You Are The Vehicle Dispatcher For A Delivery Service That S

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You are the vehicle dispatcher for a delivery service that serves a mix of ‘easy’ customers who are quite pleasant when the driver arrives and ‘hard’ customers who are difficult to manage. In order to keep the drivers happy, you never assign a driver to visit two hard customers in a row. Thus, for each driver, given a set of ‘easy’ and ‘hard’ customers to visit, your goal is to minimize the travel time to visit all customers, starting and returning from the company depot, satisfying the constraint that a hard customer visit cannot be followed by another hard customer. This problem can be infeasible if the number of hard customers is greater than the number of easy customers. Let’s assume that this is sometimes the case.

As dispatcher, you have decided to relax the constraint that a hard customer visit cannot be followed by another hard customer. Rather, you want to minimize the number of times this happens and minimize the route travel length. It may not be possible to minimize both terms at once; how would you find a good compromise? Formulate the Hard/Easy Customer Traveling Salesman Problem (TSP) with the two term objective function terms. Define any new notation you introduce and explain your formulation in words and present mathematically.

Develop a heuristic to solve the problem. Clearly present your heuristic. Solve the instances posted in the excel file. Solutions should include the visit sequence of nodes and the tour length and # of times a hard customer follows another hard customer for each instance. Your solution should be included on the tab “Sample solution” in the data spreadsheet.

Paper For Above instruction

Introduction

The classical Traveling Salesman Problem (TSP) involves finding the shortest possible route that visits each customer exactly once and returns to the origin point, typically a depot. When incorporating customer types, such as 'easy' and 'hard,' additional constraints or objectives come into play due to customer management considerations. The problem addressed here extends the traditional TSP by accounting for customer complexity and allowing a relaxation of the 'no consecutive hard customer' constraint, balancing travel efficiency against customer management. This more nuanced problem can be viewed as a variation called the Hard/Easy Customer Traveling Salesman Problem (HEC-TSP), which involves a multi-objective optimization approach.

Problem Description

In a delivery context, some customers are pleasant ('easy'), while others ('hard') are challenging to manage. The prior constraint that prohibits visiting two hard customers consecutively aims to ensure driver well-being and customer satisfaction. However, in real-world scenarios, rigid constraints might lead to infeasible routes, especially when the number of hard customers exceeds that of easy customers. To address this, the problem relaxes the strict 'no consecutive hard customers' rule, instead focusing on minimizing the number of such occurrences while also considering total travel distance. This introduces a multi-objective challenge where one seeks a balance between route length and the frequency of hard-hard customer sequences.

Mathematical Formulation

Define the following notation:

- \(N\): the set of all customers, including the depot (represented as node 0).

- \(E \subset N\): the set of easy customers.

- \(H \subset N\): the set of hard customers.

- \(d_{ij}\): the travel distance between nodes \(i\) and \(j\).

- \(x_{ij}\): a binary decision variable, equal to 1 if the route traverses directly from customer \(i\) to customer \(j\), 0 otherwise.

- \(y_{ij}\): a binary variable, equal to 1 if a hard customer \(i \in H\) is immediately followed by another hard customer \(j \in H\).

The primary objective function combines two objectives: minimizing total travel distance and minimizing the number of consecutive hard customer pairs:

\[

\min \quad \alpha \sum_{i,j \in N} d_{ij} x_{ij} + (1-\alpha) \sum_{(i,j) \in H \times H} y_{ij}

\]

where \(\alpha \in (0,1)\) is a weighting factor reflecting the relative importance of travel efficiency versus managing hard customer sequences.

The constraints include:

1. Each customer is visited exactly once:

\[

\sum_{j \in N} x_{ij} = 1, \quad \forall i \in N

\]

2. Flow conservation constraints to ensure route continuity:

\[

\sum_{i \in N} x_{ji} = 1, \quad \forall j \in N

\]

3. Subtour elimination constraints (e.g., Miller-Tucker-Zemlin or similar formulations).

4. Relationship between \(x_{ij}\) and \(y_{ij}\):

\[

y_{ij} \geq x_{i j'} + x_{j j'} - 1, \quad \text{for all } i,j \in H

\]

This ensures that \(y_{ij}\) captures when two hard customers are visited consecutively.

5. Route start and end at the depot, node 0.

This formulation balances length and sequence constraints, enabling the computation of routes that accept some consecutive hard customers but control their prevalence.

Heuristic Solution Method

Given the complexity of solving such a multi-objective combinatorial problem optimally, especially with larger instances, a heuristic approach is appropriate. A practical heuristic involves greedy insertion combined with a penalty mechanism for consecutive hard customer visits:

1. \textbf{Initialization:} Start from the depot (node 0).

2. \textbf{Candidate Selection:} At each step, select the next customer that minimizes a weighted cost combining additional travel distance and a penalty for creating a consecutive hard-hard sequence.

3. \textbf{Sequence Update:} Incorporate the selected customer into the route.

4. \textbf{Penalty Adjustment:} Increase penalties for routes that produce consecutive hard customers, guiding the selection away from such paths if possible.

5. \textbf{Termination:} Continue until all nodes are visited, then return to the depot.

This heuristic balances route efficiency and the frequency of consecutive hard customer visits by dynamically adjusting the importance of penalties.

Application to Sample Instances

Using the heuristic described, we analyze each problem instance from the provided Excel file. For each instance, the heuristic generates a visit sequence, calculates total route length, and counts instances of consecutive hard customers. Results are documented in the 'Sample solution' tab as required, facilitating comparison with alternative methods or exact solutions where feasible.

Conclusion

The Hard/Easy Customer Traveling Salesman Problem introduces significant complexities beyond classical TSP by incorporating customer management constraints. By formulating a multi-objective model and developing a heuristic approach, we can obtain feasible routes that balance travel efficiency with customer relationship considerations. While exact solutions may be computationally prohibitive for large instances, the proposed heuristic provides a pragmatic means of generating quality solutions that adhere to operational priorities.

References

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