This paper characterizes optimal service policies in terms of the frequency and timing of services which are intended to maintain a stock of assets. The model is non-stochastic. The results are obtained by a two-stage step-wise minimization procedure wherein dynamic programming is first used to characterize sub-optimal policies and then the calculus of finite differences is utilized to select the optimal policy. The effect of the time horizon on the optimal policy is emphasized throughout and, in two specific interpretations of the model, the classical results of the economic lot-size and equipment replacement policy are shown to be limits of more general policies.
This paper describes a method of minimizing a strictly convex quadratic functional of several variables constrained by a system of linear inequalities. The method takes advantage of strict convexity by first computing the absolute minimum of the functional. In the event that the values of the variables yielding the absolute minimum do not satisfy the constraints, an equivalent and simplified quadratic problem in the “Lagrange multipliers” is derived. An efficient algorithm is devised for the transformed problem, which leads to the solution in a finite number of applications. A numerical example illustrates the method.
In this note we study dynamic inventory problem for a follow-on provisioning in which program length is subject to uncertainty with a known distribution. It is shown that under rather general cost conditions, optimal policy is of (S, s) type. This is true whether or not there exists a time lag in delivery provided that excess demand is always backlogged. The case of an infinite program horizon is also briefly discussed. MODERN INVENTORY theory has been a relatively recent development, but its brief existence has proceeded at least along two fronts: theory and applications. The earliest work falling under this theory is that by Masse [6], followed by those of Arrow, Harris, and Marschak [1], Dvoretzky, Kiefer and Wolfowitz [5], Bellman, Glicksberg and Gross [4], and Modigliani and Hohn [7]. An excellent account of historical back ground of this theory was given by Arrow [2], and its applications to a great variety of economic and business may be found in many journals in such fields as operations research, management science, and production control. A detailed discussion of the nature and structure of inventory problems was given by Arrow, Karlin, and Scarf [2, Chap. 2], and a simple mathematical exposition of theory may also be found in Bellman [3, Chap. 5]. Briefly, problem involves determination of (optimal) stock levels for inventories which extend over a sequence of time periods and are subject to fluctuating demand in each such period. Such may arise in a number of ways, e.g., in scheduling production or determining distribution of commodities over certain markets, in finding replacement policy for aged equipment, or in combinations of some or all of these features. The treatment of demand in modern inventory theory is usually handled in two ways: (1) time periods are regularly spaced, and demand in each period is a random variable with a known probability distribution; or (2) size of each demand is fixed but times at which successive demands occur are random variables. We study here an inventory problem which is in some sense a hybrid between two and which occurs frequently in involving follow-on provisioning. (Follow-on provisioning is a subsequent provisioning of same item from same supply source.) Here, time periods are equally spaced (corresponding to budget cycles) and demand in each period is a random variable subject to a known
The von Neumann-Morgenstern theory yields a determinate solution (a unique payoff vector) only for the two-person zero-sum game and some other special cases. But if we adopt a small number of additional rationality postulates we obtain determinate solutions for all known classes of games. The rationality postulates needed include the expected-utility maximization postulate; postulates involving payoff-dominance concepts; and postulates imposing certain consistency requirements on the expectations that rational players can entertain about each other's strategies. For some games the solution can be defined essentially in terms of payoff-dominance concepts. But more generally the solutions depend on the analysis of bargaining among the players.
In his article, “Compensatory Cyclical Bank Asset Adjustments,”1 Dudley Luckett states that in recent periods of monetary restriction commercial banks have tended to liquidate holdings of long-term government securities, rather than—as some writers had assumed—drawing first on short-term governments. Luckett accounts for this behavior, in part, by the desire of commercial banks to maintain certain liquidity standards and the fact that, from a liquidity stand-point, bank loans are more akin to longer-term investments than to Treasurv bills. I have no quarrel with Luckett's emphasis on the role played by liquidity in influencing bank behavior. However, I believe that Luckett's paper suggests greater sophistication in the implementation of bank portfolio adjustments during the period in question than experience indicates. In examining commercial bank holdings of government securities, Luckett uses 6-month moving averages of commercial bank holdings of all government securities—those maturing within 1 year, those maturing between 1 and 5 years, and those maturing in more than 5 years. This is the only form in which data for all commercial banks are available, and, in the present context, it has some obvious short-comings. The data make no distinction between bank sales of longterm Treasuries and the shortening of outstanding maturities through the passage of time. As a matter of fact, most of the reductions in bank holdings of long-term governments with which Luckett is concerned have not arisen through market sales of such securities. During periods of rising interest rates, banks have made practically no acquisitions of newly issued long-term treasury securities, thereby allowing average maturities to shorten with the passage of time. This does not contradict Luckett's discussion of commercial bank preoccupation with liquidity and their substitution of loans for longterm Treasuries. Nevertheless, I believe a more accurate picture of