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# p19 | UN Humanitarian Disaster Response Hub Location (UNHDR)

```{seealso}
This problem is sourced from [Ferchtandiker2025](https://github.com/nathan-ferchtandiker/LLMs-For-Optimization-Reformulations).
```

```{note}
- We add implicit assumptions that all sets are non-empty and $a_c, C_{hc}, t_{hc} \geq 0$.
```


## Description

Strategic Hub Network Design for Disaster Response Logistics

In the aftermath of disasters such as earthquakes, floods, or cyclones, the rapid delivery of life-saving relief supplies—tents, food, medical kits, and hygiene items—is critical to reducing human suffering. However, humanitarian organizations face a logistical paradox: pre-positioning supplies in strategic hubs accelerates response times but incurs significant storage and operational costs. Balancing these trade-offs becomes even more complex when operating in politically unstable regions, under budget constraints, and with unpredictable disaster patterns.

Context and Challenges:

Storing relief items in advance near disaster-prone areas can slash delivery times from weeks to days. However, maintaining hubs is expensive, especially in remote or high-risk regions.

Organizations like the United Nations Humanitarian Response Depot (UNHRD) mitigate this by offering free storage space in global hubs (e.g., Dubai, Panama, Italy), enabling smaller NGOs to share infrastructure.

The HLSP Advantage:

Humanitarian Logistics Service Providers (HLSPs) like UNHRD pool resources across agencies, leveraging economies of scale. For example, consolidating shipments from a hub in Malaysia to cyclone-hit Philippines reduces per-unit transportation costs.

However, HLSP services for transportation are not free—costs must be carefully optimized to avoid draining limited aid budgets.

Operational Constraints:

Fixed Hubs: Some hubs (e.g., UNHRD facilities) are non-negotiable; they must be used due to pre-existing agreements or strategic importance.

Time Sensitivity: Deliveries must reach disaster zones within a strict time window (e.g., 72 hours) to prevent secondary crises like disease outbreaks.

Budget Limits: Organizations can only afford to operate a limited number of additional hubs beyond fixed ones.

Strategic Decisions
Humanitarian organizations must design a global hub network that:

Minimizes Transportation Costs:

Prioritize routes with lower per-person costs (e.g., shipping from Dubai to South Asia vs. airlifting from Europe).

Leverage cargo consolidation through HLSPs to reduce expenses.

Guarantees Coverage for All Disaster Regions:

Ensure every disaster-prone area (e.g., flood-vulnerable Bangladesh, earthquake-prone Nepal) is assigned to at least one hub.

Respects Time and Capacity Limits:

Avoid overloading hubs: the hubs have a limit on storage.

Meet strict delivery deadlines: deliveries have a strict time limit to ensure that goods arrive in time.

## Formulations
### Formulation `a` (valid)

```{seealso}
This formulation is sourced from [Ferchtandiker2025](https://github.com/nathan-ferchtandiker/LLMs-For-Optimization-Reformulations) (formulation id: efficient).
```

#### Parameters

| Name | Description | Type | Shape |
|---|---|---|---|
| `nH` | Total number of candidate hub locations | integer | *scalar* |
| `nC` | Number of disaster-prone regions | integer | *scalar* |
| `a` | Number of people affected in each disaster region | integer | `[nC]` |
| `C` | Cost of transporting relief items for one affected person from each hub to each region | continuous | `[nH, nC]` |
| `t` | Transportation time from each hub to each region | continuous | `[nH, nC]` |
| `T` | Maximum allowed transportation time from any hub to any region it serves | continuous | *scalar* |
| `n` | Maximum number of hubs that can be opened | integer | *scalar* |
| `Hf` | Indicator for each hub: 1 if the hub is fixed and must remain open, 0 otherwise | binary | `[nH]` |


#### Variables

| Name | Description | Type | Shape / Indices |
|---|---|---|---|
| `x` | Amount of supply (as a fraction of demand) transported from hub h to region c | continuous | `[nH, nC]` |
| `y` | 1 if hub h is opened, 0 otherwise | binary | `[nH]` |


#### Assumptions

| Description | Formulation | Implicit |
|---|---|---|
| The number of candidate hubs is positive. | $nH > 0$ | yes |
| The number of disaster regions is positive. | $nC > 0$ | yes |
| The number of people affected in each region is non-negative. | $a_c \geq 0 \quad \forall c \in C$ | yes |
| Transportation cost per person from each hub to each region is non-negative. | $C_{hc} \geq 0 \quad \forall h \in H, c \in C$ | yes |
| Transportation time from each hub to each region is non-negative. | $t_{hc} \geq 0 \quad \forall h \in H, c \in C$ | yes |


#### Constraints

- Each hub can only serve regions if it is open: total supply shipped from hub h is at most nC times its open indicator.

