Solving the multi-commodity capacitated multi-facility Weber problem using Lagrangean relaxation and a subgradient-like algorithm

Abstract

The Multi-commodity Capacitated Multi-facility Weber Problem (MCMWP) is concerned with locating I-capacitated facilities in the plane in order to satisfy the demands of J customers for K commodities so that the total transportation cost is minimized. We propose a Lagrangean relaxation scheme and a subgradient-like algorithm based on the relaxation of the capacity and commodity bundle constraints. The resulting subproblem is a variant of the well-known Multi-facility Weber Problem and it can be solved by using column generation and branch-and-price on an equivalent set covering formulation, which is accurate but extremely inefficient. Therefore, we devise different strategies to increase the efficiency. They mainly benefit from the effective usage of the lower and upper bounds on the optimal value of the Lagrangean subproblem. On the basis of extensive computational tests, we can say that they increase the efficiency considerably and result in accurate Lagrangean heuristics.

Appendix

A.1. Subgradient-like algorithm

Algorithm 1 SL Algorithm

• 1 Initialization: Set β _ik ≔ 0, μ _ij ≔0 for all i , j and k , π ≔2, z _LB ^best ≔−∞, z _UB ^best ≔∞, Repeat steps 2–5 until the algorithm converges,

• 2 Find heuristic bound z _H ( β , μ ) with Lagrange multipliers β _ik and μ _ij . If z _H ( β , μ )> z _UB ^best , then find a valid lower bound z _LB with β _ik and μ _ij on the optimum z ^* and update z _LB ^best if necessary (ie, z _LB ^best ≔max { z _LB , z _LB ^best }). Otherwise, set z _LB = z _H ( β , μ )

• 3 Find an upper bound z _UB on z ^* If z _UB < z _UB ^best , then set z _UB ^best := z _UB ,

• 4 Update multipliers by setting and where and Set π ≔ π/2 if z _LB did not improve within the last 30 iterations,

• 5 Find a final valid lower bound z _LB with the best β _ik and μ _ij on z ^* , update z _LB ^best as necessary and output z _LB ^best and z _UB ^best .

The current lower bound values z _LB computed by the SL algorithm can be either an optimum solution of the LR subproblem z _LR ^*(β, μ) or a lower bound satisfying z _LB ⩽z _LR ^*(β, μ) for given multipliers β and μ. The upper bounds are determined by an Alternate Location Allocation (ALA)-type heuristic (ie, Cooper, 1964, 1972) which is initialized with the allocation values w _ijk =α _ijk × q _jk obtained from the solution of the LR subproblem.

When using heuristic solution values z _H (β, μ), we set z _LB =z _H (β, μ) through the steps of the algorithm. In case z _H (β, μ)>z _UB ^best occurs at some step of the algorithm, we switch back to previous multipliers and solve one of the recovery algorithms (ie CG or block norm-based lower bounding procedures) to re-update multipliers and obtain a valid lower bound on z ^*. Note that, this procedure reduces to the classical SL algorithm when z _H (β, μ)>z _UB ^best holds at each step. Otherwise, z _H (β, μ) mimics as if it is a true lower bound and the algorithm converges as usual. However, according to our computational experiments we have not observed such a case where z _H (β, μ) has exceeded z _UB ^best. At the end of the SL algorithm, we have performed one of the recovery procedures (ie CG with DC, CG with WPLD, ℓ ₁ UDAP and ℓ _∞ UDAP) in order to ensure that the algorithm produces a valid lower bound on z ^*.

A.2. Multi-commodity-capacitated alternate location allocation heuristic

Pursuing the basic ideas proposed by Cooper (1972), the MCALA heuristic is described below. Observe that once a feasible transportation plan is given, MCMWP reduces to solving I Weber problems

for each facility i=1, …, I. Here . Note that each of these I Weber problems can be solved by Weiszfeld's algorithm (1937) and its generalizations. Although the summation is taken over all customers, it considers only I _i of them, which is the size of the set defined by I _i =∣{(j, k) : w _ijk >0}∣. Clearly, ∑ _i=1 ^I I _i ⩾J × K holds since a customer can be served by more than one facility. In short, when a feasible assignment of w _ijk variables is given, the problem reduces to optimally locating single facilities with respect to I _i customers. Consequently, it is possible to tailor Cooper's (1964) ALA heuristic to produce a good feasible solution for MCMWP. We give below a formal outline of the MCALA heuristic:

Algorithm 2 MCALA Heuristic

• 1 Locate I facilities at randomly selected points x _i =( x _{i 1} x _{i 2} ) ^T for i =1, …,  I ;

• 2 For each facility i and customer j calculate the distance d ( x _i , a _j ) between them, and set the new unit transportation costs as c̄ _ijk = c _ijk d ( x _i , a _j );

• 3 Determine feasible allocations w _ijk ^* by solving the Multi-commodity Transportation Problem (MTP):

  
  
  
  

• 4 Solve I Weber problems (A.1) to relocate I facilities;

• 5 Repeat Steps 3 and 4 until either facility locations x _i =(x _i1 x _i2 ) ^T for i=1, …, I or allocations w _ijk * for i=1, …, I, j=1, …, J, and k=1, …, K remain unchanged.

In order to observe and compare the performance of our LR strategies and the lower bounds they produce, we have adapted the Sherali et al's (2002) RLT-based lower bounding approach to the multi-commodity extension of the CMWP, that is, the MCMWP. The resulting RLT formulation is presented in the following.

A.3. Derivation of the lower bounding RLT formulation

The MCMWP formulation given by (1)–(5) can be rewritten as

where the allocation variables have lower and upper bounds which can be set as l _ijk =0 and u _ijk =min {s _ik , q _jk , u _ij } for i=1, …, I; j=1, …, J; k=1, …, K and the new defined variables α _ij measure the distance between facility i and customer j. The RLT approach is applied in order to get a lower bounding linear programming formulation on the formulation given by (A.2)–(A.7), which has bilinear objective function and non-linear constraints. The non-linear constraint set is approximated by generating some supports on it.

Let the facets of the convex hull of the customer locations be defined by a set of inequalities of the form

Let us define the following bounds on α _ij :

The inequalities given above are linearized which result in the following set of constraints:

Reformulation:

• 1 Multiply each inequality given by (A.12) with each of inequality (A.7).

• 2 Multiply each inequality given by (A.8), (A.9), (A.10) by each of inequality (A.7).

• 3 Generate the following equalities:

Linearization:

The resulting constraints are linearized by substituting the bilinear terms which are defined with the following new decision variables:

Note that the notation used by Sherali et al (2002), where additional details can be found within the context of the CMWP, is maintained here. We present next the RLT-based lower bounding formulation for the MCMWP:

RLT: