# Collaborative Geodesign and Spatial Optimization for Fragmentation-Free Land Allocation

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## Abstract

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## 1. Introduction

## 2. General Problem Definition: Fragmentation-Free Land Allocation

#### 2.1. Basic Concepts

**Watershed:**A basin where all surface water converges to a single outlet on its boundary.

**Unchangeable landscape:**Places in a watershed where the land use and land cover cannot be changed (e.g., water bodies and roads). These places are not within the scope of land allocation.

**Spatial location:**In this paper, a watershed consists of a finite collection of non-overlapping spatial locations. Each location is a region. Examples of spatial locations are cells in an orthogonal grid or hydrological response units [8].

**Ecosystem services:**Benefits from an ecosystem, such as water, carbon sequestration, habitat, economic market return, etc.

**Land management practices (LMP):**Methods of managing agricultural land, such as conventional tillage, conservation tillage, low-phosphorus application and grassland. A management practice directly interacts with properties of a farm (e.g., soil type, water quality) and affects the quality of ecosystem services. Geographic variation plays an important role in the interaction. For example, allocating grassland at places near water bodies can significantly filter out fertilizer residues and protect water quality.

**Profit and cost:**In this research, the value of an ecosystem service is quantified using the Soil and Water Assessment Tool (SWAT) and InVEST modeling [9,10]. Multiple ecosystem service objectives are linearly combined into a single objective (profit). The cost includes economic losses from decreases in food production and investment needed for re-purposing lands (e.g., change management practices).

**Land fragmentation:**Small or irregularly shaped patches of land management practices, where a patch is a contiguous region with homogeneous choice of land management practices. Land fragmentation prohibits efficient use of large farm equipment (e.g., 10- to 20-m-wide combine harvester heads) [3,4].

**Spatial constraints:**Spatial constraints are used to avoid land fragmentation. In this research, we use a minimum patch area to guarantee each patch of land management practice is large enough to be farmed efficiently. In addition, we use a regular-shaped constraint, where the shape is defined based on user preference (e.g., rectangle).

#### 2.2. Formal Problem Definition

**Inputs:**

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- A watershed (study area) containing a set of spatial locations $lo{c}_{ij}$ in a two-dimensional space, where $i,j$ are coordinates on two mutually-orthogonal directions;
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- Boundary of unchangeable landscapes within the watershed;
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- A set L of choices of land management practices {$lm{p}_{k}$};
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- A list of ecosystem objectives {$ob{j}_{l}$} and a weight ${w}_{l}$ of each objective;
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- For each objective $ob{j}_{l}$, its profit value ${v}_{ijkl}$ of LMP choice $lm{p}_{k}$ at location $lo{c}_{ij}$;
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- A cost value ${c}_{ijk}$ of LMP choice $lm{p}_{k}$ at location $lo{c}_{ij}$;
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- A budget limit ${c}_{tot}$;
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- A minimum area $\alpha $ and regular shape (e.g., rectangle);

**Output:**A land allocation map with assignment of land management practices for all spatial locations. The solution at each location is denoted by ${s}_{ijk}$, where ${s}_{ijk}=1$ if $lm{p}_{k}$ is selected; otherwise 0.

**Objective:**Maximize overall profit in the watershed.

**Constraints:**

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- Each spatial location can only have one choice of LMP:$${s}_{ijk}\in \{0,1\}$$$$\sum _{k}{s}_{ijk}=1,\phantom{\rule{0.166667em}{0ex}}\forall i,\phantom{\rule{0.166667em}{0ex}}\forall j$$
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- The budget for the land allocation is less than or equal to ${c}_{tot}$:$$\sum _{i}\sum _{j}\sum _{k}{c}_{ijk}\xb7{s}_{ijk}\u2a7d{c}_{tot}$$
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- Every patch formed by a homogeneous LMP in the land allocation map satisfies the minimum area and width constraint, and the regular-shape constraint.

## 3. Challenges

## 4. Related Work and Novelty

#### 4.1. Limitations of Related Work

#### 4.2. Novelty

## 5. Frameworks for Fragmentation-Free Land Allocation

#### 5.1. Overall Architecture

#### 5.2. Collaborative Geodesign

**Publicly available implementation :**We have fully implemented and documented the collaborative geodesign tools and made them publicly available. The newest web-based tool is available at: http://maps.umn.edu/geodesign/, where users can interact with the collaborative geodesign tools in a web-browser. The instructions and overall description are available at: http://newagbioeconomy.umn.edu/collaborativegeodesign/. The desktop versions implemented using Java (both 32-bit and 64-bit) are available at: http://www-users.cs.umn.edu/~xiexx347/cg.

