We note that the Single Stage Single Period Multi Commodity Warehouse Location Problem (SSSPMCWLP) has been first attempted by Geoffrion and Graves [1], and that they use the weak formulation (in context of contribution of this paper). We give for the first time “strong” formulation of SSSPMCWLP. We notice advantages of strong formulation over weak formulation in terms of better bounds for yielding efficient Branch and Bound solutions. However, the computation time of “strong” formulation was discovered to be higher than that of the “weak” formulation, which was a major drawback in solving large size problems. To overcome this, we develop the hybrid strong formulation by adding only a few most promising demand and supply side strong constraints to the weak formulation of SSSPMCWLP. So, the formulations developed were put to test on various large size problems. Hybrid formulation is able to give better bound than the weak and takes much less CPU time than the strong formulation. So, a kind of trade off is achieved allowing efficiently solving large sized SSSPMCWLP in real times using hybrid formulation.
SSUWLP SSSPMCWLP Strong and Weak Constraints Hybrid Formulation RMIP1. Introduction and Literature Survey
Warehousing is a key component of today’s business supply chain. It is not just a place to store finished goods but involves various value added functions like consolidation, packing and shipment of orders to customers. Organizations produce several commodities at various manufacturing sites, and generally it is not economical to get these commodities directly from plants to markets owing to distant market locations, associated demand disparities and uncertainties. Therefore, warehouses are installed at various locations between plants and markets to consolidate the localized customer demand. There can be more than one stage of the warehouse located between plants and the markets (Multistage Warehouse Location Problem). Among all considerations for securing a warehouse premise, the most important features are the space the warehouse can offer and its location. Generally, the sole objective in warehouse allocation problem is to minimize the total cost of transporting the goods from the manufacturing sites to the end customers via these warehouses.
We develop in this work the strong constraints for the multi-commodity case of Single Stage Single Period Capacitated Warehouse Location Problem (SSSPMCWLP). Here, “Capacitated” means that the capacity of a warehouse at a particular location is fixed and limited. This problem can be compared with a real life problem faced by Food Corporation of India (FCI) [2] . In India, FCI distributes a variety of food grains (wheat, rice, etc.) all over the country on a massive scale, purchasing them from various assigned locations called “MANDIS” directly from the farmers. FCI has a central office and five zones (North, South, East, North-East, and West). Each zone is further divided into regions, and each region is divided into several states and each state has several districts in it. From “MANDIS”, these grains are taken to central warehouses of each region and then transported to district warehouse as per the demand. From district warehouse, food grains are moved to depots, which distribute food grains directly to the public at a fair price. So, the MANDIS acts as plants or sources of food grain and district warehouses as demand points. Regional warehouses serve the intermediate transhipment points. The problem faced by FCI is to choose an optimum number of regional warehouse locations with sufficient capacity so as to minimize the sum of location and transportation costs of distributing food grains from MANDIS to these large Regional warehouses, and subsequently to smaller warehouses or district distribution centres. This results in SSSPMCWLP.
The importance of allocation of warehouses/facility between the production sites and the market can be depicted from the fact that this has been studied in its variant forms by long lists of researchers from last four decades and still remains the crucial decision of today’s supply chain design. Researchers have used several methods/approaches to reach the optimal solution. Warehouse allocation problem can be classified as:
1.1. Uncapacitated (SPLP) or Capacitated (CPLP)
Uncapacitated or Simple Plant Location Problems are those where it is assumed that facility has infinite storage space/handling capacity while the Capacitated Plant Location Problem implies that the capacity of the warehouse is limited and known in advance.
1.2. Single Commodity or Multi-Commodity
Single commodity means that there is only a single type of commodity and multi-commodity problems dealing with more than one type of commodities to be distributed.
1.3. Single Stage or Multi-Stage
Single stage refers to the real life problems where commodities are stored at one intermediate stage between plants and markets, while in the multi-stage problems, food grain distribution system commodities are stored at multiple stages (Figure 1 shows the schematic representation of single stage).
Our problem is multi-commodity single stage single period capacitated warehouse location problem (SSSPMCWLP), whose variants have been attempted by many researchers. An overview of the work done in the field of facility allocation can be found in review work of ReVelle and Eiselt [3] . Geoffrion and Graves [1] , Sharma [4] [5] , Drezner et al. [6] and Kouvelis et al. [7] have used weak formulation to address single stage capacitated ware house location problem (SSCWLP). Keskin and Uster [8] have attempted to solve SSCWLP problem with capacity limit at the manufacturing plant, and not on the warehouse. Geoffrion and Graves [1] and Sharma [4] have attempted the SSUWLP and have given completely different formulations considering two different real life problems. Geoffrion and Graves [1] gave a formulation for multi-commodity capacitated single- period version and successfully applied it to a real problem for a food firm using solution technique based on bender decomposition. Hindi and Basta [9] and Hindi et al. [10] solved the similar production-distribution pro-
Multi-commodity distribution from plants to markets via warehouses
blem by a branch-and-bound algorithm. Sharma [4] formulated a fertilizer production-distribution system problem as a mixed zero-one integer linear programming problem (MILP).Sharma and Verma [11] gave strong and weak formulation for SSUWLP considering single commodity; they also gave hybrid formulation to get a better bound than the weak formulation along with lesser CPU time than the strong formulation. Montoya-Torres et al. [12] developed three-echelon un-capacitated facility location problem (TUFLP) where they minimized the total cost of warehouse location, production, and distribution of products and proposed a greedy randomized adaptive search procedure (GRASP) to solve the multi-item version of the TUFLP. Elson [13] proposed a different multi- commodity version of the plant-warehouse facility logistics problem, concentrating on a single echelon of transshipment/stocking points. Many researchers have worked on developing both heuristic solution methods and exact algorithms to solve capacitated plant location problem. Pirkul and Jayaraman [14] proposed a heuristic based on the Lagrangian relaxation of the model to solve the two-stage production-distribution system design problem (TSPDSDP) with multi-commodity. Keskin and Üster [8] proposed a scatter search algorithm with hybrid improvements including local search and path-relinking to solve a TSPDSDP in which a fixed number of intermediate facilities points are to be located between the plants and customer locations. Verma and Sharma [15] applied vertical decomposition and Lagrangian relaxation approach to solve SSCWLP. Sharma and Aggarwal [16] solved two-stage capacitated warehouse location problem (TSCWLP) by vertical decomposition method, which was attained by relaxing the associated flow balance constraints.
