Model for distribution of warehouses in the commercial network in optimising transportation of goods

Abstract A model for distribution of warehouses in the commercial network in optimising transportation of goods has been established. The target function in the model includes: fixed prices for construction of big and small warehouses; variable prices for transportation of raw materials and goods between warehouses and commercial points. The algorithm was created for optimisation of local roads network between commercial points. As substantiated, if density of distribution of target function values in the zone of extreme is fairly high, the independent random search is sufficiently effective even when comparing it with regular methods.


Introduction
The current commercial giants, e.g. Hyper RIMI (or MAXIMA) have located their commercial points in various places of the country. They have been accumulating and producing certain goods in big warehouses, and storing other goods in warehouses located in certain points. Goods are being transported between the above points by road transport means of certain capacity. On the one hand these increase the flows of transport means by burdening the roads, on the other hand they pollute the environment with the exhaust gases and, moreover, their price increases due to irrational transportation of goods. Therefore it is necessary to optimise distribution of big and small warehouses within the network of these commercial giants. Thus, it is necessary to define the number and place of big and small warehouses the construction and transportation of which could incur the lowest total goods costs (z). The argument that it is necessary to optimise distribution of warehouses within the commercial network, has been received via the synthesis of a typological structure of the regional system of the freight road transport, see researches by Baublys andIšoraitė (2006), andBaublys (2003).
Having carried out the synthesis of a typological structure of the road transport freight regional system, it was substantiated that it is necessary to optimise distribution of warehouses within the network, as well as to optimise the local roads network between commercial points.

Optimisation model
Formally the task looks as follows: It is necessary to minimise Under the following conditions where f i − construction price of a warehouse in place i, beside which manufacturing of certain goods is executed; g j − construction price of a small warehouse in j place; c ijk − price of delivery of goods to commercial point k from a warehouse-manufacturer, located in place i to a small warehouse located in point j; y i − binary variable which is equal to 1, in case if a warehouse-manufacturer is in place i, and equal to 0 in other case; z j − binary 23(1): -9 variable which is equal to 1, in case if a warehouse is in station j, and equal to 0 in other case; x ijk − incessant variable which equals the part of demand in commercial point k, warehouse-manufacturer i of supplied goods via warehouse j. Target function (1) consists of a fixed construction price f i and g j correspondingly of warehouses-manufacturers and small warehouses, and variable prices c ijk (of raw-material and goods to be transported to a warehousemanufacturer i), transportation costs between stations i and j and transportation costs between warehouse j and commercial point k. Equality (2) means that demand of each commercial point should be fulfilled. Terms (3) and (4) mean that demand of a commercial point k can be fulfilled only for goods from the warehouse located in territory i, via a small warehouse located in territory j. Inequality (5) guarantees that a big warehouse is also a manufacturer, and term (6) − that construction of big and small warehouses has been completed, and that part of supplied goods is a plus.
In case if values of variables y i and z j in expressions (1)−(6) are known, optimal values of remaining variables are estimated in the following way: variable x ijk of each variable k corresponds to the minimal c ijk value, in case if condition y i = z j = 1 is fulfilled, and variables x ijk are considered as equal to 0. Then the task (1−6) can be solved by selecting vectors Y = (y i ) and Z = (z i ). Each of iteration respective variables is fixed 0, others -1, and some of them remain unfixed. Let mean the set of variables y i , having corresponding values 0, 1 and free variables, and 1, 2, ..., mean corresponding sets of variables z j . After having changed fixed variables by their values in the formulas (1−6), we get the following task which has to be minimised: under the following conditions, 1; 1, 2, ..., ; 1, 2, ..., { } ∈ ∈ ∈ 2 2 , 0, 1 , , 1, 2, ..., .
Let c mk mean the minimal goods transportation price to k commercial point from a big warehouse ∈ 1 The main condition for the delivery of goods from the warehouse-manufacturer i via the warehouse j in case of any optimal answer (solution) is c ijk ≤ c mk . It allows to get upper margins m i and n j of the number of commercial points, to which goods from warehouses i and j are delivered, i.e.
{ }  1 2 : { }  1 2 : Further c ijk to all ∈ ∈   1 2 1 2 , I I I I i K K j K K and k = 1, 2, ..., n is defined as follows: And ′ rk c because of k=1, 2, …, n is found from The lower margin of the answer of the task (13−19) is: The value of optimal solution (13)−(19) cannot be higher than If a warehouse-manufacturer is in point i, then value (26) will decrease (or increase) by values ( ) 1 2 1 max 0, If a warehouse is in point j, then value (26) will decrease (or increase) by value ( ) 1 2 1 max 0, Since total (26) decrease due to a possible assumption on location of several big and small warehouses cannot exceed decreases of the sum, corresponding to the location of each big and small warehouse, the following result is valid: (13)−(19) the lower margin of solution is de ned as follows: Another algorithm, based on the method of branches and margins, was tested.
1st block. Let z opt. be value of the best solution which corresponds at the beginning to a free big number.
2nd block. e rst check for optimality. We estimate values of m i , n j , ijk c and rk c according to (21)−(24). Find z´ according to (25). If z´ ≥ z opt. , we proceed to 5th block.
3rd block. Checking of solution. If variables y i and z i are xed under 0 or 1, then we x the found answer, resume value z opt. and proceed to 8th block.
4th block. e rst conditional checking of optimality. We estimate values p of all 5th block. e second conditional checking of optimality. We estimate 1 I j K g j of all according to (28). If g j < 0, z j = 0. 6th block. e second checking of optimality. We estimate z˝ according to (29). If z˝ > z opt. , then we proceed to 8th block. 7th block. Selection of a move. If during checking at least one variable is xed in 4th and 5th blocks, we proceed to 2nd block. Otherwise If p K ≥ g l , then y K = 1, otherwise y K = 0. We proceed further to 2nd block. 8th block. Regression. We will search for the last variable y K or z l , xed under 1. If there is no such variable, estimation is nalised. Otherwise all free a er y K or z l variables are equal to 0; we x y K or z l under 0 and proceed to 2nd block. Algorithm ends a er the nite number of steps. We return back to block (8) only when a variable is xed or when value of a variable changes.