#### 5.3. Spatial Optimization

**Inputs:**

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- A grid partition G of the study area, where each grid cell is identified by its row i and column j.
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- A list L of LMP choices.
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- A full list Z of combinations of tile, LMP choice and the corresponding profit p and cost c:$Z=\{{Z}^{t}=({i}_{0}^{t},{j}_{0}^{t},{i}_{1}^{t},{j}_{1}^{t},LM{P}_{k}^{t},{p}^{t},{c}^{t})\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}{i}_{0}^{t},{j}_{0}^{t},{i}_{1}^{t},{j}_{1}^{t}\in G,LM{P}_{k}^{t}\in L\}$, where $({i}_{0}^{t},{j}_{0}^{t})$ defines the top-left grid cell of the tile and $({i}_{1}^{t},{j}_{1}^{t}$) defines the bottom right.
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- A budget limit ${c}_{tot}$, a minimum area $\alpha $ and a minimum width $\beta $;

**Output:**A subset ${Z}^{\prime}$ of Z (tile-partitioning and LMP assignment)

**Objective:**To maximize overall profit in the watershed:

**Constraints:**

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- Binary choice on ${s}_{p}$:$${s}_{p}\in \{0,1\}$$
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- The cost is less than or equal to ${c}_{tot}$:$$\sum _{t=1}^{\left|Z\right|}{c}^{t}\xb7{s}^{t}\u2a7d{c}_{tot}$$
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- Each tile belonging to ${Z}^{\prime}$ satisfies the minimum area $\alpha $ and minimum width $\beta $:$$({i}_{1}^{t}-{i}_{0}^{t}+1)\xb7({j}_{1}^{t}-{j}_{0}^{t}+1)\ge \alpha ,\phantom{\rule{3.33333pt}{0ex}}\forall {Z}^{t}\in {Z}^{\prime}$$$${i}_{1}^{t}-{i}_{0}^{t}+1\ge \beta ,\phantom{\rule{3.33333pt}{0ex}}\forall {Z}^{t}\in {Z}^{\prime}$$$${j}_{1}^{t}-{j}_{0}^{t}+1\ge \beta ,\phantom{\rule{3.33333pt}{0ex}}\forall {Z}^{t}\in {Z}^{\prime}$$
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- The tiles of selected elements in ${Z}^{\prime}$ are non-overlapping. For all (${Z}^{t1},{Z}^{t2}$) in ${Z}^{\prime}$:$$(({i}_{0}^{t1}-{i}_{0}^{t2})\xb7({i}_{0}^{t1}-{i}_{1}^{t2})>0)\phantom{\rule{3.33333pt}{0ex}}\wedge \phantom{\rule{3.33333pt}{0ex}}(({i}_{1}^{t1}-{i}_{0}^{t2})\xb7({i}_{1}^{t1}-{i}_{1}^{t2})>0)=1$$$$\mathbf{or}(({j}_{0}^{t1}-{j}_{0}^{t2})\xb7({j}_{0}^{t1}-{j}_{1}^{t2})>0)\phantom{\rule{3.33333pt}{0ex}}\wedge \phantom{\rule{3.33333pt}{0ex}}({j}_{1}^{t1}-{j}_{0}^{t2})\xb7({j}_{1}^{t1}-{j}_{1}^{t2})>0)=1$$

**Phase 1: Tile growth:**At each step, the size of the new tile and the corresponding LMP choice is determined by a local guidance function. The local optimizer will enumerate all the potential new tiles that can be grown and select the one that locally maximizes the guidance function value. In [11], we pre-computed a linear programming solution in which the spatial constraints and binary choice constraints are relaxed. The relaxed linear programming solution was used as an upper-bound on the optimal solution of FF-LA. For the rectangular region defined by a tile, denote the profit and cost achieved in the linear programming solution as ${P}^{*}$ and ${C}^{*}$ (the LMP choices in a tile may not be the same in the relaxed linear programming solution), and denote those achieved by a homogeneous choice of $LM{P}_{k}$ as ${P}_{k}$ and ${C}_{k}$. The score of the guidance function is computed as $\frac{{P}_{k}/{P}^{*}}{{C}_{k}/{C}^{*}}$. The local optimizer selects the new tile and LMP choice by maximizing the score. The aim of this guidance function is to make a decision to locally best approach the upper-bounding solution.

**Phase 2: LMP rearrangement:**After Phase 1 is completed, a tile-partitioning is obtained with initial LMP assignment. The goal of Phase 2 is to rearrange the LMP assignment on tiles to further improve the objective function. In [11], we used a heuristic pair-wise trading method. In each step, we enumerate a pair of tiles and explore potential LMP changes on the two tiles that can improve the objective function without violating the cost constraint. The trading continues until no further improvement can be made.

