Lecture 24: Location Routing Problem

General Description

Precise Modeling

A typical Location Routing Problem is defined on a directed graph \(G=(D,C,A)\), where \(D\) represents set of depot nodes, node set \(C\) represents customer nodes, and arc set \(A\) represents the set of arcs connecting these nodes. Each depot node \(d \in D\) has an associated set of delivery vehicles \(V_d\), capacity \(q_d\), as well as a fixed cost \(\pi_f^d\) . Further, each customer node \(c \in C\) has a demand \(q_c\) that must be fulfilled from one of the depot nodes via its delivery fleet, wherein each vehicle has a fixed cost \(\pi_f^v\), capacity \(q_v\) and speed \(s_v\). Given a traversal cost of \(\pi_o d_{ij}\) on arc \((i,j) \in A\), the objective of a Location Routing Problem is to develop least cost routes from select distribution facilities using select vehicles such that every customer node is visited exactly once while satisfying all logistic constraints.

Objective:

\[ \min z = \sum_{d \in D} y_d(\pi_f^d + \sum_{v \in V_d} \sum_{c \in H(d)} \pi_f^v x^v_{dc} + \eta \sum_{v \in V_d} \sum_{(i,j) \in A} \pi_od_{ij}x^v_{ij}) \]

Subject to:

\[\begin{split} \begin{aligned} \sum_{d \in D}\sum_{v \in V_d}\sum_{j \in H(c)} x^v_{cj} & = 1 & \ \forall \ c \in C \\ \sum_{i \in T(j)} x^v_{ij} & = \sum_{k \in H(j)} x^v_{jk} & \ \forall \ j \in N, \ v \in V_d, \ d \in D \\ \sum_{c \in H(d)} x^v_{dc} & \leq 1 & \forall \ v \in V_d, \ d \in D \\ \sum_{v \in V_d}\sum_{j \in H(d)} x^v_{dj} & \leq |V_d| & \ \forall \ d \in D \\ \sum_{c \in C}\sum_{v \in V_d}\sum_{j \in H(c)} x^v_{cj}q_c & \leq q_dy_d & \ \forall \ d \in D \\ z^v_k & \leq x^v_{ck}(z^v_c - q_c) + (1 - x^v_{ck})\text{M} & \forall \ c \in C, \ k \in H(c), \ v \in V_d, \ d \in D \\ z^v_c & \leq q_v & \forall \ c \in C, \ v \in V_d, \ d \in D \\ z^v_d & = 0 & \forall \ v \in V_d, \ d \in D \\ x^v_{ij} & \in \{0,1\} & \ \forall \ (i,j) \in A, \ v \in V_d, \ d \in D \\ y_d & \in \{0,1\} & \ \forall \ d \in D \\ z^v_c & \in \mathbb{R}_+ & \forall \ c \in C, \ v \in V_d, \ d \in D \end{aligned} \end{split}\]

Here, \(x^v_{ij}\) represents traversal of arc \((i,j)\) by vehicle \(v\), and \(y_d\) represents depot node \(d\) use. Note, \(\eta\) factorises the operational costs in the form of net present value.

Note,

• first constraint is the customer service constraint, requiring every customer node to visited exactly once by one of the vehicles

• second constraint is the flow conservation constraint, necessitating incoming flow for vehicle \(v\) at node \(j\) to match the outgoing flow

• third constraint is the vehicle use constraint, ensuring that each vehicle only makes at most one delivery tour

• fourth constraint is the fleet size constraint (note that this is redundant due to the third constraint)

• fifth constraint is the depot capacity constraint, ensuring that the demand satisfied via a particular depot is less than its capacity

• sixth constraint is the load balance constraint, managing vehicle load on all the nodes based on the order of customer visit

• seventh constraint is the vehicle capacity constraint

• eight constraint is the boundary condition constraint, ensuring that vehicle arrives back empty at the depot node

Note

Just like the Vehicle Routing Problem (VRP), the Location Routing Problem (LRP) has numerous variants, each addressing specific constraints or objectives. Specifically, in addition to all the variants discussed for VRP, LRP variants could include multiple facilities or even multiple echelons (hierarchy of facilities). To address scuh sophisiticated variants of LRP, we typically deploy metaheuristic algorithms for practical purposes (discussed in the next module).

