Optimization Based Method for Supply Location Selection ... - CiteSeerX

> T-ASE-2008-253
T-ASE-2008-253 < extension of this problem considering random arc capacity, supply, and demand is proposed in [23]. Based on VRP formulation, a chance constrained programming model with the objective function of minimizing unsatisfied demand is reported in [24] which addresses the vehicle scheduling problem for supplying large-scale disaster areas. For emergency material delivery by air, the problems for scheduling helicopters and crews are studied and formulated hierarchically based on VRP with the objective to minimize unsatisfied demand or maximize the demand satisfied [25], [26]. In addition, a hybrid problem integrating the multi-commodity network flow problem with vehicle routing problem is studied in [27]-[30] to plan the coordinated wounded evacuation and material distribution. The objective is to minimize a weighted sum of unsatisfied demands and time delay. Due to the distinctive features of emergency supply problems, it is desirable to consider the supply location selection synergistically with the vehicles scheduling and routing problem (SLSR) to meet the critical demands in specified time windows as required in many practical applications. Furthermore, every supply source should be able to provide a commodity to any customer such that the supply system can make full use of distributed warehouses. Each demand should be satisfied within a time window, and the congestion caused by heavy traffic should also be considered. The above described SLSR problem is NP-hard [31]. General integer programming methods such as that used in CPLEX software package can only solve small to medium size problems in practical computational time. It is even difficult to find a feasible solution for SLSR due to its complicated constraints and the problem size. This causes another dimension of difficulties since finding a feasible solution is often the first necessary step for many traditional methods. Lagrangian relaxation (LR) is one of the most efficient and effective method for solving complex optimization problems with decomposable structures [36]-[43]. The main idea of this approach is to relax the system wide coupling constraints and to form a two-level optimization framework. The decomposed subproblems are solved individually at low level with much less computational efforts and the Lagrange multipliers are updated at the high level. Since the dual solution is generally infeasible. That is, the once relaxed constraints are not satisfied, one of the most challenging issues for Lagrangian relaxation based approaches is to obtain a feasible solution based on the optimal dual solution in good quality. To address the above issue, a new method is developed in this paper based on the successive subproblem solving approach [33], [34] in LR framework. The main idea of this approach is to relax the coupling location selection constraint and flow capacity constraints, and to solve individual sub-problems successively. A salient feature of the new method is that the near optimal solution is obtained systematically by adding the once relaxed constraints back into the dual problem successively in feasibility iterations. Convergence proof of the new algorithm is provided and its properties are presented in the paper. Numerical test results for

2 the problems with transportation networks up to 180 nodes and 6 commodities show that the new method is effective and the near optimal solutions are obtained consistently. In comparison with the general integer programming method, the new method is much more efficient. The rest of this paper is organized as follows. Section II presents the mathematical formulation for the supply location selection and routing problem. The new method is developed in Section III and its convergence proof is given in the Appendix. The numerical test results are provided in Section IV. A short discussion on the results and concluding remarks then follow. II. PROBLEM FORMULATION Given a network topology represented by a directional graph whose arcs associate with nonnegative capacity, transit time, and cost is shown in Fig. 1. The available sources and sinks are labeled as a set of nodes in the network. Without losing generality, assume there is no direct link between the source nodes (or sink nodes) since this case can be equivalently transformed by adding virtual nodes and virtual arcs [35]. Our goal is to select a set of source nodes as the supplying warehouses and to schedule the vehicle routing to minimize the total transportation distances (or time) while the supply of demand quantities under certain deadlines is guaranteed. The assumptions in our formulation are listed as follows: 1) each source can accommodate any type of commodity, and can supply any sink; 2) each source is assumed to have adequate capacity; 3) no storage or buffering delay at intermediate nodes; 4) the transportation capacity is sufficient. ( u ,τ , c )

Fig. 1: Map Topology. Available source nodes: 1-3; sink nodes: 7-9; intermediate nodes: 4-6. Each arc is with capacity u, transit time τ and cost c.

