# A Risk-Based Location-Allocation Approach for Weapon Logistics

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## Abstract

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## 1. Introduction

## 2. Literature Review

#### 2.1. Facility Location Problems

#### 2.2. Overview of Military Optimization Problems

## 3. Mathematical Model

- N is the set of one or more candidate supply locations.
- M is the set of one or more demand points.
- P is the set of one or more product types.
- ${\mathrm{d}}_{\mathrm{jk}}$ is the quantity of product type $\mathrm{k}\in \mathrm{P}$ demanded by demand point $\mathrm{j}\in \mathrm{M}$.
- ${\mathrm{e}}_{\mathrm{ij}}$ is the distance between candidate location $\mathrm{i}\in \mathrm{N}$ and demand point $\mathrm{j}\in \mathrm{M}$.
- ${\mathrm{x}}_{\mathrm{ijk}}$ is the number of units of ${\mathrm{d}}_{\mathrm{jk}}$ supplied by candidate location $\mathrm{i}\in \mathrm{N}$; this variable is a decision variable in the model.
- ${\mathrm{y}}_{\mathrm{ijk}}$ is the number of units of ${\mathrm{d}}_{\mathrm{jk}}$ that would have been supplied by candidate location $\mathrm{i}\in \mathrm{N}$ if there was sufficient stock of product type $\mathrm{k}\in \mathrm{P}$; this variable is also a decision variable in the model.
- ${\mathrm{s}}_{\mathrm{k}}$ is the available stock of product type $\mathrm{k}\in \mathrm{P}$; that is, the number of units available for allocation to demand points.
- ${\mathrm{s}}_{\mathrm{jk}}$ is the amount of stock available for supplying point $\mathrm{j}\in \mathrm{M}$’s demand of product type $\mathrm{k}\in \mathrm{P}$.
- ${\mathrm{z}}_{\mathrm{k}}$ is equal to 1 if there is sufficient stock to supply the entire demand of product type $\mathrm{k}\in \mathrm{P}$; and is equal to 0 if not.
- ${\mathrm{f}}_{\mathrm{jk}}$ is the number of units of ${\mathrm{d}}_{\mathrm{jk}}$ that could not be supplied because of insufficient stock.$${\mathrm{f}}_{\mathrm{jk}}={\mathrm{d}}_{\mathrm{jk}}-{\mathrm{s}}_{\mathrm{jk}}$$
- ${\mathrm{r}}_{\mathrm{ij}}$ is the risk of allocating a unit of demand of demand point $\mathrm{j}\in \mathrm{M}$ to candidate location $\mathrm{i}\in \mathrm{N}$. This parameter is used as a weight that increases the cost of risky locations. Thus, if one location is riskier than another location, the cost of allocation increases, and thus the location becomes less likely to be allocated any weapons.
- ${\mathrm{u}}_{\mathrm{ik}}$ is the setup cost of allocating one unit of product type $\mathrm{k}\in \mathrm{P}$ to candidate location $\mathrm{i}\in \mathrm{N}$.
- ${\mathrm{p}}_{\mathrm{ik}}$ is the maximum quantity of units of product type $\mathrm{k}\in \mathrm{P}$ allowed to be allocated to candidate location $\mathrm{i}\in \mathrm{N}$.$$\begin{array}{r}\mathrm{Min}{\displaystyle \sum _{\mathrm{i}\in \mathrm{N}}\sum _{\mathrm{j}\in \mathrm{M}}\sum _{\mathrm{k}\in \mathrm{P}}}{\mathrm{x}}_{\mathrm{ijk}}{\mathrm{e}}_{\mathrm{ij}}{\mathrm{r}}_{\mathrm{ij}}+{\displaystyle \sum _{\mathrm{i}\in \mathrm{N}}\sum _{\mathrm{j}\in \mathrm{M}}\sum _{\mathrm{ik}\in \mathrm{P}}}{\mathrm{x}}_{\mathrm{ijk}}{\mathrm{u}}_{\mathrm{ik}}\\ +{\displaystyle \sum _{\mathrm{i}\in \mathrm{N}}\sum _{\mathrm{j}\in \mathrm{M}}\sum _{\mathrm{k}\in \mathrm{P}}}{\mathrm{y}}_{\mathrm{ijk}}{\mathrm{e}}_{\mathrm{ij}}{\mathrm{r}}_{\mathrm{ij}}+{\displaystyle \sum _{\mathrm{k}\in \mathrm{P}}\sum _{\mathrm{j}\in \mathrm{M}}\sum _{\mathrm{k}\in \mathrm{P}}}{\mathrm{y}}_{\mathrm{ijk}}{\mathrm{u}}_{\mathrm{ik}}\end{array}$$Subject to the following:$$\begin{array}{c}{\displaystyle \sum _{\mathrm{j}\in \mathrm{M}}}{\mathrm{x}}_{\mathrm{ijk}}+{\displaystyle \sum _{\mathrm{j}\in \mathrm{M}}}{\mathrm{y}}_{\mathrm{ijk}}\le {\mathrm{p}}_{\mathrm{ik}};\forall \mathrm{i}\in \mathrm{N},\forall \mathrm{k}\in \mathrm{P}\end{array}$$$$\begin{array}{c}{\displaystyle \sum _{\mathrm{j}\in \mathrm{M}}}{\mathrm{x}}_{\mathrm{ijk}}=\left(1-{\mathrm{z}}_{\mathrm{k}}\right){\mathrm{d}}_{\mathrm{jk}}+{\mathrm{z}}_{\mathrm{k}}{\mathrm{s}}_{\mathrm{jk}};\forall \mathrm{i}\in \mathrm{N},\forall \mathrm{k}\in \mathrm{P}\end{array}$$$$\begin{array}{c}{\displaystyle \sum _{\mathrm{i}\in \mathrm{N}}\sum _{\mathrm{j}\in \mathrm{M}}\sum _{\mathrm{k}\in \mathrm{P}}}{\mathrm{y}}_{\mathrm{ijk}}={\mathrm{z}}_{\mathrm{k}}{\mathrm{f}}_{\mathrm{jk}};\forall \mathrm{i}\in \mathrm{N},\forall \mathrm{j}\in \mathrm{M},\forall \mathrm{k}\in \mathrm{P}\end{array}$$$$\begin{array}{c}{\mathrm{x}}_{\mathrm{ijk}},{\mathrm{y}}_{\mathrm{ijk}}\mathrm{integer};{\mathrm{x}}_{\mathrm{ijk}},{\mathrm{y}}_{\mathrm{ijk}}\ge 0\end{array}$$