**Solution quality improvement:**Here we use an exact solver to replace the heuristic pair-wise trader in Phase 2. Since the tile-partitioning is already fixed after Phase 1, we are left with a set of fixed tiles. Thus, the spatial constraints (e.g., minimum area, shape, non-overlapping) are no longer binding or effective. This change makes the problem much easier to solve since we can treat each tile as an independent variable. In addition, since the number of tiles in the final solution is always smaller than or equal to the number of grid cells, the scale of the problem is also reduced. In practice, the problem can be efficiently solved by standard integer programming solvers (e.g., CPLEX solver [24] or dynamic programming [5]). The use of exact solvers guarantees the optimality of LMP assignment in Phase 2 and thus can always produce a higher objective function value than our previous Phase 2 approach in [11].

#### 5.4. Towards the Future: A Hybrid Framework

**Stakeholder intervention:**Stakeholders are allowed to intervene in the solution of the optimization algorithm. They can constrain both space partitioning and choices of land management practices at certain locations based on their discussion and expert knowledge, which is part of a social learning process and is hard to know beforehand using the optimization algorithm. The algorithm then guarantees that solutions at those places are fixed and not changed. Alternatively, stakeholders can directly choose a specific sub-region where decisions are difficult to make and seek suggestions from the optimization algorithm only for that region.

**Sub-problem optimization:**With stakeholder intervention, the optimization algorithm needs to solve a specific sub-problem (e.g., at a certain region) instead of the original problem (e.g., entire watershed). In order to solve the sub-problem, the algorithm needs to determine the feasibility of the sub-problem and identify a solution. In scenarios where stakeholder interventions create a non-feasible problem (e.g., insufficient budget or space to satisfy the constraints), the algorithm should suggest several modifications to stakeholder interventions so stakeholders can discuss and reconfigure the interventions to make them feasible.

**Interactivity:**The use of an optimization algorithm in collaborative geodesign requires interactions between stakeholders and the algorithm. In a geodesign session, this requires real-time or quick (e.g., several seconds to minutes) feedback/suggestions from the algorithm. To meet the processing speed requirement, the algorithm needs to be accelerated or it can optimize at a coarse resolution to allow interactive decision-making. Between sessions (e.g., overnight), the optimization algorithm is given plenty of time to search a larger space of solutions, and it can run on fine-resolution to help identify new opportunities for stakeholders to discuss in the next session.

## 6. Case Study: Seven Mile Creek Watershed

#### 6.1. Stakeholder Solution with Collaborative Geodesign

#### 6.2. Computer-Suggested Solutions with Spatial Optimization

#### 6.3. Comparison of Land Allocation Approaches

## 7. Conclusions and Future Work

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Aerial images showing farm shapes in Google Maps. (

**a**) shows rectangular-shaped farms and (

**b**) shows circular-shaped farms.

**Figure 2.**Reduction from the multiple-choice knapsack problem (MCKP) to the fragmentation-free land allocation (FF-LA). LMP: land management practices.

**Figure 4.**Overall architecture of fragmentation-free land allocation. SWAT: Soil and Water Assessment Tool. GIS: Geographic Information System. DBMS: Database Management System.

**Figure 6.**Software developed to engage stakeholders in creating and evaluating land allocations. (

**a**) Stakeholders work together to hand-craft designs using the interactive interface. (

**b**) Design performance is evaluated and returned to stakeholders for discussion. (

**c**) An example design with charts showing performances. (

**d**) Stakeholders can compare multiple designs and evaluate trade-offs.

**Figure 9.**Example of land fragmentation in integer programming solution without spatial constraints.

**Figure 10.**Fragmentation-free land allocation results with different spatial constraints. The minimum tile area and width gradually increase from (

**a**–

**c**). The total sediment of (

**a**) is 1508 ton/year, (

**b**) is 1597 ton/year and (

**c**) is 1664 ton/year.

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**MDPI and ACS Style**

Xie, Y.; Runck, B.C.; Shekhar, S.; Kne, L.; Mulla, D.; Jordan, N.; Wiringa, P. Collaborative Geodesign and Spatial Optimization for Fragmentation-Free Land Allocation. *ISPRS Int. J. Geo-Inf.* **2017**, *6*, 226.
https://doi.org/10.3390/ijgi6070226

**AMA Style**

Xie Y, Runck BC, Shekhar S, Kne L, Mulla D, Jordan N, Wiringa P. Collaborative Geodesign and Spatial Optimization for Fragmentation-Free Land Allocation. *ISPRS International Journal of Geo-Information*. 2017; 6(7):226.
https://doi.org/10.3390/ijgi6070226

**Chicago/Turabian Style**

Xie, Yiqun, Bryan C. Runck, Shashi Shekhar, Len Kne, David Mulla, Nicolas Jordan, and Peter Wiringa. 2017. "Collaborative Geodesign and Spatial Optimization for Fragmentation-Free Land Allocation" *ISPRS International Journal of Geo-Information* 6, no. 7: 226.
https://doi.org/10.3390/ijgi6070226