Optimisation of local roads network between commercial points
e given network under establishment N with m nonoriented branches and n hubs. e initial information is presented in distribution density g(X) s-meter vector X area Y. Components X are lengths z i of limits N and transport connections p ijt between hubs (i, j ) in pair years t during the planning period [Q, T].
e trajectory f has to be found in which , min Total expenditure F(f, X) − an additive (according to network branches) function, basically depending on branch loading vector ( ) Q t , the components of which are , where p v − probability in accepting one or other routing criteria (distance, time of transportation, transport costs); v lijt − supplementary binary values, de ned by a complex of tasks about the shortest road for each connection: where v(q lt ), b(q lt ) − average speed of transportation and cost price, when technical level is q lt ; 1, when , 0, when . e set of trajectories G, in which functional (1) is minimised, is predetermined by the following conditions: (37) − indicate that any reconstruction measure leads to technical level which is not lower than required by design standards; (38) − de ne whether there are social restrictions: among them -restrictions of the state, reachable in year t, guarantee connection between all points of a region; (39) − de ne whether there are any resource restrictions, de ned for each resource d for all years t.
When network is optimised, then technical level of l branch in year t can be conditionally de ned as follows:

5-9
i.e. anticipate an even change of the technical level from the initial q l0 up to the final q lT , which is a controllable variable.
The task of optimisation of the current network (this is the totality of branches, by which transport connections can be maintained at any moment) − the search for the status ( ) q T network N, which could minimise total expenditure: Here markings are the same as in (32)−(34). Technical level is found simply: where N i-1 , N i − technical level of the defined mark i will be applied a priory. In expression (41) functions f 1 , f 2 , f 3 evaluate all costs, depending and not depending on traffic, as well as their discounting. These functions do not depend on the decision and, therefore, can be estimated approximately in the form of value tables, and size of tables should not exceed 10×10. Of course, traffic intensity at the beginning N t (0), as well as the initial technical levels q l0 and lengths r l are known. Since the final state ( ) Simulation procedure of optimizing (141−(46) in fact is discrete and multi-extreme.
When newly designing or developing the current network, the pair τ ( ) q and θ ( ) q of network N has to be found (usually τ = s and θ = 0), minimising the costs: