Localization of Warehouses in Distribution Companies

Localization of Warehouses in Distribution Companies FILIP EXNAR, OTAKAR MACHAČ Department of economics and management of chemical and food industry University of Pardubice Studentská 95, 532 10 Pardubice CZECH REPUBLIC Filip.Exnar@upce.cz Otakar.Machac@upce.cz Abstract: - Localization models are an important part of tools to manage and optimize supply chains. In the first part of the article, we deal with the description of basic localization models and variables occurring in them. In the second part, we address a case study of a distribution company related to locating a new warehouse. First, an optimal warehouse location was found in terms of transport distances and subsequently also in terms of costs. The result of the model is determination of savings in transportation costs in the case of the use of the new warehouse as against the existing transport from the central warehouse. Key-Words: - Localization Models, Variables and Cost Functions in LM, Optimal Warehouse Localization, Economic Efficiency of New Warehouse. 1 Introduction With increasing concentration and globalization of production capacities, demands are increasing for efficient distribution and transport of products to the final customers. The tools for the management and optimization of distribution systems include, among others, localization models. It is a wide variety of models that seek optimal geographic distribution of suppliers, manufacturers, distribution centres and other entities in relation to the final customers. At the strategic level, this can be an optimal deployment of cooperating production plants, distribution and assembly centres in a pre-defined area of their interest. Frequent situation which is solved in companies is that the period which is required for realization of all activities from purchase, through production up to distribution is longer than is acceptable for customer [1]. In these cases where customers require very short delivery terms, we must solve the problem of how many detached warehouses to build and where. With localization models, however, we can solve problems such as the location of machines in the workshops so that the material flow is smooth and as simple as possible, arrangement of goods on the shelves of stores, etc. Localization models are used in many different areas and under very different conditions. Therefore, they also have many different modifications. In this article, we restrict ourselves to the cases of localization of objects in a plane that have most real applications, and are related to our investigated problem. 2 Variables for the design of location models Schwartz [2], in his survey defined ten most powerful factors in location decisions. In order of their appearance: reasonable cost for property, roadway access for trucks, nearness to customers, cost of labour, low taxes, tax exemptions, tax credits, low union profile, ample room for expansion, community disposition to industry. By Ravindran only two or three of these (property and labour cost and perhaps low taxes) could by directly incorporated into one of the classic facility location models [3]. Formulating an optimization problem associated with the location of desirable or undesirable facilities varies with respect to several criteria including the feasible region (discrete or continuous whether we have a choice of pre-defined locations or whether we can choose any place), number of places (single or multiobjective), the feasible solution space (acyclic or network that allows cycles) and the interactions considered (distances between facilities or the distances between facilities and customers). Regardless of the diversity of specific models, there are certain variables that occur in all models and shape the criterion function for optimization. ISBN: 978-960-474-324-7 87

2.1 Number of new objects and existing sites In all optimization models there is variable i for the number of new objects and variable j for the number of already existing objects (customers). In all the models, the number of existing objects assume values j = 1, 2,... n, the variable i can assume values i = 1,2,... m, but this value varies from model to model. Chopra and Meindl [4] suggest that supply chain network design includes four phases: SC Strategy, Regional Facility Configuration, Desirable Sites and Location Choices. They also suggest a number of factors that enter into these decisions. Ravindran [3] add to this that the primary strategic choice in the treatment of supply chain network design that follows will be how centralized or decentralized the network design should be. A choice on this centralized-decentralized spectrum will lead to a sense of how many facilities (warehouses) there should be at each echelon of the network. The simplest case is locating only one new object. Besides the number of newly located objects models, they distinguish whether an object(s) can be placed anywhere (continuous models) or whether we have accurately determined set of possible locations (discrete models). 