  $$ \sum_{c \in C} x_{hc} \leq |C| \, y_h \quad \forall h \in H $$

- Each disaster region's demand must be fully supplied.

  $$ \sum_{h \in H} x_{hc} = 1 \quad \forall c \in C $$

- The total number of opened hubs must not exceed the maximum.

  $$ \sum_{h \in H} y_h \leq n $$

- Fixed hubs must be opened.

  $$ y_h = 1 \quad \forall h \in H^f $$

- Per-region weighted travel time must not exceed the maximum allowed time.

  $$ \sum_{h \in H} t_{hc} \, x_{hc} \leq T \quad \forall c \in C $$

- Supply fractions are non-negative. _(implicit)_

  $$ x_{hc} \geq 0 \quad \forall h \in H, c \in C $$

#### Objective

Minimize total per-person-weighted transportation cost across all hubs and regions.

$$ \min \sum_{h \in H} \sum_{c \in C} a_c \, C_{hc} \, x_{hc} $$

### Formulation `b` (valid)

```{seealso}
This formulation is sourced from [Ferchtandiker2025](https://github.com/nathan-ferchtandiker/LLMs-For-Optimization-Reformulations) (formulation id: inefficient).
```

```{note}
- The inefficient formulation adds an assignment variable $z_{hc}$.
- The original formulation was identified by `FLARE` to be invalid. It includes the constraint $\sum_{h \in H} z_{hc} = 1$. This constraint enforces that every disaster region $c \in C$ is assigned to *exactly one* response hub $h \in H$. The efficient formulation allows a disaster region to be supplied by multiple hubs simultaneously. To make this a valid reformulation, we omit this constraint.
```


#### Parameters

| Name | Description | Type | Shape |
|---|---|---|---|
| `nH` | Total number of candidate hub locations | integer | *scalar* |
| `nC` | Number of disaster-prone regions | integer | *scalar* |
| `a` | Number of people affected in each disaster region | integer | `[nC]` |
| `C` | Cost of transporting relief items for one affected person from each hub to each region | continuous | `[nH, nC]` |
| `t` | Transportation time from each hub to each region | continuous | `[nH, nC]` |
| `T` | Maximum allowed transportation time from any hub to any region it serves | continuous | *scalar* |
| `n` | Maximum number of hubs that can be opened | integer | *scalar* |
| `Hf` | Indicator for each hub: 1 if the hub is fixed and must remain open, 0 otherwise | binary | `[nH]` |
| `M` | Big-M constant used to link the demand-fraction variable q to the assignment indicator z | continuous | *scalar* |


#### Variables

| Name | Description | Type | Shape / Indices |
|---|---|---|---|
| `q` | Fraction of region c's demand served by hub h | continuous | `[nH, nC]` |
| `z` | 1 if hub h contributes any supply to region c, 0 otherwise | binary | `[nH, nC]` |
| `y` | 1 if hub h is opened, 0 otherwise | binary | `[nH]` |


#### Assumptions

| Description | Formulation | Implicit |
|---|---|---|
| The number of candidate hubs is positive. | $nH > 0$ | yes |
| The number of disaster regions is positive. | $nC > 0$ | yes |
| The number of people affected in each region is non-negative. | $a_c \geq 0 \quad \forall c \in C$ | yes |
| Transportation cost per person from each hub to each region is non-negative. | $C_{hc} \geq 0 \quad \forall h \in H, c \in C$ | yes |
| Transportation time from each hub to each region is non-negative. | $t_{hc} \geq 0 \quad \forall h \in H, c \in C$ | yes |
| The big-M constant is positive. | $M > 0$ | no |


#### Constraints

- Big-M link: the demand fraction from hub h to region c is zero unless z_hc = 1.

  $$ q_{hc} \leq M \, z_{hc} \quad \forall h \in H, c \in C $$

- Each disaster region's demand must be fully supplied.

  $$ \sum_{h \in H} q_{hc} = 1 \quad \forall c \in C $$

- A hub can only be assigned to regions if it is open: total assignments from hub h is at most nC times its open indicator.

  $$ \sum_{c \in C} z_{hc} \leq |C| \, y_h \quad \forall h \in H $$

- The total number of opened hubs must not exceed the maximum.

  $$ \sum_{h \in H} y_h \leq n $$

- Fixed hubs must be opened.

  $$ y_h = 1 \quad \forall h \in H^f $$

- Per-region weighted travel time must not exceed the maximum allowed time.

  $$ \sum_{h \in H} t_{hc} \, q_{hc} \leq T \quad \forall c \in C $$

- Demand fractions are non-negative. _(implicit)_

  $$ q_{hc} \geq 0 \quad \forall h \in H, c \in C $$

#### Objective

Minimize total per-person-weighted transportation cost across all hubs and regions.

$$ \min \sum_{h \in H} \sum_{c \in C} a_c \, C_{hc} \, q_{hc} $$