## 4. Case Study

#### 4.1. Data Sources

#### 4.2. Assumptions

#### 4.2.1. Attack Scenarios

#### 4.2.2. Nodes and Weapon Types

#### 4.2.3. Supply

#### 4.2.4. Demands

#### 4.2.5. Risks

#### 4.2.6. Setup Costs

#### 4.2.7. Supply City Capacities

## 5. Solution Method

## 6. Results and Discussion

#### 6.1. Solving Different Attack Scenarios

#### 6.2. Limitations and Potential Improvements

#### 6.3. Military Perspective Insights

## 7. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Map of Turkey; for higher resolution, please visit: https://commons.wikimedia.org/wiki/File:Turkey,_administrative_divisions_-_de.svg.

Abbreviation | Weapon Type | |
---|---|---|

Category | Weapon | |

AFV | MBT | Main Battle Tanks |

AIFV | Armored Infantry Fighting Vehicle | |

APC | Armored Personnel Carrier | |

ARV | Armored Recovery Vehicles | |

RECCE | Reconnaissance | |

AT | MSL | Missiles |

MSL SP | Self-propelled Missiles | |

RCL | Recoilless Launchers | |

GUNS | Guns | |

ARTY | SP | Self-Propelled Artillery |

TOWED | Towed Artillery | |

MOR | Mortars | |

MRL | Multiple Rocket Launchers |

**Table 2.**Turkey’s capacity of weapon types. For more information on data sources, please refer to Section 4.1 (Data Sources).