It may be noted that Sharma [17] , Sharma and Murali [18] , Sharma and Berry [3] and Sharma and Verma [12] have attempted single commodity problem and have used “Normalized” variables for developing their strong formulation. We attempt for the first time to give strong formulation for multi-commodity version of the location-distribution problem (SSSCPMCWLP) and have not used ‘Normalized’ variables due to obvious difficulty.
The remainder of this paper is organized as follows. Section 2 gives mathematical formulation of the SSSCPMCWLP based on the existing literature and our modifications. Section 3 provides theoretical framework for using strong formulation, followed by empirical justification of hybrid formulation trade off in relaxed bound values versus computation time. The results from empirical investigation are followed by conclusions in Section 4.
2. Mathematical Formulation of SSSCPMCWLP2.1. Indices Used
: Set of the supply points (plants);
: Set of the potential warehouse points;
: Set of the markets;
: Set of the commodities.
2.2. Definition of Constants
: Fixed cost of establishing and maintaining the warehouse at location “”;
: Supply capacity of plant “” for commodity “”;
: Demand for commodity “” at market “” ;
: Capacity of the warehouse at location “” for all commodities (assumed that all the commodity are of same density and occupy same space**);
: A very large number, here taken as two times the maximumsupplyor maximum warehouse capacity;
: Cost of shipping one unit from plant “” to warehouse “” of the commodity “”;
: Cost of shipping one unit from warehouse “” to market “” for commodity “”;
**: Even if the volume or density are different for commodities a conversion factor can be introduced to account for them in warehouse capacity constraint (say for commodity “”).
2.3. Definition of Decision Variables
: Number of units shipped from plant “” to warehouse “” of the commodity type “”;
: Number of units shipped from warehouse “” to market “” of commodity type “”;
: Binary variable which is 1 if it is decided to locate a warehouse at location “” and 0 otherwise;
: Total cost of transporting commodities from plants to warehouses, warehouses to demand points or markets and fixed cost of locating the warehouses.
2.4. Mathematical Model
Minimize
Subjected to:
Linking constraints
Positive and binary constraints on decision variables:
3. Theoretical Framework and Empirical Verification
Theorem 1: LP relaxation of strong formulation (formulation II) gives superior bounds than the LP relaxation of weak formulation (formulation I) of SSSPMCWLP.
Proof: we see the feasible region of LP relaxation of formulation II is much smaller than feasible region of LP relaxation of formulation I, since. Tighter feasible region for a minimization problem obviously result in higher objective function value i.e. formulation II objective function value will be higher (proved).
Though LP relaxation of weak formulation gives inferior bounds but are computed very fast (as it has linear number of linking constraints). LP relaxation of strong formulation gives superior bounds but it takes more time (as number of linking constraints are of order). Therefore computational time of LP relaxation of strong formulation is much higher than the LP relaxation of weak formulation of SSSPMCWLP.
The hybrid formulation is developed to take advantage of both (better bound of strong formulation and better computation time of weak formulation) by adding most promising constraints of the strong formulation to the weak formulation of the SSSPMCWLP. Those few ‘strong’ constraints added may be binding in strong formulation, and can carefully be obtained by sorting the demand and supply values and gradually iterating to get the desired result. We started with 20% of strong constraints then reduced to5%, by step size of 5% and later reduced with step size of 1%. In each step, we observed decrease in the bounds obtained by LP relaxation but also simultaneous significant improvement in computational time. After some iteration we arrived at desired/satis- factory level and decided that the top 2% of total “strong” demand and supply constraints were to be added to the “weak” formulation of SSSPMCWLP to get the required hybrid formulation. At this level, the bounds were only a little inferior from weak bounds, but took substantially less computational time compared to strong computational time; significantly improving the overall computational times for solving original SSSPMCWLP. This motivated us to develop the following procedure.
Procedure to build hybrid formulation constraints
Step 1: Sort demand points values in increasing order of demand for each.
Set cut off number, where K is number of demand points.
Let the demand associated with this number N and given m be:
Prepare a set of most promising demand points
Now,
Step 2: Sort supply points values in increasing order of demand for each.
Set cut off number, where S is the number of supply points.
Let the Supply capacity associated with this number N and given m be:
GAMS programming is used to show the superiority of formulation (II) over formulation (I) and t-test has been conducted for the statistical significance of the results. To overcome the drawback of higher CPU time for strong formulation (Formulation II), a hybrid formulation is developed by adding the most promising supply and demand side strong linking constraint to the weak formulation. Here we have included 2% of the demand and strong side linking constraints to the weak formulation for which demand and supply are least.
Problem instances of the size 50 × 50 × 50 and 100 × 100 × 100 are solved in GAMS for the following four categories (Table 1) and as per assumptions given below:
1) In each problem instance demand for each commodity in each market is taken as uniformly distributed with lower bound of 100 units and upper bound of 150 units;
Problem categories for SSSPMCWLP
Category
Average warehouse capacity
Average supply capacity
A.
30% more than the average demand
30% more than the average warehouse capacity
B.
30% more than the average demand
125% more than the average warehouse capacity
C.
125% more than the average demand
30% more than the average warehouse capacity
D.
125% more than the average demand
125% more than the average warehouse capacity
2) Total Supply Capacity of each commodity at any plant and Warehouse Capacity at any location is more than any market average demand and taken as per the above four categories in Table 1;
3) Warehouse Capacity and Supply Capacity of plants are also uniformly distributed;
4) Cost of carrying one unit from plant to warehouse is normally distributed with mean of 4000 and standard deviation of 300;
5) Cost of carrying one unit from warehouse to market is normally distributed with mean of 5000 and standard deviation of 400;
6) Fixed Cost of locating a warehouse is normally distributed with a mean of 10,000 and standard deviation of 1500.
Samples problems are solved by relaxing the 0 - 1 constraints for the location variable Yj (constraint (12) in place of constraint (11) in formulation I and formulation II).
GAMS code is written to generate the random values for supply capacity, warehouse capacity, demand, costs of transportation and fixed cost of warehouse location as per the above specified criteria. The same code solves both the formulation (I) and formulation (II) with the same input data. Same data is used to solve hybrid formulation, so that we can compare the objective function values and CPU time values for these three formulations. 30 Problems are solved for each of the four categories for each problem size (total number of problem instances solved = 240, 120 of each of the size 50 × 50 × 50 and 100 × 100 × 100) using a Intel (R) Core (TM) i7-4770S CPU @ 3.10GHz.
Objective function values and CPU time for each and every problem instance is recorded and statistical analysis (t-test) is done to check the superiority of formulation (II) over formulation (I) and superiority of hybrid formulation over other formulations with a confidence level of 99.5% (Tables 2-13).
3.1.1. Statistical Analysis
Hypothesis tests are conducted as follows:
I. To check improvement in objective function values of LP relaxed Strong Bound (formulation II) from LP relaxed Weak Formulation (formulation I)
µ1: Percentage increase in objective function value of LP relaxed formulation (II) over LP relaxed formulation (I).
Null hypothesis,;
Alternate hypothesis,.
From the statistical t-tables we have the critical value for t-stats at as 2.68 for d.o.f. = 49 and 2.6264 for and d.o.f. = 99, thus we can easily reject null hypothesis.
II. To check improvement in objective function values of LP relaxed Hybrid formulation vs. LP relaxed Weak Formulation (formulation I).
µ2: Percentage increase in objective function value of LP relaxed Hybrid formulation over LP relaxed Weak formulation (formulation I).
Null hypothesis,;
Alternate hypothesis,.
Result: Comparing the calculated t-values with critical t-values, the null hypothesis µ2 = 0 is rejected.
III. To check deterioration in CPU time values of LP relaxed Strong bound (formulation (II)) vs. LP relaxed Weak bound (formulation (I)).
t1: Percentage deterioration in CPU time for Strong formulation (II) over weak formulation (I).
Null hypothesis,;
Alternate hypothesis,.
Problem size 50 × 50 × 50 of Category A
Problem instance number
Weak bound (relaxed formulation I)
Hybrid formulation bound (relaxed weak formulation with most promising strong linking constraints added)
Strong bound (relaxed formulation II)
Value
CPU time
Value
CPU time
Value
CPU time
P1
464761657.8
0.172
464,856,004
0.193
465252234.1
0.203
P2
465982540.6
0.141
466,077,974
0.160
466,479,602
0.203
P3
463307328.6
0.141
463,406,152
0.153
463794110.6
0.203
P4
463827848.1
0.14
463,921,356
0.159
464328797.5
0.187
P5
462125994.6
0.141
462,227,246
0.156
462614855.4
0.203
P6
464303782.3
0.141
464,405,186
0.158
464812299.2
0.203
P7
466278050.8
0.140
466,375,689
0.158
466787603.6
0.203
P8
463635528.2
0.141
463,730,203
0.152
464,121,359
0.187
P9
463313848.7
0.141
463,413,322
0.156
463809058.1
0.187
P10
464,843,759
0.109
464,937,239
0.120
465335659.3
0.187
P11
464899247.3
0.110
464,993,389
0.119
465404355.6
0.203
P12
463364638.6
0.109
463,457,636
0.122
463851905.1
0.203
P13
464718283.4
0.125
464,815,642
0.135