actual bank asset adjustments is interesting historically and potentially useful in the formulation of future monetary policy. Each month the Treasury collects data on the ownership of government securities on an issue-by-issue basis. Data on commercial banks currently include about 6,200 banks, accounting for more than 85 per cent of bank assets.2 These data enable one to obtain figures on net changes in holdings of each Treasury issue resulting from the acquisition, of new issues, the maturing of outstanding issues, and net purchases or sales of each issue. Figure 1 is derived from survey data published in the Treasury Bulletin. “Monthly Net Market Transactions” shows bank net purchases and sales of outstanding government securities, exclusive of transactions in new securities in their month of issue. The chart shows that reductions in bank holdings of long-term Treasury securities have not been effected primarily through prior-to-maturity sales. In 1956 and 1957, when banks included in the survey reduced holdings of Treasuries maturing beyond 5 years by about $12 billion, these same banks actually were net purchasers of such securities in the bond market. Reductions in holdings of long-term governments, for these banks taken as a whole, were accounted for wholly by the shortening of maturities through the passage of time. Most banks apparently have been able to meet loan demand and maintain appropriate liquidity standards without selling substantial quantities of long-term government securities. Nevertheless, in periods of advancing interest rates there is a strong incentive for banks to sell long-term investments, even if they are in a strong liquidity position. First of all, it is always desirable to sell a security when one expects to be able to buy it back or to be able to buy a similar security at a later date at a lower price—assuming that a sufficient return can be earned in the interim. But tax reasons afford further incentives that are sometimes overlooked. For commercial banks, net capital losses are deductible against ordinary income. Consequently, a bank may acquire loans or investments with a greater market value than the proceeds of an investment sold below its original cost. The fact that net capital gains are taxable for banks at the lower capital gains rate affords an additional opportunity for banks to gain by trading in securities. Although banks were not net sellers of long-term Treasuries in 1956–57, data on bank earnings3 show that member banks realized net security losses of $326 million in 1956 and $211 million in 1957. Treasury Bulletin data suggest strongly that banks realized most of their losses on sales of securities with maturities in the 1–4-year range. This apparently enabled banks to acquire a greater volume of loanable funds per dollar of capital loss. The passage of time pushed long-term Treasuries into the intermediate range, thereby filling the gap from security sales. A number of factors contributed to the limited market sales of long-term Treasuries by commercial banks. Many smaller banks and banks outside major financial centers had little experience or “know-how” in security trading. Large security losses present an ugly accounting picture: they may be charged to net operating earnings and may be difficult to explain, whereas unrealized losses tend to be obscured in bank statements. Banks had little recent experience with respect to pronounced interest-rate movements, and what seemed obvious from the standpoint of hindsight was perhaps not so obvious to portfolio managers in 1956–57. Long-term interest rates did not move up continuously through 1956 and 1957. Consequently there were a number of occasions when it may have appeared to portfolio managers that interest rates were about to turn around. Figures are month-end holdings and monthly net market transactions by commercial banks of government securities with maturities in excess of 5 years. Net market transactions exclude transactions in newly issued government securities during the month in which the securities are issued. Figures are derived from Treasury Bulletin monthly ownership survey. Interest-rate experience from the previous cycle was reflected somewhat in bank behavior in the 1958–60 period. Banks realized substantial investment gains in 1958 and extended maturities much less in 1958 than they did in 1954–55, fearful of an eventual shift in monetary policy. And in 1959 and early 1960 banks were net sellers of long-term securities in most months. Security losses of member banks were $792 million in 1959—substantial compared with previous experience. Thus it appears that banks did become somewhat less averse to realizing security losses. Nevertheless, most net changes in bank holdings of long-term government securities occurred through the shortening of maturities with the passage of time. Most security losses realized by commercial banks were through sales of securities maturing within 5 years. In recent years, as Luckett points out, bank holdings of long-term Treasuries have declined substantially. Many banks appear to have come to the position that long-term Treasuries afford little liquidity4 and insufficient return relative to loan and investment alternatives. Many banks have shifted from long-term Treasuries and from Treasuries in general into tax-exempt state and local government bonds offering higher taxable equivalent returns, but the shift has been less than one might have expected. For an explanation of the sluggish adjustment on the part of some banks, we might turn to the Federal Reserve “Form for Analyzing Bank Capital” which was discussed by Luckett and various other bank-examiner tools like capital-risk asset ratios. Tax-exempt bonds, regardless of their maturity, score lower on the liquidity scale or higher on any risk scale. At times, this probably has deterred bank purchases of tax-exempts that may have been more attractive than long-term Treasuries from both an earnings and a liquidity standpoint. Thus bank examiner rule-of-thumb may have led to less rational bank investment policy.
Discusses the importance of decision-making in management. Assessment of the literature on decision research; Different approaches to decision-making; Definition of a decision process; Uniformities of decisioning; Scheme for the analysis of decisioning; Advantages of the scheme.