Approaches to Solve Mid_cplp Problem: Theoretical Framework and Empirical Investigation

In two-stage warehouse location problem, goods are moved from plants to warehouses at stage-1 (which are larger sized warehouses), and from there to warehouses at stage-2 (which are smaller sized warehouses); and finally to the markets. We aim to minimize the sum of location costs of the warehouses at stage-1 and stage-2; plus the total distribution cost of goods to the markets. In this paper two-stage capacitated warehouse location problem (TSCWLP) is vertically decomposed into the smaller problems, which is attained by relaxing the associated flow balance constraints. This leads to three different versions of Capacitated Plant Location Problem (CPLP) referred as RHS_CPLP, MID_CPLP and LHS_CPLP (Verma and Sharma [1]). In this paper MID_CPLP is reduced to RHS_CPLP and a single constraint 0-1 Knapsack problem by relaxing a difficult constraint. Interesting results and conjectures are given. Later two more valid constraints are added to MID_CPLP which are relaxed further to get additional results.


Introduction and Literature Survey
In country like large geographical area, such as India, distribution of goods such as fertilizer, cement, edible oil, sugar, food grains etc, is close in stages.From large sized plants, goods are moved to larger sized warehouses, from larger sized warehouses to smaller sized warehouses and finally to markets/customers.It is to be decided that where to locate warehouses (among potential locations for large sized warehouses as well as smaller sized warehouses).It is required to minimize sum total of cost of location of warehouse and cost of distribution of goods.Stage wise location-distribution has been attempted by Geoffrion and Graves [2], Sharma [3], Sharma and Berry [4] and Verma and Sharma [5].In this paper we attempt a two-stage warehouse location problem and alongside give a novel method to solve MID_CPLP that results due to vertical decomposition developed for solution.Facility location problems form an important class of integer programming problems, with application in the distribution and transportation industries.In any organization, one of the most important strategic decisions is locating facilities viz.factories, plants or warehouses and their suitable stages.While locating a facility, the most prioritized criterion is a good service level.However, achievement of an economic optimality is also a key decisive factor.Based on the number of stages between the producing facility and the market, there are different types of facility location problems like plant location problem, single-stage/two-stage warehouse location problem.A review of the literature on location problems can be found in (Sahin and Sural [6], ReVelle and Eiselt [7], ReVelle, Eiselt and Daskin [8] and Brandeau and Chiu [9]).Multistage warehouse location problems are frequently occurring in real life; see (Geoffrion and Graves [2], Sharma [3] and Sharma [10]).TSCWLP problem has been studied by Verma and Sharma [1].In Section 2, we give formulation of TSCWLP, few relaxations of MID_CPLP; and also give a few results.In the Section 3 we give empirical investigation and related statistical significance.Finally in Section 4 we conclude.

Problem Formulation of TSCWLP
In this section, we propose the formulation of TSCWLP using the style of Sharma [3], Sharma and Namdeo [11], Sharma and Sharma [12].Verma and Sharma [5] developed a variety of constraints that link real and 0-1 integer variables.They have also developed some strong constraints based on Sharma and Berry [4].

j
: set of potential warehouse points at stage 2; 1, ,

Definition of Constants
D k : Demand for the commodity at market "k" ∑ , Demand at market "k" as a fraction of total market demand S h : Supply available at plant "h" ∑ , Supply available at plant 'h' as a fraction of the total market demand fws1 i : Fixed cost of locating a warehouse at "i" fws2 j : Fixed cost of locating a warehouse at "j" CAPWS1 i : Capacity of a stage 1 warehouse "i" CAPWS2 j : Capacity of a stage 2 warehouse "j" ∑ , Capacity of whs-1 at "i"as a fraction of the total market demand ∑ Capacity of whs-2 at "j" as a fraction of the total market demand ∑ goods from plant "h" to whs-1 "i" ∑ goods from whs-1 "i" to whs2 "j" ∑ goods from whs-2 "j" to market "k"

Definition of Variables
XPWS1 hi : Quantity of commodity transported from plant "h" to whs-1 "i" ∑ , Quantity transported from "h" to "i" as fraction of total demand XWS1WS2 ij : Quantity of commodity transported from whs-1 "i" to whs-2 "j" ∑ , Quantity transported from "i" to "j" as fraction of total demand XWS2M jk : Quantity of commodity transported from whs-2 "j" to market "k" ∑ , Quantity transported from "j" to "k" as fraction of total demand yws1 i : 1 if stage 1 warehouse is located at "i", 0 otherwise yws2 j : 1 if stage 2 warehouse is located at "j", 0 otherwise

Mathematical Formulation
The cost minimization problem for the TSCWLP can be written as mixed 0- This can be written as: Minimize  ( ) , constraints 2, 2(a) and 2(b) ensure that flow across stages is equal to total demand by all the markets.Constraints 3(a) and 3(b) are strong linking constraints (see Sharma and Berry [4] and Verma and Sharma [5]).Equations 4(a) and 4(b) are weak linking constraints.Equations 5, 5(a-i), 5(a-ii) and 5(b) strong capacity constraints (see Sharma and Berry [4]).Equation 6 ensures the flow from plant is less than supply available at that plant.Equations 6(a-i) and 6(a-ii) ensure that throughput form a warehouse is less than or equal to its capacity.Equation 6(b) ensures that the quantity received at a market is equal to its demand.Equations 7, 7(a) and 7(b) are non-negativity restrictions on real variables.Equations 8(a) and 8(b) are 0-1 restrictions on binary location variables.Equations 9(a) and 9(b) ensure that located capacity is more than or equal to the total demand of the markets.Equations 10(a) and 10(b) ensure that total average capacity is more than market demand at each stage.Equations 11(a) and 11(b) are flow balance constraints (inflow is equal to outflow) at each of the warehouses.By relaxing different constraints, various relaxations can be obtained as Lagrangian relaxation (LR).LR is a relaxation technique, which works by moving hard constraints into the objective to impose a penalty on the objective if they are not satisfied.This is easier to solve than the original problem.An optimal objective value of the Lagrangian relaxed problem, for a given set of multipliers, provides a lower bound (in the case of minimization) for the optimal solution to the original problem.The best lower bound can be derived by updating the multipliers by a dual ascent procedure.An upper bound on the optimal solution of the original problem can be derived by using the information obtained from the LR to construct a feasible solution to the original problem.This is normally done by applying some heuristic.In the next section, we present vertical decomposition approach for solving TSCWLP and variety of LR and LP relaxations.

Lagrangian Relaxation of TSCWLP
Proceeding in a manner similar to Verma and Sharma [13], TSCWLP is also vertically decomposed into three sub problems viz.LHS_CPLP, MID_CPLP and RHS_CPLP.In this approach, stages of warehouses are vertically decomposed to get smaller sized problems, which are relatively easier to solve.1st LR of this formulation is obtained by relaxing flow balance constraints (11(a)) and (11(b)) with suitable multipliers 1 λ and 2 λ respectively.In main problem, we note that constraint set (11(a)) connects the "xpws1 hi " and "xws1ws2 ij " variables.Similarly, constraint set (11(b)) connects the "xws1ws2 ij " and "xws2m jk " variables.If these two sets of constraints are relaxed, main problem will be separated into three sub-problems.One problem with "xpws1 hi " and "yws1 i ", called as LHS_CPLP, the other with "xws1ws2 ij ", "yws1 i " and "yws2 j " to be called as MID_CPLP, and the last problem with "xws2m jk " and "yws2 j " called as RHS_CPLP.Verma and Sharma [13] have already attempted RHS_CPLP and LHS_CPLP.Here different relaxations of MID_CPLP are developed and relationship is obtained for them.3), ( 4), ( 5), ( 6), ( 7), (8(a)), (9(a)), (10(a)).

Relaxation of CPLP_R (λ3 j ) Now
( ) is attempted in literature by three different relaxations.They are R1, R2 and R3 respectively.When integrality restriction on yws1 i is relaxed we obtain (12(a)).Result 1: R1 is equal to strong LP relaxation of ( ) (Sharma and Berry [4], Verma and Sharma [5]).Here the restriction on yws1 i is relaxed by (12(a)).
Proof: It is easy to see.

2) CPLP_R2 (λ3 j )
In this LR a constraint introducing upper (P U1 ) and lower (P L1 ) limits on the number of open plants is added up, Cornuejols, Sridharan and Thizy [14] i.e.
Result 2: ( ) ( ) ( ) If we add the objective function value of the 0-1 knapsack problem in the objective value of CPLP_R then we get the objective value of the MID_CPLP_R.Hence, the same relaxation along with their relative strength is used for MID_CPLP here in form of theorem 2 below.
where OFV stands for objective function value of ( )

Hence we have
Theorem 1: For any value of ( ) . This theorem provides the relative effectiveness of the bounds that may be obtained for these relaxations of MID_CPLP.Verma and Sharma [5] have shown that difference between R2-R1 and R3-R2 is significant.Hence we have the result given below.
Result 4: Difference between ( ) We get the modified and a valid formulation of MID_CPLP i.e., MO_MID_CPLP as given below.14) and ( 15).Now we relax the constraints (5(a-ii)), ( 14) and ( 15) by attaching multiplier λ3 j , λ4 ij and λ5 ij respectively to get the following.

MO_MID_CPLP
1) MO_MID_CPLP_R (λ3 j , λ4 ij , λ5 ij ) ( ) In R2: we solve single constraint bounded variable LP to get a factor that is added to 0-1 single constraint knapsack problem (which is solved by inspection) (see Christifides and Beasley [15]).A priori it is observed that the process is complicated enough and is NOT amenable to theoretical result (for determining the difference between MID_CPLP (R2) and MO_MID_CPLP (R2).
In R3: we solve several continuous knapsack problems & their objective function value is fed to a 0-1 knapsack problem (see Nauss [16]).This is also complicated enough; and theoretical result may NOT be possible (for determining the difference between MID_CPLP (R3) and MO_MID_CPLP (R3)).
Hence, we may have to determine differences empirically (by coding R2 and R3 for different Lagrangian Relaxations).However, as warehouse capacities are more constrained, MO_MID_CPLP for both R2 and R3 will give significantly superior lagrangian lower bounds at little computational expense.However, we do have a theoretical result for R1 (for difference between MID_CPLP and MO_MID_CPLP) which a LP relaxation.Hence we have the following Conjectures.

Conjecture 1:
( ) ( ) Conjecture 2: ( ) ( ) If only ( 14) is added to MID_CPLP then we get MO1_MID_CPLP.Several similar results are given without much discussion as they are obvious.

MO2_MID_CPLP
is attempted in literature by three different relaxations.They are R1, R2 and R3 respectively as before.When integrality restriction on yws1 i is relaxed we obtain (12(a)).( ) ( ) By the similar logic we have Since the effect of λ4 ij and λ5 ij is known only after it is through two combinatorial optimization problems, we believe a priori that it will difficult theoretically to prove the conjectures given in this paper.Hence we propose to carryout empirical investigation to support these conjectures.
1 integer linear programming problem as given in the formulation below.

Table 1 .
T-stat values for difference in performance of Lagrangian Relaxations on Model(j) -Model(i).

Table 2 .
T-stat values for difference in performance of Lagrangian Relaxations on Model(j) -Model(i).

Table 3 .
T-stat values for difference in performance of Lagrangian Relaxations on Model(j) -Model(i).