465205666.3
0.203
P14
461971206.5
0.125
462,068,220
0.142
462461413.5
0.203
P15
466460412.4
0.109
466,557,763
0.124
466968490.2
0.187
P16
466,121,646
0.125
466,215,197
0.140
466623984.1
0.203
P17
466,613,596
0.109
466,708,832
0.118
467117668.5
0.187
P18
457,455,118
0.125
457,547,981
0.141
457,942,972
0.203
P19
465677481.6
0.125
465,778,580
0.141
466196407.9
0.203
P20
464276924.6
0.125
464,370,012
0.142
464,767,951
0.203
P21
462941205.5
0.125
463,035,738
0.140
463438858.3
0.203
P22
461101747.4
0.125
461,195,628
0.136
461599000.9
0.203
P23
465161010.4
0.125
465,258,415
0.135
465650542.7
0.203
P24
461900311.9
0.125
461,998,743
0.135
462396333.7
0.203
P25
463755031.4
0.125
463,856,362
0.141
464258661.5
0.203
P26
459684460.9
0.125
459,780,121
0.134
460163496.2
0.235
P27
466716094.5
0.110
466,811,865
0.125
467213751.9
0.219
P28
465410474.9
0.109
465,503,929
0.123
465913015.5
0.203
P29
465,546,559
0.125
465,645,488
0.141
466045992.9
0.219
P30
459281474.2
0.141
459,377,556
0.156
459785502.4
0.203
Problem size 50 × 50 × 50 of Category B
Problem instance number
Weak bound (relaxed formulation I)
Hybrid formulation bound (relaxed weak formulation with most promising strong linking constraints added)
Strong bound (relaxed formulation II)
Value
CPU time
Value
CPU time
Value
CPU time
P1
462154595.8
0.125
462,252,526
0.168
462647776.5
0.203
P2
463922755.7
0.125
464,021,386
0.172
464424535.6
0.219
P3
454902314.7
0.094
454,995,979
0.130
455391806.2
0.234
P4
462729689.5
0.11
462,824,549
0.156
463237509.7
0.203
P5
463264246.2
0.109
463,357,548
0.159
463756427.2
0.203
P6
459871109.2
0.109
459,967,544
0.149
460359595.1
0.203
P7
463005259.3
0.109
463,102,583
0.150
463499775.1
0.203
P8
461,750,540
0.11
461,846,630
0.154
462250609.3
0.203
P9
462513368.3
0.109
462,607,027
0.142
463008757.8
0.203
P10
464652116.8
0.109
464,746,116
0.157
465,123,264
0.203
P11
465,808,805
0.125
465,907,370
0.174
466,306,867
0.25
P12
466659753.5
0.14
466,754,205
0.181
467165643.8
0.25
P13
465859563.6
0.141
465,953,947
0.177
466,367,333
0.234
P14
460683422.9
0.125
460,784,405
0.158
461175212.4
0.234
P15
459791614.3
0.125
459,888,906
0.174
460282546.8
0.188
P16
460724495.9
0.109
460,819,912
0.155
461231699.5
0.172
P17
461385093.5
0.125
461,480,185
0.158
461887513.4
0.203
P18
463032455.3
0.125
463,126,358
0.173
463528485.4
0.188
P19
461040271.5
0.109
461,141,147
0.152
461544498.2
0.25
P20
461,943,881
0.093
462,038,256
0.134
462455999.9
0.187
P21
461975515.3
0.11
462,077,104
0.139
462458306.3
0.188
P22
459544615.7
0.109
459,645,210
0.152
460,037,834
0.187
P23
461886169.8
0.11
461,983,905
0.146
462402129.2
0.203
P24
459427161.6
0.109
459,522,677
0.147
459,901,863
0.188
P25
465536812.4
0.125
465,631,596
0.161
466,037,225
0.234
P26
461721331.5
0.125
461,819,032
0.173
462220540.5
0.25
P27
460304848.7
0.11
460,396,956
0.148
460808541.2
0.234
P28
458865037.7
0.125
458,963,051
0.167
459365142.4
0.218
P29
461224194.5
0.109
461,321,005
0.153
461736086.6
0.203
P30
462191950.1
0.109
462,287,670
0.149
462677404.9
0.219
Problem size 50 × 50 × 50 of Category C
Problem instance number
Weak bound (relaxed formulation I)
Hybrid formulation bound (relaxed weak formulation with most promising strong linking constraints added)
Strong bound (relaxed formulation II)
Value
CPU time
Value
CPU time
Value
CPU time
P1
460156810.2
0.125
460,249,210
0.157
460647316.2
0.25
P2
461928500.6
0.109
462,025,737
0.154
462422695.2
0.187
P3
460184521.6
0.094
460,277,847
0.118
460670115.5
0.188
P4
458523854.3
0.109
458,623,812
0.154
459015839.3
0.188
P5
461210372.4
0.109
461,310,363
0.146
461695917.4
0.172
P6
466736925.2
0.109
466,839,421
0.134
467227158.5
0.188
P7
465040652.4
0.109
465,135,474
0.160
465533649.4
0.188
P8
465,711,634
0.11
465,806,825
0.132
466203952.5
0.188
P9
462,663,898
0.093
462,758,652
0.118
463143978.5
0.171
P10
461994675.3
0.11
462,089,430
0.155
462501565.4
0.188
P11
463137094.7
0.11
463,237,318
0.158
463654765.4
0.187
P12
460644100.8
0.11
460,736,460
0.149
461147401.4
0.187
P13
459979852.8
0.109
460,080,404
0.144
460480501.1
0.172
P14
465645551.9
0.125
465,743,431
0.180
466,147,369
0.171
P15
459540857.4
0.109
459,641,773
0.143
460034510.6
0.172
P16
457524145.7
0.094
457,617,435
0.116
458003132.4
0.187
P17
463560066.4
0.094
463,658,619
0.129
464066716.1
0.172
P18
461800417.3
0.125
461,896,010
0.169
462290440.1
0.187
P19
461310140.9
0.094
461,402,864
0.132