Assumption 1) ensures the sufficient use of the commodities in sources; Assumptions 2) and 3) are derived from the emergency response requirement; e.g., the temporary warehouses are constructed in open field, and the emergent delivery may go through an unfriendly environment. Assumption 4 means that there is sufficient transportation capacity (number of available shipping vehicles) so that a feasible solution can be found to meet the demands for various commodities in the required time windows. For convenience of presentation we define the following notations. Parameters: , the sets of nodes and arcs of the network respectively, where S = {s | 1 ≤ s ≤ S } , the set of source (supply) nodes; D = {d || N − D | + 1 ≤ d ≤ D } ,

the set of disaster areas;

> T-ASE-2008-253 < K = {k | 1 ≤ k ≤ K

3

} , the set of commodities;

bs = 0,1

τij, the transit time over arc ∈A; uij, the capacity or throughput of arc ∈A; cijk , the unit cost of commodity k over arc i , j ∈ A , possibly the traveling time or distance; rdk, the demand for commodity k from disaster area d; tdk, the deadline of providing commodity k for disaster area d, for convenience, we denote T = Max ( tdk ) ; d ∈D , k∈K

bmax, the maximal number of source nodes that can be selected from S; M: a sufficiently large constant; Decision variables: xijk ( t ) , the quantity of commodity k in terms of shipping vehicles entering arc i , j ∈ A at time t ∈ {0, , T } ; x is the vector for all the commodities while xk is the vector for commodity k; bs , location selection variable, which is set to be 1 if source s is selected, 0 otherwise; b is the vector for all the locations. The total transportation distance (or time) is used to represent the effectiveness of locations selected and the vehicle routing and scheduling decisions from the selected sources. The objective of the problem is formulated as min x ,b

T

∑ ∑ ∑ cijk xijk (t ) .

k ∈K i , j ∈ A t = 0

(1)

∀s ∈ S

,

(t ) ∈ Z

(7) ,

xijk

( t ) ≥ 0,

xijk

(t ) = 0

∀ i , j ∈ A , k ∈ K , t > T − τ ij ,

xijk

(t ) = 0

∀ k ∈ K , i , j ∈ A, i ∈ D ∨ j ∈ S , t ≤ T

xijk

∀ k ∈ K , i , j ∈ A, t ≤ T

(8) (9) .

(10)

By referring to the three characteristics of the SLSR problem listed in Section 1, constraints (2) - (5) guarantee to rapidly establish the supply chain, constraint (4) guarantees timely supply of the materials to the disaster areas, and constraint (3) takes into account the congestion caused by transportation conditions and traffic. The fleet routing and scheduling scenarios can be constructed by a simple heuristic. It should be noted that practically the model is extendable to incorporate the following considerations and the methodology developed in this paper will be still applicable: 1. If transportation capacity is not sufficient, constraints (4) are removed, and the objective function (1) is replaced by minimizing total weighted unsatisfied demand and transportation distance (or time). 2. If the commodities in a location are limited, then constraints (5) are modified as ∑ j∈N − S ∑ t ≤T xsjk ( t ) ≤ wsk bs , ∀s ∈ S , k ∈ K , s, j ∈A

where wsk is the quantity of commodity k available at location s. III. THE SUCCESSIVE SUBPROBLEM SOLVING BASED METHOD IN LAGRANGIAN RELAXATION FRAMEWORK

The constraints are: Lagrangian relaxation framework is applied for solving the 1) Dynamic flow conservation similar to those in network above SLSR problem. Clearly the cost function (1) is flow problem. Buffers in intermediate nodes are not allowed. decomposable in terms of decision variables. The system-wide coupling constraints (3), (5), and (6) are additive and also ∑ x kji ( t ) − ∑ xijk ( t − τ ij ) = 0 , (2) i∈ N − S i∈ N − D decomposable. In considering the possible relaxation schemes, j ,i ∈ A i, j ∈ A if the location constraints (5) and (6) are relaxed, the dual ∀j ∈ N − S − D, k ∈ K , t ≤ T 2) Capacity constraint. We define arc capacity as the problem is decomposable with respect to location selection maximal number of vehicles allowed to traverse the arc in the variable. However, the dual subproblem in this case is a multi-commodity dynamic network flow problem that needs transit time window τij. significant computational effort. On the other hand, if t +τ ij −1 (3) constraints (3), (5), and (6) are all relaxed, the dual problem will xijk ( t ' ) ≤ u ij ∀ i , j ∈ A , t ≤ T − τ ij , ∑ ∑ k∈K t ' = t be decomposed in terms of location selection and commodities, 3) Demand. These constraints ensure that the demand for and all subproblems will become much easier to solve. The commodity k with deadline tdk at each disaster area d must be feasible solution of the primal problem, however, is difficult to satisfied, hence expressed as construct from the dual solution obtained since constraints (5) t dk and (6) are hard to satisfy simultaneously by heuristics due to the k , (4) ∑ ∑ xid ( t ) = rdk ∀ d ∈ D , k ∈ K strong connection between (5) and (6). i∈ N − D t = 0 To obtain a systematic and scalable solution, we develop a 4) Limits on warehouse selection. The constraints indicate that a location with outflows must be selected but the maximum new method based on the successive subproblem solving approach (SSS) in Lagrangian relaxation (LR) framework allowable number of locations to be selected is bmax. T proposed in our previous work [33], [34]. The main idea is to , (5) k ∑ ∑ x sj ( t ) ≤ Mbs ∀ s ∈ S , k ∈ K relax the coupling flow capacity constraints (3) and location j∈ N − S t = 0 s, j ∈A selection constraint (6), and to solve the individual subproblems S successively. In this way, the subproblems coupled by the . (6) ∑ bs ≤ bmax location selection constraints can still be solved individually. s =1 The new method actually includes two algorithms: 5) Flow constraints. These constraints ensure that the flows after deadline are of no use, and there should be no outflow Algorithm 1 and Algorithm 2. Algorithm 1 is applied in our earlier work to deal with non-decomposable dual problem, and from sinks and no inflow to sources.

> T-ASE-2008-253
0 for ∀ k < n based on Algorithm 2. Therefore with (28) and (29)

=

⎡μin ⎣

+s

n

n

+ g in ⎤⎦

=

= μid + ∑ s k g ik ≥ k =0

Since

g ik

μin

+s

n

∑ sk

n

(33)

k =0

is integer and μid ≥ 0 . If g in > 0 for any i with

n → +∞ , by (33) we have n n n n n k k ∑ μ i g~i ≥ ∑ μ i ≥ ∑ ∑ s ≥ ∑ s → +∞ . i∉I nf

i∉I nf

i∉I nf k =0

(34)

k =0

Therefore, the first two items of equation (32) are bounded, and the third one goes to infinity for n → +∞ . QED

{ } 0≤n
0 for n→+∞, that is, g~ n ≤ 0 cannot be satisfied in Algorithm 2, then

Thus we have lim I nf = {1,..., m1 + 1} .

n

+∞ generated in Algorithm 2 converges and is bounded.

(37)

QED Therefore, all the once relaxed constraints will be added back to the feasibility problem (FP-k), and a feasible solution of primal solution will be obtained. Theorem 2: The optimal primal solution x* can be obtained by Algorithm 2 under the condition that the surrogate dual cost * * n L ≤ J for 0≤n T-ASE-2008-253
0

and g i ( x(μ)) < 0 . Then the multipliers are updated for the next iteration based on (28) as: μi′ = [μi + s g i ( x(μ ))]+ < μi .

(42)

This contradicts with (41). Hence, (39) holds.

{ }

Let the surrogate dual cost series L n 0≤n T-ASE-2008-253