts with fixed counts, suitable as:
  • A ground-truth basis for validating heuristic or approximate optimization solutions.
  • A way to benchmark how layout interacts with the spatial coupling represented by W.

Even at this stage, I can combine these components to:

• Inspect how different radius and sigma values change the effective “neighborhood” of each slot.
• Study how specific packing patterns (e.g., clustering vs. mixing of produce types) might interact with that neighborhood, paving the way for future objective functions (e.g., minimizing “risk” computed from W and pattern labels).

Lessons Learned

• Sparse linear algebra is essential: For grid-based models, applying sparsity early (and explicitly) keeps simulations tractable and fast.
• Kernel parameters matter: The choice of sigma and truncation radius r fundamentally changes the structure of the system. Modeling assumptions should be clearly tied to physical intuition (e.g., cooling radius, handling radius).
• Enumeration vs. optimization: Exhaustively listing patterns via combinatorics is powerful for understanding the search space but becomes infeasible quickly; in a production setting, this pushes you toward optimization or sampling approaches.
• Project structure helps scalability: Using MATLAB Project infrastructure, even for a small assignment, makes it easier to grow the codebase into a more complex simulation framework.

Links

• GitHub Repository
• Demo / write-up: TBD