461809878.3
0.172
P20
463826134.1
0.109
463,921,172
0.137
464324355.3
0.188
P21
461729092.8
0.125
461,826,795
0.158
462207481.3
0.172
P22
459986245.2
0.109
460,084,636
0.148
460465903.5
0.187
P23
466476622.5
0.094
466,576,728
0.126
466958243.4
0.219
P24
461750643.4
0.094
461,844,887
0.114
462241797.1
0.172
P25
461884416.2
0.11
461,978,687
0.149
462377121.9
0.172
P26
462809272.1
0.11
462,905,999
0.162
463300162.2
0.188
P27
462,954,846
0.109
463,050,076
0.140
463446331.4
0.172
P28
462717628.6
0.11
462,814,105
0.157
463207321.1
0.172
P29
461110049.8
0.109
461,211,171
0.160
461601038.8
0.187
P30
461842013.8
0.125
461,938,493
0.170
462355526.8
0.187
Problem size 50 × 50 × 50 of Category D
Problem instance number
Weak bound (relaxed formulation I)
Hybrid formulation bound (relaxed weak formulation with most promising strong linking constraints added)
Strong bound (relaxed formulation II)
Value
CPU time
Value
CPU time
Value
CPU time
P1
465681644.6
0.109
465,779,158
0.125
466164697.7
0.188
P2
457544301.9
0.109
457,638,693
0.128
458054151.7
0.203
P3
459974575.7
0.093
460,067,583
0.106
460478771.5
0.188
P4
458538365.5
0.094
458,633,100
0.116
459024202.6
0.187
P5
460692904.2
0.094
460,793,612
0.111
461176844.4
0.203
P6
459,533,942
0.094
459,626,170
0.124
460005875.9
0.203
P7
460283567.4
0.109
460,376,361
0.141
460778713.3
0.187
P8
461,788,231
0.109
461,882,251
0.131
462277760.5
0.188
P9
456723909.6
0.125
456,816,899
0.150
457209892.4
0.172
P10
459648712.5
0.125
459,744,963
0.158
460163566.7
0.172
P11
456480416.3
0.11
456,579,336
0.127
456977922.9
0.188
P12
465421105.4
0.109
465,520,659
0.127
465917396.2
0.172
P13
463411618.9
0.109
463,506,016
0.138
463903987.7
0.188
P14
462087761.9
0.125
462,188,543
0.160
462585844.7
0.172
P15
464393272.7
0.109
464,494,557
0.135
464873431.8
0.187
P16
459898122.6
0.125
459,995,391
0.147
460381579.4
0.172
P17
460720255.7
0.109
460,814,657
0.145
461222610.1
0.187
P18
460516332.1
0.125
460,613,547
0.153
461019806.6
0.188
P19
460753982.8
0.109
460,850,649
0.131
461254252.7
0.187
P20
459873402.2
0.109
459,966,067
0.144
460379360.6
0.172
P21
461661335.6
0.125
461,757,407
0.149
462144676.9
0.218
P22
464025496.2
0.109
464,123,545
0.124
464,520,012
0.172
P23
456996616.6
0.141
457,093,226
0.184
457461190.2
0.172
P24
463178689.2
0.125
463,280,311
0.156
463674506.7
0.172
P25
460206278.2
0.11
460,301,633
0.147
460708601.5
0.188
P26
462363155.8
0.11
462,459,743
0.130
462850313.3
0.172
P27
464172352.1
0.11
464,268,389
0.137
464682393.6
0.203
P28
457850163.7
0.11
457,942,192
0.135
458357345.2
0.188
P29
462003319.6
0.125
462,098,030
0.148
462497026.1
0.172
P30
460635912.7
0.11
460,733,936
0.133
461120649.7
0.187
Problem size 100 × 100 × 100 of Category A
Problem instance number
Weak bound (relaxed formulation I)
Hybrid formulation bound (relaxed weak formulation with most promising strong linking constraints added)
Strong bound (relaxed formulation II)
Value
CPU time
Value
CPU time
Value
CPU time
P1
1,365,932,733
0.969
1,366,215,071
1.085
1,366,941,579
1.562
P2
1,356,964,622
1.062
1,357,249,992
1.144
1,357,960,054
1.572
P3
1,366,670,095
1.093
1,366,946,572
1.248
1,367,675,025
1.65
P4
1,364,344,024
1.047
1,364,630,536
1.145
1,365,354,097
1.578
P5
1,359,252,277
1.109
1,359,550,905
1.265
1,360,243,625
1.578
P6
1,362,703,027
1.031
1,362,989,604
1.137
1,363,687,229
1.547
P7
1,357,648,342
1.094
1,357,931,004
1.248
1,358,646,263
1.532
P8
1,361,734,440
1.078
1,362,015,230
1.234
1,362,771,012
1.578
P9
1,363,335,394
1.094
1,363,626,057
1.226
1,364,326,431
1.578
P10
1,354,662,978
1.062
1,354,940,548
1.126
1,355,672,184
1.625
P11
1,359,010,784
1.031
1,359,297,671
1.111
1,360,015,221
1.578
P12
1,360,290,820
1.031
1,360,566,551
1.118
1,361,275,760
1.484
P13
1,359,813,107
1.015
1,360,106,147
1.136
1,360,804,455
1.516
P14
1,360,229,705
1.094
1,360,524,059
1.190
1,361,194,499
1.563
P15
1,357,065,375
1
1,357,352,530
1.080
1,358,068,651
1.485
P16
1,366,116,379
1.063
1,366,414,875
1.185
1,367,101,877
1.797
P17
1,354,295,387
1.016
1,354,575,049
1.166
1,355,307,736
1.547
P18
1,362,453,698
1.093
1,362,753,165
1.196
1,363,466,065
1.656
P19
1,351,212,750
1.047
1,351,506,098
1.133
1,352,228,221
1.594
P20
1,354,546,726
1.062
1,354,825,085
1.156
1,355,567,185
1.593
P21
1,361,671,542
1.078
1,361,968,932
1.167
1,362,652,718
1.515
P22
1,365,090,355
1.031
1,365,366,240
1.125
1,366,081,015
1.531
P23
1,353,711,255
1.015
