The capital supply curve in capacity expansion models: Some economic and algorithmic aspects

Capacity expansion models typically minimize the discounted cost of acquisition and operation over a given planning horizon. In this article we generalize this idea to one in which a capital supply curve replaces the usual discount rate. A capital supply curve is a means to model financial outlook, investment limits, and risk. We show that when such a curve is included in a capacity expansion model, it will, under certain conditions, provide a less capital intensive solution than one which incorporates a discount rate. In this article, we also provide an algorithm that solves capacity expansion models that incorporate a capital supply curve. The attractive feature of this algorithm is that it provides a means to utilize the “discount rate” models efficiently. Throughout, we give applications in power generation planning and computational experience for this application is also presented.     

A cost-benefit analysis of military aircraft replacement policies

This paper describes a method of solving aircraft service life problems. The particular application concerns aircraft in the Naval Advanced Jet Training Command. The method of solution is comparative present value analysis of alternative replacement policies. The likely risks of estimation errors are reflected in the comparisons of present values. Differences are noted in the benefits associated with each policy, but external to Naval Aviation. Since the values of these benefits can be determined only at a higher level of decision-making, the result of the study is not a conclusive selection among policies, but a schedule of present values on the basis of which, together with values of the external benefits, a decision can be reached. This paper discusses replacement policies for aircraft used in the Naval Advanced Jet Pilot Training mission. Taking engineering technology and the training syllabus as given, four feasible plans for introducing replacement aircraft into service are evaluated in terms of the present values of differential costs associated with the plans and in terms of the likely errors in cost estimates used in calculation of the present values. The trade-off between present value of costs and planning flexibility is emphasized in choosing a recommended time pattern of aircraft replacement. The specific aircraft mixes considered are the TF–9J/TAF–9J and the TA–4F/A–4B. The first is the currently employed mix; the second is the proposed replacement. The problem is to select an optimal time-pattern of replacement of F–9's by A–4's, given technological differences favoring the A–4 and increasing costs of maintaining squadrons of F–9's. Replacements by aircraft types other than the A–4 are considered impractical. Four feasible plans for introducing A–4's through a 5-year period are evaluated in terms of current best estimates of the related costs of the plans and in terms of the flexibility of modifying each plan given future better information concerning the relevant costs. The method of analysis is comparative present value of expected costs.     

On the theory of semi-infinite programming and a generalization of the kuhn-tucker saddle point theorem for arbitrary convex functions

We first present a survey on the theory of semi-infinite programming as a generalization of linear programming and convex duality theory. By the pairing of a finite dimensional vector space over an arbitrarily ordered field with a generalized finite sequence space, the major theorems of linear programming are generalized. When applied to Euclidean spaces, semi-infinite programming theory yields a dual theorem associating as dual problems minimization of an arbitrary convex function over an arbitrary convex set in n-space with maximization of a linear function in non-negative variables of a generalized finite sequence space subject to a finite system of linear equations. We then present a new generalization of the Kuhn-Tucker saddle-point equivalence theorem for arbitrary convex functions in n-space where differentiability is no longer assumed.     

The formulation of some allocation and connection problems as integer programs

In this paper a component placement problem and a digital computer backboard wiring problem are formulated as integer linear programs. The component placement problem consists of making a unique assignment of components to column positions such that wireability is maximized. The backboard wiring problem consists of three interrelated subproblems, namely, the placement, the connection, and the routing problems. The placement and connection problems are combined and solved as one, thereby giving the optimal circuit connections as well as minimizing the total lead length. It is shown that under certain assumptions, the number of inequalities and variables in the problem can be greatly reduced. Further simplifying assumptions lead to a near optimal solution. Examples of other allocation problems to which the models presented here are applicable are given. The following concepts are formulated as linear inequalities: (1) the absolute magnitude of the difference between two variables; (2) minimize the minimum function of a set of functions; and (3) counting the number of (0, 1) adjacent component pairs in a vector.     

On some stocxastic tactical antisubmarine games

Mathematical models of tactical problems in Hntisubmarine Warfare (ASW) are developed. Specifically, a game of pursuit between a hunter-killer force. player 1, and a possible submarine, player 2 is considered. The game consists of a sequence of moves and terminates when player 2 is tcaught or evades player 1. When the players move they observe the actual tactical configuration of the forces (state) and each player choosa-s a tactical plan from a finite collection. This joint choice of tactical plans determines an immediate payoff and a transition probability distribution over the states. Hence an expected payoff function is defined, Formally this game is a Terminating Stochastic Game (TSG). Shapley demonstrated the existence of a value and optimal strategies (solution), An iterative technique to approximate the solution to within desired accuracy is proposed. Each iteration of the technique is obtained by solving a set of linear programs. To introduce more realism into the game several variations of the TSG are also considered. One variation is a finite TSG and linear programming techniques are employed to find the solution.     

Some approximations in multi-item,multi-echelon inventory systems for recoverable items

The optimization problem as formulated in the METRIC model takes the form of minimizing the expected number of total system backorders in a two-echelon inventory system subject to a budget constraint. The system contains recoverable items – items subject to repair when they fail. To solve this problem, one needs to find the optimal Lagrangian multiplier associated with the given budget constraint. For any large-scale inventory system, this task is computationally not trivial. Fox and Landi proposed one method that was a significant improvement over the original METRIC algorithm. In this report we first develop a method for estimating the value of the optimal Lagrangian multiplier used in the Fox-Landi algorithm, present alternative ways for determining stock levels, and compare these proposed approaches with the Fox-Landi algorithm, using two hypothetical inventory systems – one having 3 bases and 75 items, the other 5 bases and 125 items. The comparison shows that the computational time can be reduced by nearly 50 percent. Another factor that contributes to the higher requirement for computational time in obtaining the solution to two-echelon inventory systems is that it has to allocate stock optimally to the depot as well as to bases for a given total-system stock level. This essentially requires the evaluation of every possible combination of depot and base stock levels – a time-consuming process for many practical inventory problems with a sizable system stock level. This report also suggests a simple approximation method for estimating the optimal depot stock level. When this method was applied to the same two hypotetical inventory systems indicated above, it was found that the estimate of optimal depot stock is quite close to the optimal value in all cases. Furthermore, the increase in expected system backorders using the estimated depot stock levels rather than the optimal levels is generally small.