Weapon Type | Stock |
---|---|

Turkey: AFV: AIFV | 650 |

Turkey: AFV: APC | 3643 |

Turkey: AFV: ARV | 0 |

Turkey: AT: GUNS | 0 |

Turkey: AFV: MBT | 2504 |

Turkey: ARTY: MOR | 5813 |

Turkey: ARTY: MRL | 146 |

Turkey: AT: MSL | 1363 |

Turkey: AT: MSL SP | 365 |

Turkey: AT: RCL | 3869 |

Turkey: AFV: RECCE | 320 |

Turkey: ARTY: SP | 1133 |

Turkey: ARTY: TOWED | 760 |

Turkey: Land Forces | 402,000 |

AĞRI | ARDAHAN | ARTVİN | EDİRNE | GAZİANTEP | HAKKARİ | HATAY | IĞDIR | KARS | KİLİS | KIRKLARELİ | MARDİN | ŞANLIURFA | ŞIRNAK | VAN | TOTAL | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

AIFV | 32 | 24 | 24 | 478 | 408 | 324 | 408 | 236 | 57 | 408 | 80 | 408 | 408 | 529 | 203 | 4027 |

APC | 43 | 63 | 63 | 2614 | 250 | 868 | 250 | 257 | 106 | 250 | 64 | 250 | 250 | 905 | 214 | 6447 |

ARV | 0 | 0 | 0 | 0 | 0 | 108 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 107 | 0 | 215 |

GUNS | 0 | 15 | 15 | 63 | 0 | 0 | 0 | 0 | 20 | 0 | 63 | 0 | 0 | 0 | 0 | 176 |

MBT | 36 | 41 | 41 | 1394 | 990 | 824 | 990 | 591 | 78 | 990 | 40 | 990 | 990 | 990 | 554 | 9539 |

MOR | 4 | 21 | 21 | 2427 | 68 | 2141 | 68 | 1671 | 26 | 68 | 108 | 68 | 68 | 544 | 1666 | 8969 |

MRL | 17 | 12 | 12 | 159 | 83 | 495 | 83 | 509 | 30 | 83 | 12 | 83 | 83 | 84 | 492 | 2237 |

MSL | 0 | 10 | 0 | 0 | 730 | 0 | 365 | 0 | 0 | 365 | 0 | 365 | 365 | 365 | 0 | 2565 |

MSL SP | 7 | 0 | 0 | 612 | 68 | 0 | 68 | 7 | 8 | 68 | 12 | 68 | 68 | 69 | 0 | 1055 |

RCL | 0 | 0 | 0 | 4508 | 0 | 66 | 0 | 67 | 0 | 0 | 0 | 0 | 0 | 0 | 66 | 4707 |

RECCE | 0 | 1 | 1 | 239 | 588 | 48 | 0 | 12 | 2 | 0 | 10 | 0 | 0 | 36 | 11 | 948 |

SP | 12 | 22 | 22 | 611 | 83 | 121 | 83 | 110 | 36 | 83 | 24 | 83 | 83 | 108 | 97 | 1578 |

TOWED | 44 | 23 | 23 | 565 | 338 | 706 | 338 | 720 | 67 | 338 | 12 | 338 | 338 | 369 | 676 | 4895 |

Force | 13,950 | 5916 | 5916 | 101,650 | 36,666 | 143,666 | 36,666 | 130,616 | 19,867 | 36,666 | 8150 | 36,666 | 36,666 | 63,667 | 116,667 | 793,395 |

Ankara | Bursa | Batman | |
---|---|---|---|

MBT | 658 | 366 | 88 |

SP | 109 | 61 | 15 |

TOWED | 338 | 188 | 45 |

Problem Description and Solution Results | |

Solution Time | ~14 min |

# Decision Variables | 27,720 variables |

# Bordering Cities | 15 |

# Supply Cities | 66 |

Total Cost | 755,210,884.00 |

Solution Infrastructure | |

Operating System | Windows 7 SP1 |

RAM | 3 GB |

CPU | Intel Core 2 Duo |

Solver | CBC Solver |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Çetinkaya, C.; Haffar, S.
A Risk-Based Location-Allocation Approach for Weapon Logistics. *Logistics* **2018**, *2*, 9.
https://doi.org/10.3390/logistics2020009

**AMA Style**

Çetinkaya C, Haffar S.
A Risk-Based Location-Allocation Approach for Weapon Logistics. *Logistics*. 2018; 2(2):9.
https://doi.org/10.3390/logistics2020009

**Chicago/Turabian Style**

Çetinkaya, Cihan, and Samer Haffar.
2018. "A Risk-Based Location-Allocation Approach for Weapon Logistics" *Logistics* 2, no. 2: 9.
https://doi.org/10.3390/logistics2020009