2.2 Transport distances d ij and d ik. To locate objects in a plane, we need to find their rectangular coordinates (x j, y j ) in appropriately selected units. Transport distance d ij represents the distance from the new site i and to the existing site j. The variable d ik represents the distance between the new objects, where k = 2,3,..., n. In the case of the discrete model, we need to use the actual distance. For continuous models, Gros mentions 4 types of distances [5].: - direct = + - direct-corrected = + - by axis = - quadratic = + The direct distance is an approximation of actual values and suits only for air transport. Therefore, in the second case, distances are adjusted with the correcting coefficient k > 1, by means of which we respect, for example, curvature of roads, etc. The third case assumes transport via routes perpendicular to each other (e.g. warehouses, etc.). The quadratic distance is used, for example, for locating and determining power of transmitters. 2.3 Transported quantities x ij and x ik. Another important set of information is the transported quantity. The variable x ij expresses the total quantity transported from the new object i to the existing location j for a longer period (usually a year). The variable x ik expresses the quantity transported between new objects. The total quantity transported for a period of time is usually transported in certain more or less regular deliveries, which we refer to as q ij. They may always be the same or may differ in their size. In this case, under certain conditions we may use in the models an average size of delivery q ij =x ij /n ij where n ij is the number of deliveries per year. 2.4 Shipping costs N ij and N ik and freight rates c ij and c ik. According to Gros [5], transportation costs from the i-th to the j-th object can be written as: = = for connecting existing and new objects = = for links between new objects to each other Where c ij and c ik are the freight rates in monetary units per unit of distance and weight (e.g. CZK to t and km) and w ij and w ik are referred to as weights of points, being essentially the rates for the total quantity transported per unit of distance / as CZK / km). The aggregate criterion cost function can then be written as: min= + In this case, the function is linearly dependent on the weight and the transport distance in the units of weight and distance used. In our experience, however, such a model is unusable in practice since transportation costs are usually counted differently. Freight rates of carriers are generally defined as discrete rates for each specific category (range) of weight in combination with a certain distance range. However, tariffs for the category of weight do not grow with transport distances linearly, but degressively and in a leap. If we term c ij as freight rates of one shipment of a certain weight, transported over a distance d ij from the i-th to the j-th object, then the sub-criterion cost function N ij will be dependent only on the number of transports carried out n ij : =. The aggregate criterion function to be minimized can then be written: ISBN: 978-960-474-324-7 88

min= = í It was Exnar, for example, who worked with a model for this type [6]. The price of transport assumed discrete values based on the weight of the shipment and transport distance. In some other models (for example [7]), we can choose mileage distance as the cost function. The distance d ij and the number of trips made n ij from the i-th to the j-th object then enter the function as its components. 2.5 Object Construction and Maintenance Costs These costs should be taken into account especially when making decisions about building a new warehouse, that is, whether to build a new warehouse or not. The cost of construction will then be affected mainly by the size and type of the warehouse, but also by the future location of the warehouse (for example, the price of land, different property taxes in different locations). Possible types of warehouse are described by Gros, for example [8]. According to Rushton [9], the costs of maintenance and operation of new objects include costs of: staff (45-50% of costs), building (25%, including rent or depreciation on building), building services (15%, including heat, light, power, building maintenance, insurance and rates), equipment (10-15%, including rental or depreciation, equipment maintenance and running costs) and information technologies (5-10%, including systems and data terminals). Ravindran [3], for example, includes costs of the construction and maintenance of new objects in his model (marking them f i ). These costs shall be included in the criterion function. 