Approximation

The conventional approach to modelling routing problems involves use of the discrete formulation method rendering a representative mathematical model. This approach, while detailed, necessitates the use of advanced solution techniques due to inherent model complexities. Such an intensive approach is justified for operational planning, wherein decision-makers have exact information and thus require a precise plan to execute specific last-mile distribution tasks. However, for strategic planning, wherein the primary goal is to develop broad plan based on representative rather than exact information, the level of precision offered by the conventional approach becomes redundant. To this end, continuous approximation (CA) method offers a practical alternative with use of continuous density functions that estimate parameters approximately. In particular, by abstracting discrete information into continuous function forms, this approach strikes a balance between estimation accuracy and computational effort, facilitating agile and efficient decision-making. To do so, CA-based routing frameworks assume a simplified distribution environment. Specifically, these frameworks consider a logistic operator catering to customers having uniformly dispersed random locations and fixed unitary demand from a distribution facility located sufficiently far from the service region, rendering total delivery tour length of

\[ L = (2\rho + kc/\sqrt{\delta})mf \]

where,

• \(\rho\) : average distance between distirbution facility and customers

• \(c\) : number of customers per delivery tour

• \(m\) : number of delivery tours

• \(f\) : number of delivery vehicles

• \(\delta\) : customer density

• \(k\) : constant

Objective:

\[ \min z = \sum_{d \in D} y_d(\pi_f^d + \pi_f^vf_d + \eta\pi_o(2\rho_d + kc_d / \sqrt{\delta})m_df_d) \]

Subject to:

\[\begin{split} \begin{aligned} f_d & \leq |V_d| & \ \forall \ d \in D \\ m_d & \leq m_v & \ \forall \ d \in D \\ c_d & \leq q_v & \ \forall \ d \in D \\ c_dm_df_d & \leq q_d & \ \forall \ d \in D \\ \sum_{d \in D}c_dm_df_dy_d & \geq |N| \\ c_d, m_d, f_d & \in \mathbf{Z}_+ & \ \forall \ d \in D \\ y_d & \in \{0,1\} \end{aligned} \end{split}\]

Example

Amazon plans to serve 12000 customers in a service region of size 369.34 \(\text{km}^2\) from the following potential distribution facilities (see Table below) using a fleet of diesel vans with a purchase cost of ₹6L, operational cost of ₹35 per km, and a capacity of 200 customers, each, over a period of 7 years, each with 330 working days. Given that a delivery vehicle can do at most 3 tours per day, owing to temporal constraints (customer time-windows, driver work hours, distribution facility operational hours, etc.), which facility should Amazon choose to operate from?

Table 1. Potential Distribution Facility Locations

Location	Fixed Cost (in ₹cr)	Distance from Service Region (in km)	Capacity (in customers)
Location #1	75	1	3000
Location #2	50	5	10000
Location #3	10	20	30000

1. Formulate the objective function.

\[\begin{split} \begin{aligned} \eta & = ((1 - 1.03 ^ {-7})/ 0.03) * 330 = 2056 \\ \sqrt\delta & = \sqrt{12000/ 369.34} = 5.7 \end{aligned} \end{split}\]

\[ \min z = y_1(75 \times 10^7 + 2056 \times 35(2 \times 1 + 0.57c_1/5.7)m_1f_1) + y_2(50 \times 10^7 + 2056 \times 35(2 \times 5 + 0.57c_2/5.7)m_2f_2) + y_3(10 \times 10^7 + 2056 \times 35(2 \times 20 + 0.57c_3/5.7)m_3f_3) \]

Rendering,

\[ \min z = y_1(75 \times 10^7 + 6 \times 10^5 f_1 + (143920 + 7196c_1)m_1f_1) + y_2(50 \times 10^7 + 6 \times 10^5 f_2 + (719600 + 7196c_2)m_2f_2) + y_3(10 \times 10^7 + 6 \times 10^5 f_3 + (2878400 + 7196c_3)m_3f_3) \]

2. Formulate the constraints.

\[\begin{split} \begin{aligned} m_1 & \leq 3 \\ m_2 & \leq 3 \\ m_3 & \leq 3 \\ c_1 & \leq 200 \\ c_2 & \leq 200 \\ c_3 & \leq 200 \\ c_1m_1f_1 & \leq 3000 \\ c_2m_2f_2 & \leq 10000 \\ c_3m_3f_3 & \leq 30000 \\ c_1m_1f_1y_1 + c_2m_2f_2y_2 + c_3m_3f_3y_3 & \geq 12000 \\ c_d, m_d, f_d & \in \mathbf{Z}_+ \ \forall d \in D \\ y_d & \in \{0,1\} \\ \end{aligned} \end{split}\]

3. Solve the above optimisation model using a spreadsheet to find the optimal solution.

Location	\(y_d\)	\(f_d\)	\(m_d\)	\(c_d\)
1	0	0	0	0
2	0	0	0	0
3	1	20	3	200