A Warehouse Location Routing Problem Jirawan Niemsakul (Jossef Perl: University of Maryland Presentation By: Mark S.Daskin: Northwestern University) & 23 November 2007Sripatum University Chonburi Campus

Introduction This paper defines the WLRP as solving the DC location and vehicle routing problem. They presented a mixed integer programming formulation and a heuristic solution method for WLRP.

Mathematical models Route Components C ji = the cost per unit weight of shipping from DC j to customer i X ji = the quantity shipped from DC j to customer i M = the number of DC sites N = the number of customers

WLRP : Warehouse Location-Routing Problem As shown by Perl (1983) Including the Transportation Location Problem (TLP), the General Transportation Problem (GTP), the Multi-Depot Vehicle Dispatch Problem (MDVDP) and the Traveling Salesman Problem (TSP). The location and expected requirements of a set of N customers are given.

WLRP : Warehouse Location-Routing Problem (cont.) Each customer is to be assigned to a regional warehouse which will supply the customer’s expected requirement. Also given is a set of M potential sites for the warehouse and the warehousing costs at each potential site.

Objective Determine the number, size and locations of the warehouse, the allocation of customers to warehouse, and the delivery routes. Minimize the total system cost (warehousing cost and transportation cost) Warehousing cost include both fixed and variable costs, while the transportation costs consist of the trunking and delivery cost.

Mathematical Model for WLRP (i) Subscripts: h, g = “point” index (customer or DC site) (1 ≤ h ≤ N+M), (1 ≤ g ≤ N+M) i = customer index (1 ≤ i ≤ N) j = DC site index (N+1 ≤ j ≤ N+M) k = route index (1 ≤ k ≤ K) s = supply source index (1 ≤ s ≤ S)

Mathematical Model for WLRP (cont.) (ii) Parameters: d ij = distance between points i and j q i = requirement of customer i FC j = fixed cost of establishing DC j VC j = variable cost per unit throughput at DC j T j = maximum throughput at DC j

Mathematical Model for WLRP (cont.) (ii) Parameters: (cont.) CT sj = unit cost of trunking from supply source s to DC j C k = capacity of vehicle (or route) k D k = maximum allowable length of route k CM k = cost per mile of delivery vehicle on route k

Mathematical Model for WLRP (cont.) (iii) Variables: X ghk = 1 if point g precedes h on route k 0otherwise Y ij = 1 if customer i is allocated to DC j 0otherwise Z j = 1 if customer i is allocated to DC j 0otherwise f sj = quantity shipped from supply source s to DC j

Mixed-integer Programming Model Objective Function:

Subject to: (i) (ii) (iii) (iv)

Subject to: (cont.) (v) (vi) (vii) (viii)

Case Study Example 1.Customer’s Demand (Q i ) = 20 units; i = 1…55 2.Vehicle Capacity (C) = 120 units 3.Fixed Warehousing Cost (FC) = $240; j = 1…15 4.Variable Warhousing Cost (VC j ) = $0.74/unit; j = 1…15 5.Warehouse Capacity (T j ) = 550 units; j = 1…15 6.Cost Per Vehicle Mile (CM) = $1.0 7.Maximum Number of Routes (K) = 11

Test Problem Results Travel distance = 9, miles Total weekly cost = $14,639

Test Problem Results Heuristics Method: In a test on a small problem, the solution provided by heuristic method was 5.2 % higher than a lower bound.