2.6 Basic localization models Typical problems by Ravindran [5] are the p-median and p-center problems. In p-median problem, the location of p facilities is determined so that the average travel distance for all customers is minimalized. Brito et al. [10] for example solve this problem by Particle Swarm Optimization model and Martínez et al. [11] by genetic algorithms. The objective function in the p-center problem is to minimize the maximum distance (or time) travelled by a customer to the nearest facility. For locating one object with an unlimited set of possible locations (the object can be placed anywhere), Gros recommends the following procedure [5]. The linear distance with correction is used as the most appropriate expression of distances. The objective function is: min= + The value of the coefficient k does have an impact on the value of the objective function, but does not affect the optimal location. For finding a solution, he recommends the following iterative method. First, he defines the auxiliary function,= + + Where ε is a very small value close to zero. Expressions for the sought coordinates can then be written as =, =,,, The algorithm can then be divided into several steps: - It is necessary to choose an input solution as a starting point, a place found using a different method can be used (e.g. using a quadratic distance [5], p. 120) - We choose a sufficiently small ε > 0, or alternatively a decrease of the objective function, which we consider to be irrelevant, z min. - In the k-th step we find new location coordinates of the object. - We go from the coordinates (x (k),y (k) ) to the coordinates (x (k+1),y (k+1) ), which we calculate using the formulas =,, =,, - Then we calculate z = z (k) z (k+1). If z min < z, we finish the calculation, we consider the found coordinates to be optimal. If not, go to the previous step. Searching for curves for the value of the objective function greater than the optimum location (e.g. because the site found is unsuitable) we need to look for points of the curve using an iterative process. We choose a new value z and randomly change coordinates, always leaving one coordinate constant and calculating the other one. It is advantageous to first estimate the extreme values in horizontal and vertical direction from the optimum point. If it is, for any reason, not possible to ensure the desired level of service from one object (e.g. longer time for execution of orders), we need to look for the location of a number of distribution centres. In this case, Gros created the following algorithm [12]: ISBN: 978-960-474-324-7 89

- In the first step, we locate one object using the method described above. - We check whether we can ensure the required level of service from this point. If an object does not suit, we move on to the next step. - We find the furthest point from the already localized object and we place a new object in the half of the linear distance. We place a second object at the same distance on the opposite mirror-image side of the original object. For each supplied place, we calculate product d ij x ij for both new locations and assign it to the object for which the product will be smaller. We thus divide the set of locations in two. We return to the first step and we again localize both new objects in each area. Thus, we find the optimal location of the object for each area individually. Albright [7] describes exactly on several practical examples how to find and calculate the optimal solution in a spreadsheet environment using MS Excel Solver tool. In first example he wants to design a hub system in the United States for Western Airlines. It is about finding the optimal number and location of centres so that each airport serviced was 1000 miles at furthest from a centre. The centre then must be placed on some of those airports. In terms of localization task classification, it is a discrete and multi-objective problem. Furthermore, a sensitivity solution analysis is dealt with, i.e. how the solution changes if the required distance from the airport to the centre changes (e.g. instead of 1000 miles the limit will be 800 or 1200 miles). Next example is location and assigning service centers at United Copiers. United Copiers sells and services copy machines to customers in 11 cities throughout the country. The company wants to set up service centers in three cities. They want to minimize the total annual distance travelled by its service representatives. Again, it is a discrete and multi-objective location problem, but in this case the limiting conditions and the objective function have changed. Again, the example includes a sensitivity solution analysis, i.e. how the total distance travelled varies depending on changes in the number of service centres. Albright also describes problem with location of one facility at any place [5]. Using the nonlinear programming method, it tries to find the location of a warehouse so that the total distance to customers is as minimized as possible. For the calculation, it is necessary to know the coordinates of customers served and the number of deliveries to these customers. It then results in coordinates of the location of the new warehouse and the related total mileage distance when serving customers. Again, the example includes a sensitivity solution analysis. Here, it addresses how the location of the warehouse changes if the number of deliveries to the individual customers changes. Ravindran [3] describes algorithm for solving single-echelon location-allocation problem. This problem is used to determine which sites to include in a network from a set of m production or distribution sites that serve a set of n customers. The fixed cost of including site i in the network and operating it across the planning period is given by f i, and each unit of flow from site i to customer j results in variable cost c ij. Site i has capacity K, and the total planning period demand at customer j is given by D j. Therefore, our objective is to + =,, =1,, =1,, 0,1, =1,, 0, =1,,;=1,, Where decision variables y i (i = 1,,m) determine whether site i is included in the network (y i = 1) or not (y i = 0) and x ij (i = 1,,m; j = 1,,n) determine the number of units shipped from site i to customer j. As parameter f i and c ij are constants, this problem is a mixed-integer program (MIP). 3 Warehouses localization solution in a distribution company While addressing the inventory optimization task in a distribution company in the Czech Republic, we encountered the following problem: the company has customers across the entire Czech Republic, supplying them from a single central warehouse. The company uses a certain number of shipment warehouses, but only exclusively for individual key customers and in their premises. In an effort to achieve savings in distribution costs, a question was asked whether it would be appropriate to establish additional local warehouse for selected products, then where would be the best location for this warehouse and whether and under what conditions it would be economically advantageous to set it up. ISBN: 978-960-474-324-7 90

Figure 1: Customers and warehouse location. 70 60 50 40 30 20 10 0 0 20 40 60 80 100 120 140 160 180 customers central warehouse warehouse by distance warehouse by costs 3.1 Description of the baseline situation Based on the distribution of the current customer on a common map, Figure 1 was made. It shows positions of individual customers in the coordinate system, the existing warehouse and optimal location of the new warehouse, calculated by the mentioned procedure on the basis of both optimization functions. These coordinates can be determined via GPS. Just from looking at Figure 1, it is clear that the location of the warehouse in this area could be advantageous. It needs to be proved with a number of calculations whether it is economically efficient. With the previous evidence of deliveries we know the total required amount x ij of products required by customers and number of deliveries n ij. From this we can estimate the average size of one delivery q. Our input values are in Table 1. 3.2 New cost optimization model To optimize localization costs, we need to know the shipping charges and their properties. When analyzing this task, we found that the model described by Gros (see above) cannot be applied to find solution because linear tariffs c ij per unit of quantity and distance are hardly utilized. Therefore, we constructed our new cost model, based on the tariff tables in which tariff rates are determined in combination for certain ranges of weight and distance. In our case, we obtained a basic price list for transportation of shipments from the carrier, a section of which is given in Table 2 [13]. Table 1: Input values. customer x i y i x ij [t] n ij q [kg] A 127,3 26,8 5 50 100 B 115,5 35,3 8,4 24 350 C 139,5 13,3 60 100 600 D 123,3 17,5 10 75 133 E 160,3 35,8 56 80 700 F 106,3 28,3 8 32 250 G 159,7 20,0 81 90 900 H 112,3 14,8 27 60 450 I 135,2 31,3 18 45 400 Actual 24,75 63,8 x x x warehouse Table 2: Pricelist of transportation in CZK. Weight Distance tarifs [km] [kg] to 50 100 200 300 400 500 50 378 416 517 555 580 616 75 466 510 636 689 742 785 100 542 605 775 845 882 948 150 680 781 1014 1105 1159 1238 200 767 882 1210 1247 1328 1439 300 979 1129 1497 1652 1733 1894 400 1159 1328 1787 1975 2114 2293 500 1310 1510 2046 2275 2426 2646 700 1534 1818 2505 2795 3015 3292 1000 1866 2198 3067 3477 3754 4123 1500 2079 2587 3696 4256 4620 5072 ISBN: 978-960-474-324-7 91

