OPERATIONS RESEARCH SYMPOSIUM
I.
n most tankage systems in the chemical industry, it is impossible to keep all inventory levels constant in the face of uncertain supply and demand rates. This unhappy situation occurs whenever there are more levels to be controlled than there are adjustable production rates affecting them. But since in practice inventories need not be constant as long as fluctuations are small compared to storage capacities, one often can achieve acceptable performance by manipulating whatever production rates are free to be adjusted. If operations are considered satisfactory as long as all inventories are within bounds, then a reasonable measure of control dfectiveness would be the probability of satisfactory operation. A previous article (4, using the concept of partial control, gave necessary conditions for achieving maximum probability of satisfactory operation. To simplify the derivations, that article assumed no interaction between the various manipulated variables, a situation seldom encountered in practice. The present work removes this restriction, giving the necessary conditions in a form which, although having a more complicated notation, is more straightforward for engineering computations, because no preliminary change of variable is required. The main idea of partial control is to partition the state space generated by the controlled variables into two orthogonal subspaces, one completely controllable and the other completely uncontrollable. If nature is kind enough to place the uncontrollable variables within certain readily computable bounds, then the control system must drive the controllable variables to keep the inventories inside the tanks. The preceding article gave upper and lower bounds on the controllable variables which must be satisfied if operations are to be satisfactory, but these bounds are not sharp enough to give a unique control law, or even to exclude unsatisfactory values. Consequently it was suggested to drive each controllable variable to a point halfway between its least upper bound and its greatest lower bound, a policy which happens to guarantee satisfactory operation when there are no more than two manipulated variables. Since the previous article gave a numerical example involving only one manipulated variable, the 60
INDUSTRIAL A N D ENGINEERING CHEMISTRY
Auto matic Partial Inventory Control with Interaction DOUCLASS J. WILDE
The manipulation of two interacting variables implements methods for regulating inventory on a chemical plant without a need for forecasting upsets or errors
present paper works a problem with two maniptilated variables to illustrate the ideas in greater, more realiitic, detail. The earlib paper mentioned that often one can achieve suitably high, although not maximum, probability of satisfactory operation simply by holding all controllable variables at their initial values. This is sometimes better than the control policy more widely used at pnsent, namely, to hold the original manipulated variables (the production rates) constant. Such control laws being much easier to implement than optimal ones, they deserve consideration in any practical situation, especially since the computations are much simpler. The style of this article is less formal than that of the earlier research paper in order to make the methods and ideas accessible to practicing chemical engineers and operations analysts concerned with inventory control. First, a specific numerical problem involving five inventories affected by only two interacting production rates is described (Figure 1). Then the results of three dflerent control policies-no control, simple control, and optimal control-are presented. Three aspects of the policies are compared in Table I: storage capacity required, probability of Satisfactory operation, and length of planning horizon possible. Finally the detailed computations are given, mathematical derivations being relegated to an appendix. The theory should be useful in existing systems for improving the probability of satisfactory operation or extending the planning horizon, or else for decreasing the storage requirements in new plants.
Problem: Two Flows for Five Tanka Figure 1 is the flow diagram for the specific numerical problem studied. For every 3 units produced in plant 1, shown as a control valve with flow rate ui, 1 unit from tank 1 and 2 units from tank 5 are consumed, yielding 2 units for tank 2 and 1 for tank 3. The second plant takes 1 unit from tank 3 and 2 from tank 4 for every 3 produced and sent to tank 5. The quantities stored in each tank at a given instant of time are XI, . . ., xy. No generality is lwt in simplifying by taking the initial values of the x, to be zero. The system is considered to be operating satisfactorily as long as the x i s are within
Figwe 1. Morniolflow cliogrcrrn