1,353,984,975
1.125
1,354,692,174
1.594
P24
1,359,167,606
1.078
1,359,454,798
1.239
1,360,142,833
1.547
P25
1,354,759,090
1.078
1,355,052,259
1.185
1,355,756,040
1.546
P26
1,360,824,145
0.984
1,361,122,302
1.101
1,361,829,294
1.594
P27
1,357,519,457
1.047
1,357,817,161
1.210
1,358,490,606
1.516
P28
1,355,437,009
1.047
1,355,724,226
1.148
1,356,416,967
1.562
P29
1,360,832,833
0.985
1,361,119,561
1.126
1,361,815,955
1.578
P30
1,349,511,932
1.062
1,349,797,353
1.191
1,350,507,838
1.578
Problem size 100 × 100 × 100 of Category B
Problem instance number
Weak bound (relaxed formulation I)
Hybrid formulation bound (relaxed weak formulation with most promising strong linking constraints added)
Strong bound (relaxed formulation II)
Value
CPU time
Value
CPU time
Value
CPU time
P1
1,347,812,101
0.984
1,348,043,655
1.099
1,348,824,104
1.437
P2
1,353,943,093
0.875
1,354,165,275
1.037
1,354,916,518
1.5
P3
1,353,589,922
0.797
1,353,818,814
0.918
1,354,572,437
1.406
P4
1,348,651,591
0.875
1,348,874,253
0.988
1,349,651,010
1.438
P5
1,353,236,978
0.859
1,353,454,443
0.970
1,354,223,675
1.422
P6
1,345,291,759
0.859
1,345,512,522
0.954
1,346,286,225
1.5
P7
1,354,089,798
0.875
1,354,306,453
0.976
1,355,101,717
1.578
P8
1,350,526,131
0.89
1,350,746,131
1.032
1,351,547,781
1.562
P9
1,347,779,924
0.844
1,348,003,925
0.939
1,348,783,579
1.563
P10
1,352,052,432
0.968
1,352,282,416
1.141
1,353,075,218
1.547
P11
1,348,474,977
0.875
1,348,704,892
0.998
1,349,452,863
1.531
P12
1,358,050,071
0.859
1,358,285,014
1.010
1,358,999,314
1.61
P13
1,360,492,873
0.875
1,360,727,422
0.989
1,361,471,188
1.563
P14
1,347,055,192
0.86
1,347,287,155
0.949
1,348,053,275
1.578
P15
1,346,813,104
0.86
1,347,032,769
0.957
1,347,825,533
1.409
P16
1,351,598,681
0.859
1,351,818,451
0.990
1,352,597,768
1.556
P17
1,352,634,141
0.844
1,352,873,693
0.938
1,353,614,996
1.438
P18
1,347,143,013
0.828
1,347,368,659
0.928
1,348,101,467
1.422
P19
1,347,578,203
0.844
1,347,811,739
0.996
1,348,549,057
1.375
P20
1,354,172,979
0.86
1,354,405,219
1.024
1,355,188,299
1.407
P21
1,353,131,970
0.844
1,353,350,636
0.974
1,354,106,286
1.407
P22
1,352,195,677
0.875
1,352,423,658
1.000
1,353,198,611
1.437
P23
1,354,067,995
0.875
1,354,309,967
1.036
1,355,054,115
1.532
P24
1,348,354,628
0.828
1,348,587,758
0.950
1,349,338,895
1.478
P25
1,348,863,378
0.86
1,349,093,359
1.030
1,349,868,041
1.578
P26
1,350,507,863
0.984
1,350,725,025
1.156
1,351,509,090
1.516
P27
1,351,696,314
1
1,351,938,538
1.191
1,352,686,083
1.453
P28
1,348,216,115
1
1,348,458,524
1.139
1,349,190,279
1.469
P29
1,349,155,465
0.969
1,349,378,346
1.123
1,350,167,144
1.406
P30
1,355,404,108
0.968
1,355,636,560
1.078
1,356,383,297
1.422
Problem size 100 × 100 × 100 of Category C
Problem instance number
Weak bound (relaxed formulation I)
Hybrid formulation bound (relaxed weak formulation with most promising strong linking constraints added)
Strong bound (relaxed formulation II)
Value
CPU time
Value
CPU time
Value
CPU time
P1
1,356,765,481
1
1,356,988,805
1.028
1,357,747,787
1.375
P2
1,348,578,687
0.906
1,348,784,480
0.934
1,349,566,313
1.375
P3
1,350,776,977
0.89
1,351,005,393
0.910
1,351,787,791
1.391
P4
1,356,692,279
0.89
1,356,918,304
0.913
1,357,678,393
1.337
P5
1,359,305,911
0.953
1,359,517,283
0.975
1,360,284,488
1.39
P6
1,349,187,512
0.922
1,349,390,565
0.951
1,350,173,750
1.359
P7
1,347,497,209
0.953
1,347,721,702
0.979
1,348,495,827
1.406
P8
1,351,561,595
0.922
1,351,780,684
0.943
1,352,539,359
1.39
P9
1,352,908,377
0.906
1,353,135,936
0.935
1,353,917,629
1.375
P10
1,355,180,785
0.906
1,355,411,030
0.928
1,356,176,036
1.359
P11
1,349,471,170
0.968
1,349,679,258
0.990
1,350,467,442
1.422
P12
1,347,863,661
0.922
1,348,079,993
0.948
1,348,867,681
1.422
P13
1,354,473,571
0.922
1,354,688,119
0.947
1,355,463,506
1.359
P14
1,350,205,726
0.922
1,350,433,235
0.948
1,351,193,519
1.344
P15
1,349,060,862
0.922
1,349,290,067
0.945
1,350,041,009
1.375
P16
1,347,587,320
0.859
1,347,789,863
0.886
1,348,577,355
1.453
P17
1,351,522,612
0.813
1,351,733,179
0.839
1,352,521,657
1.359
P18
1,348,302,161
0.828
1,348,510,204
0.847
1,349,326,170
1.422
P19
1,357,334,766
0.828
1,357,545,560
0.852
1,358,323,179
1.406
P20
1,346,200,443
0.844
1,346,428,489
0.867
1,347,193,192
1.485
P21
1,356,509,204
0.828
1,356,734,656
0.852
1,357,494,571
1.391
P22
1,350,745,322
0.829
1,350,970,356
0.852
1,351,735,380
1.438
P23
1,356,703,210
0.812
1,356,919,469
0.832
1,357,678,909
1.422
P24
1,357,978,256
0.844
1,358,191,187
0.864
1,358,962,351
1.5
P25
1,349,813,482
0.828
1,350,026,212
0.848
1,350,818,186
1.422
P26
1,354,216,441
0.937
1,354,434,470
0.963
1,355,225,303
1.516
P27
1,354,135,174
0.844
1,354,343,981
0.871
1,355,144,688
1.344
P28
1,345,665,627
0.844
1,345,879,453
0.869
1,346,652,596
1.407
P29
1,353,034,625
0.859
1,353,261,123
0.883
1,354,003,682
1.375
P30
1,352,589,054
0.844
1,352,793,971
0.869
1,353,593,395
1.343
Problem size 100 × 100 × 100 of Category D
Problem instance number
Weak bound (relaxed formulation I)
Hybrid formulation bound (relaxed weak formulation with most promising strong linking constraints added)
Strong bound (relaxed formulation II)
Value
CPU time
Value
CPU time
Value
CPU time
P1
1,349,044,304
0.953
1,349,310,336
1.076
1,350,052,277
1.328
P2
1,341,330,308
0.891
1,341,585,965
1.079
1,342,364,193
1.344
P3
1,357,367,491
0.875
1,357,634,078
1.040
1,358,357,501
1.313
P4
1,344,761,701
0.907
1,345,022,451
1.072
1,345,749,092
1.312
P5
1,353,950,432
0.906
1,354,231,377
1.065
1,354,948,263
1.343
P6
1,345,725,674
0.86
1,345,986,206
1.060
1,346,727,389
1.328
P7
1,350,064,031
0.875
1,350,340,929
0.991
1,351,070,255
1.344
P8
1,349,599,349
0.859
1,349,856,043
1.048
1,350,587,142
1.312
P9
1,347,042,608
0.875
1,347,306,898
1.071
1,348,010,275
1.328
P10
1,359,202,269
0.891
1,359,485,934
1.034
1,360,184,283
1.313
P11
1,347,843,751
0.875
1,348,111,163
1.080
1,348,840,103
1.328
P12
1,347,079,789
0.859
1,347,339,910
0.996
1,348,069,301
1.344
P13
1,345,959,280
0.875
1,346,232,645
1.005
1,346,948,448
1.36
P14
1,352,533,674
0.844
1,352,817,571
0.995
1,353,537,564
1.359
P15
1,344,792,939
0.891
1,345,071,043
1.021
1,345,811,206
1.312
P16
1,346,897,235
0.828
1,347,173,079
1.006
1,347,857,811
1.344
P17
1,335,591,579
0.765
1,335,869,115
0.905
1,336,633,363
1.344
P18
1,352,021,379
0.813
1,352,281,103
0.976
1,353,033,815
1.344
P19
1,340,538,436
0.828
1,340,813,112
0.994
1,341,534,569
1.375
P20
1,358,293,736
0.796
1,358,553,985
0.893
1,359,285,053
1.329
P21
1,349,182,387
0.828
1,349,458,565
0.983
1,350,197,272
1.343
P22
1,350,465,764
0.844
1,350,736,127
0.978
1,351,464,235
1.328
P23
1,342,712,142
0.828
1,342,975,448
0.978
1,343,717,026
1.344
P24
1,349,696,083
0.812
1,349,955,090
0.925
1,350,685,192
1.312
P25
1,344,666,743
0.813
1,344,942,265
0.967
1,345,654,739
1.281
P26
1,349,872,300
0.812
1,350,155,773
0.941
1,350,889,585
1.328
P27
1,346,328,212
0.797
1,346,606,633
0.951
1,347,349,390
1.344
P28
1,349,420,910
0.812
1,349,697,811
0.919
1,350,411,598
1.359
P29
1,350,343,978
0.828
1,350,602,029
1.010
1,351,345,376
1.313
P30
1,358,015,527
0.828
1,358,295,142
0.945
1,359,024,827
1.313
T-test results of objective function improvement of LP relaxed strong formulation over LP relaxed weak formulation
Problem category
Problem size (50 × 50 × 50)
Problem size (100 × 100 × 100)
t-values
t-values
A.
338.16
331.84
B.
264.68
281.77
C.
278.69
135.03
D.
225
289.06
T-test results of objective function improvement of LP relaxed hybrid formulation over LP relaxed weak formulation
Problem category
Problem size (50 × 50 × 50)
Problem size (100 × 100 × 100)
t-values
t-values
A.
192.01
201.35
B.
202.02
161.58
C.
215.59
247.58
D.
203.49
202.31
T-test results for CPU time deterioration of strong formulation (II) over weak formulation (I)
Problem Category
Problem size (50 × 50 × 50)
Problem size (100 × 100 × 100)
t-values
t-values
A.
17.11
40.46
B.
21..74
29.08
C.
19.85
28.51
D.
15.55
38.59
T-test results for CPU time improvement of hybrid formulation over strong formulation
Problem category
Problem size (50 × 50 × 50)
Problem size (100 × 100 × 100)
t-values
t-values
A.
17.76
37.72
B.
16.69
29.2
C.
10.45
41.63
D.
11.46
31.07
Result: Comparing the calculated t-values with critical t-values, the null hypothesis t1 = 0 is rejected.
IV. To check improvements in CPU time values of Hybrid Formulation vs. Strong Formulation (Formulation I).
As we have added only 2% most promising demand and supply side strong linking constraint to the weak formulation to form hybrid formulation, therefore we expect the improvement in CPU time i.e. CPU time for Hybrid Formulation must be less than the CPU time for Strong Formulation. We conduct the following hypothesis test:
t2: Percentage improvement in CPU time i.e. decrement in the CPU time for hybrid formulation from Strong formulation.
Null hypothesis,;
Alternate hypothesis,.
Result: Comparing the calculated t-values with critical t-values, the null hypothesis t2 = 0 is rejected.
Looking at the t-tests done on the empirical results, we see all the null hypotheses have been rejected, implying differences in relaxed formulations results; strong constraints gave the superior bounds than the weak constraints. However, the CPU time for computation for LP relaxed strong formulations were significantly higher than the CPU time for LP relaxed weak formulations, which meant that the benefit of better bounds using stronger formulations is somewhat compromised by their higher computational time. Hybrid formulation values were intermediate between strong and weak formulations with significant improvement in bound values.
4. Conclusions
The SSSPMCWLP is often encountered in distribution networks which include cross-docks with known manufacturer and customer demand points. We develop strong and hybrid formulation for the SSSPMCWLP for the first time and show its merits over prevalent weak formulation in literature. The idea behind the hybrid formulation is to get advantages of both weak and strong formulation in terms of bound and computational time.
Computational study is done for a variety of problems of different sizes. We find that hybrid formulations of SSSPMCWLP is a good compromise, as it gives better bounds while not incurring greater computation burden.
ReferencesGeoffrion, A.M. and Graves, G.W. (1974) Multicommodity Distribution System Design by Benders Decomposition. Management Science, 20, 822-844.
http://dx.doi.org/10.1287/mnsc.20.5.822FCI (Food Corporation of India) Official Website.
http://fciweb.nic.in/ReVelle, C.S. and Eiselt, H.A. (2005) Location Analysis: A Synthesis and Survey. European Journal of Operational Research, 165, 1-19.Sharma, R.R.K. (1991) Modelling a Fertilizer Distribution System. European Journal of Operational Research, 51, 24-34.
http://dx.doi.org/10.1016/0377-2217(91)90142-ISharma, R.R.K. and Berry, V. (2007) Developing New Formulations and Relaxations of Single Stage Capacitated Warehouse Location Problem (SSCWLP): Empirical Investigation for Assessing Relative Strengths and Computational Effort. European Journal of Operational Research, 177, 803-812.
http://dx.doi.org/10.1016/j.ejor.2005.11.028Drezner, T., Drezner, Z. and Salhi, S. (2002) Solving the Multiple Competitive Facilities Location Problem. European Journal of Operational Research, 142, 138-151.
http://dx.doi.org/10.1016/S0377-2217(01)00168-0Kouvelis, P., Rosenblatt, M.J. and Munson, C.L. (2004) A Mathematical Programming Model for Global Plant Location Problem: Analysis and Insights. IIE Transactions, 36, 127-144. http://dx.doi.org/10.1080/07408170490245388Keskin, B.B. and Üster, H. (2007) A Scatter Search-Based Heuristic to Locate Capacitated Transhipment Points. Computers & Operations Research, 34, 3112-3125.
http://dx.doi.org/10.1016/j.cor.2005.11.020Hindi, K.S. and Basta, T. (1994) Computationally Efficient Solution of a Multiproduct, Two-Stage Distribution Location Problem. The Journal of the Operational Research Society, 45, 1316-1323.Hindi, K.S., Basta, T. and Pienkosz, K. (2006) Efficient Solution of a Multi-Commodity, Two-Stage Distribution Problem with Constraints on Assignment of Customers to Distribution Centres. International Transactions in Operations Research, 5, 519-527.
http://dx.doi.org/10.1111/j.1475-3995.1998.tb00134.xSharma, R.R.K. and Verma, P. (2012) Hybrid Formulations of Single Stage Uncapacitated Warehouse Location Problem: Few Theoretical and Empirical Results. International Journal of Operations and Quantitative Management (IJOQM), 18, 53-69.Montoya-Torres, J.R., Aponte, A. and Rosas, P. (2011) Applying GRASP to Solve the Multi-Item Three-Echelon Uncapacitated Facility Location Problem. Journal of the Operational Research Society, 62, 397-406.
http://dx.doi.org/10.1057/jors.2010.134Elson, D.G. (1972) Site Location via Mixed-Integer Programming. The Journal of Operational Research Society, 23, 31-43.
http://dx.doi.org/10.1057/jors.1972.4Pirkul, H. and Jayaraman, V. (1998) A Multi-Commodity, Multi-Plant, Capacitated Facility Location Problem: Formulation and Efficient Heuristic Solution. Computers & Operations Research, 25, 869-878.
http://dx.doi.org/10.1016/S0305-0548(97)00096-8Verma, P. and Sharma, R.R.K. (2011) Vertical Decomposition Approach to Solve Single Stage Capacitated Warehouse Location Problem (SSCWLP). American Journal of Operations Research, 1, 100-117.
http://dx.doi.org/10.4236/ajor.2011.13013Sharma, R.R.K. and Agarwal, P. (2014) Approaches to Solve MID_CPLP Problem: Theoretical Framework and Empirical Investigation. American Journal of Operations Research, 4, 142-154. http://dx.doi.org/10.4236/ajor.2014.43014Sharma, R.R.K. (1996) Food Grains Distribution in the Indian Context: An Operational Study. In: Tripathy, A. and Rosenhead, J., Eds., Operations Research for Development, Chapter 5, New Age International Publishers, New Delhi, 212-227.Sharma, R.R.K. and Muralidhar, A. (2009) A New Formulation and Relaxation of the Simple Plant Location Problem. Asia-Pacific Journal of Operational Research, 26, 1-11.
http://dx.doi.org/10.1142/S0217595909002122