Engineering Costs and Production Economics. 15 (1988) 3 1 l-3 I5 Elsevier Science Publishers B.V., Amsterdam - Printed in Hungary 311 THE JOiNT OPT~M~ZAT~O~ OF THE INPUT FLOW AND THE SIZE OF THE STOREHOUSE” zyxwvutsrqponmlkjihgfedcba Marija Bogataj and Ludvik Bogataj Edvard KardeQ University. Ljubljana, Yugoslavia INTRODUCTION In the structure of water supply costs for the non- gravitational system the costs of energy exceed the costs of investment. Let us study the case in which the cost of electric energy differ periodically during the day and according to the total power of working pumps. We know the periodical behaviour of demand in each hour per day (more generally in each element of period). We have to control the switching of pumps in the way that we minimize the total costs of electric energy. This contro! is limited by the size of the reservoir and by the level of the investment, and therefore the size of the reservoir must be included in the optimization process. Because of stochastic demand it is advisable to vary the level of confidence and to choose the optimal solution according to this variation. FORMULATION OF THE INVENTORY PROBLEM We propose the following inventory problem: How to optimize the input of the goods which have discrete values by nature and which is con- sidered in discrete time intervals so that the total transport costs will be minimal. * This paper is a part of the research project “The models supporting decisions in building and management of communal supply systems”, carried out by Institute of Communal Economy, Faculty of Architecture, Civil Engineering and Geodesy and it is finan~ajly supported by the Research Council of Slovenia. The goods can be delivered to the central store- house (reservoir) or directly to the nsers under the following conditions: (I) There is a known probability distribution of the demand of goods (quantity of water) in each part of a period. (For example, there are known periodical indexes for each hour per day and the probability distribution of those indexes as well as the probability distribution of daily demand.) (2) The intensity of the supply flow can be ‘p,, rpl+~z,~t+~2+~3r... orcp,+92+...+ cpII. (For example, we have n pumps, each of capacity cp, each of them can be switched on or switched off. So the intensity of the supply flow can be p, 2~, 2~,. . . or ncp. Second example: We have n trucks each of capacity q, we can load r trucks per element of period (per hour) r = 1, 2, 3,. . . or n. The price of transporting one unit of goods depends on r.) (3) The control of the input flow is limited by the size of the storehouse (the reservoir) in this system which is also connected with the transporting cost of the input flow and we assume that the building expenses of the storehouse are proportional to its capacity. THE MATHEMATICAL MODEL Let us set up the scheme which ia presented in Table 1. There we have : Q,(a) demand of goods (of water) which will not