3.3 Calculating shipping costs The knowledge of the existing warehouse location allows us to calculate also the present cost of transporting goods to customers. Based on the distances of the existing warehouse and customers (the distance used is the linear distance with correction - see the Gros's method) and the known weight of a delivery to a customer, the price for transport of individual contracts c ij can be found in Table 2. We will find the total cost of transport from the thus located warehouse as =. The resulting values d ij and N ij are clearly shown in Table 3. The total current shipping costs are N = 1 407 909 CZK. Table 3: Current status. Customer Distance d ij [km] Costs N ij = c ij n ij [Kč] A 327,062 44100 B 285,360 47400 C 376,112 301500 D 326,653 86925 E 415,235 263360 F 266,826 52864 G 425,640 371070 H 300,988 145560 I 345,397 95130 sum x 1407909 3.4 Location of the warehouse with respect to the minimum distance In our case, we first searched for the optimal location of the warehouse based on the distance minimization, the distance used is again a linear distance with correction. Using the Solver tool of MS Excel 2010, we found the optimal warehouse location for the criterial function of minimization of the total distance travelled for k = 3 (in our case, this value most closely matches the real distances) at coordinates x = 134,234 and y = 21,623 (the value of the objective function z = 30767,882 km). Solver reached the same result using the GRG Nonlinear method and the genetic algorithm method. More to the calculation procedure can be found in Albright 2009. The solution includes calculation of the distances between the new warehouse and individual customers, see Table 4. Based on these distances and the known weight of a delivery to a customer, the price for transport of individual contracts c ij can be found in Table 2. We will find the total cost of transport from the thus located warehouse as =. In this calculation, N = 785 515 CZK. Table 4: Minimum distance optimization. Total Costs Distance Customer distance N d ij [km] ij = c ij n ij d ij n ij [km] [Kč] A 25,960 1297,996 27100 B 69,586 1670,055 31872 C 29,547 2954,679 153400 D 35,055 2629,126 51000 E 89,017 7121,394 145440 F 86,162 2757,177 36128 G 76,554 6889,876 197820 H 68,911 4134,637 90600 I 29,177 1312,943 52155 sum x 30767,882 785515 3.5 Warehouse location with respect to minimum costs Optimization by the total cost is more complicated. For the calculation we again used the Solver tool and its solution method using genetic algorithm. In this case, the criterion function (target cell in the Solver parameters) is defined as min= = í where n ij is known but c ij is part of optimization through the d ij values. The changed cells are x and y coordinates of the new object. When calculating using the genetic algorithm, we must specify a range of values of the changed cells, 0;170 and 0;70 in our case. The value c ij can assume only discrete values and is a function of the average weight of the shipment q ij (it is constant for the connection i and j) and the distance d ij (varies during the calculation). The calculation by Albright was adapted for our model by the authors. In this case, Solver found the location of the warehouse at coordinates x = 143,248 and y = 22,379 with a value of the criterial function z = N = 774 986 CZK. Individual distances and costs are in Table 5. The found coordinates are part of the optimal solution, but they may not be the only optimal ones. The objective function here is not continuous, but due to the discrete values c ij, it assumes discrete values for certain areas. This area can be calculated iteratively similarly to Gros [5]. In our example, we can define the area of the optimal solution approximately with 143,154;143,402 a 21,959;22,969 coordinates, which converted to kilometres is approximately 0,75 times 3,03 kilometres. ISBN: 978-960-474-324-7 92

Table 5: Minimum costs otimization. Costs Distance Customer N d ij [km] ij = c ij n ij [Kč] A 49,648 27100 B 91,826 31872 C 29,467 153400 D 61,608 58575 E 65,101 145440 F 112,258 47904 G 49,870 167940 H 95,587 90600 I 36,044 52155 sum x 774986 3.6 Assessing economic efficiency of the designed warehouse The cost criterial function is decisive for the economic evaluation of warehouse localization. By comparing current shipping costs and possible shipping costs from the new warehouse, located in the optimal area we found, we demonstrated potential savings 632 923 CZK. The question is how we can use this value to assess the appropriateness of establishing a new local warehouse. The value thus found indicates savings in transport costs to customers, but the cost of renting or establishing and maintaining the new object (see above) and shipping costs between the new and the central warehouse are not included here. It is also necessary to make a decision whether to build or rent a new warehouse. In case of rental, it can be said that the new local warehouse will pay off when the total annual cost of renting and operate a warehouse and transport of goods between this warehouse and the central warehouse is lower than our calculated savings 632 923 CZK. If it is not possible to rent a warehouse or there are attempts to establish an own warehouse, we need to assess return on investment of the construction and maintenance of a new warehouse. 4 Conclusion In the first part of our article, we pointed out the wide variety of different localization models that can be found in the professional literature. At the same time, we critically defined ourselves against some widely reported assumptions and variables that occur in the models. Particular attention was paid to freight rates that play a crucial role in cost optimization. Most of these cost models are based on a highly simplified assumption that the total shipping costs are a linear function of a constant rate c ij and weight (or volume) of freight and distance in km. While such models can be elegantly solved, they are of little use in practice. In fact, shipping costs are derived from certain rates (c ij rates), which are set for a particular range of weight (volume) of the shipment and usually in combination with a certain zone of distance in km. If the shipment is within these limits, it is transported for this constant amount c ij. The criterion cost function thus assumes a non-linear and discontinuous nature. On these assumptions, we built our new cost optimization model and showed how to solve it in Excel using the Evolutionary Solver tool. For the sake of completeness, it should be added that actual shipping costs may still differ from those calculated by us on the basis of other factors that cannot be generally included in the model. (These may include various fees, tolls, vehicle utilization discounts, etc.) Despite these problems, we believe that our model can be beneficial for more realistic estimates of the shipping costs than those provided by standard models. References: [1] Vlckova V., Patak M., Role of Demand Planning in Business Process Management, Proceedings of the 6 th International Scientific Conference Business and Management 2010 Vilnius Gediminas Technical University, Lithuania, May 13-14 2010, Vilnius, Lithuania, 2010, pp. 1119 1126, ISSN 2029-4441. [2] Schwartz B.M., Map out a site rules, Transportation and Distribution, Vol. 40, No. 4, 1999, pp. 67-69. [3] Ravindran A.R., Operations Research and Management Science Handbook, Boca Raton, FL: CRC Press, 2008, ISBN 0-8493-9721-9. [4] Chopra S., Meindl P., Supply Chain Management: Strategy, Planning, and Operation, 2 nd ed., Upper Saddle River, NJ: Pearson Prentice Hall, 2004. [5] Gros I., Matematické modely pro manažerské rozhodování [Mathematical Models for Managerial Decision Making], Vydavatelství VŠCHT Praha, 2009, ISBN 978-80.7080-709- 5. [6] Exnar F., Machač O., The travelling salesman problem and its application in logistic practice, WSEAS Transactions on Business and Economics, Vol. 8, No. 4, 2011, pp. 163-173, ISSN 1109-9526. ISBN: 978-960-474-324-7 93

[7] Albright S. Ch., Winston W.L., Management Science Modeling, South-Western Cengage Learning, 2009, ISBN -13:978-0-324-66346-4. [8] Gros I., Logistika [Logistics], Vydavatelství VŠCHT, Praha, 1996, ISBN 80-7080-262-6. [9] Rushton A. et al., The Handbook of Logistics & ditribution management, London: Kogan Page, 2010, ISBN 978-074-9457-143. [10] Brito J., Martínez F.J., Moreno J.A., Particle Swarm Optimization for the continuous p- median problem, Proceedings of the 6 th WSEAS Int. Conference on Computational Intelligence, Man-Machine Systems and Cybernetics, December 14-16 2007, Tenerife, Spain, 2007, pp. 35-40. [11] Martínez H.D., García C.A.R., Rosales A., Optimization of Distribution of Centers in the Supply Chain using Genetic Algorithms, Recent Researches in Automatic Control and Electronics, 2012, pp. 98-103, ISBN: 978-1- 61804-080-0. [12] Gros I., Hanta V., Grosová S., More than one Distribution Centres Location Problem, Economics and Business Management, No. 1/2005, VUSI Kosice, 2005, pp. 64-69, ISSN: 1336-4103. [13] Exnar F., Solving of Transport Problems in Logistics, Diploma work, University of Pardubice, 2010. ISBN: 978-960-474-324-7 94