A Warehouse Location Routing Problem

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

2. 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.

3. Mathematical models Cji = the cost per unit weight of shipping from DC j to customer i Xji= the quantity shipped from DC j to customer i M = the number of DC sites N = the number of customers Route Components

4. 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.

5. 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.

6. 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.

7. 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)

8. Mathematical Model for WLRP(cont.) (ii) Parameters: dij= distance between points i and j qi = requirement of customer i FCj = fixed cost of establishing DC j VCj = variable cost per unit throughput at DC j Tj= maximum throughput at DC j

9. Mathematical Model for WLRP(cont.) (ii) Parameters: (cont.) CTsj = unit cost of trunking from supply source s to DC j Ck = capacity of vehicle (or route) k Dk= maximum allowable length of routek CMk= cost per mile of delivery vehicle on route k

10. Mathematical Model for WLRP(cont.) (iii) Variables: Xghk= 1 if point g precedes h on route k 0 otherwise Yij= 1 if customer i is allocated to DC j 0 otherwise Zj= 1 if customer i is allocated to DC j 0 otherwise fsj= quantity shipped from supply source s to DC j

11. Mixed-integer Programming Model Objective Function:

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

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

14. Case Study Example • Customer’s Demand (Qi) = 20 units; i = 1…55 • Vehicle Capacity (C) = 120 units • Fixed Warehousing Cost (FC) = $240; j = 1…15 • Variable Warhousing Cost (VCj) = $0.74/unit; j = 1…15 • Warehouse Capacity (Tj) = 550 units; j = 1…15 • Cost Per Vehicle Mile (CM) = $1.0 • Maximum Number of Routes (K) = 11

16. Test Problem Results Travel distance = 9,290.41 miles Total weekly cost = $14,639

17. 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.

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