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A major challenge faced by marketers attempting to optimize their advertising campaigns is to deal with budget constraints. The problem is even harder in the face of multidimensional budget constraints, particularly in the presence of many decision variables involved and the interplay among the decision variables through such constraints. Concise bidding strategies help advertisers deal with this challenge by introducing fewer variables upon which to act. In this paper, we study the problem of finding optimal concise bidding strategies for advertising campaigns with multiple budget constraints. Given bid landscapes—that is, the predicted value (e.g., number of clicks) and the cost per click for any bid—that are typically provided by ad-serving systems, we optimize the value of an advertising campaign given its budget constraints. In particular, we consider bidding strategies that consist of no more than k different bids for all keywords. For constant k, we provide a polynomial-time approximation scheme to optimize the profit, whereas for arbitrary k we show how a constant-factor approximation algorithm can be obtained via a combination of solution enumeration and dependent LP rounding techniques, which can be of independent interest. In addition to being able to deal with multidimensional budget constraints, our results do not assume any specific payment scheme and can be applied on pay-per-click, pay-per-impression, or pay-per-conversion models. Also, no assumption about the concavity of value or cost functions is made. Finally, we evaluate the performance of our algorithms on real data sets in regimes with up to six-dimensional budget constraints. In the case of a single budget constraint, in which uniform bidding (currently used in practice) has a provable performance guarantee, our algorithm beats the state of the art by an increase of 1%–6% in the expected number of clicks. This is achieved by only two or three clusters in contrast with the single cluster permitted in uniform bidding. With multiple budget constraints, the gap between the performance of our algorithm and an enhanced version of uniform bidding grows to an average of 5%–6% (and as high as 35% in higher dimensions). This paper was accepted by Yinyu Ye, optimization.

This paper studies the flexible versus dedicated technology choice and capacity investment decisions of a multiproduct firm under demand uncertainty in the presence of budget constraints. The firm operates under a capital budget for financing the capacity investment, and an operating budget, which is uncertain in the capacity investment stage, for financing the production. We investigate how the tightening of the capital budget and a lower financial flexibility in the production stage (the likelihood of having a sufficient operating budget) shape the optimal technology choice. We find that the dominant regime is one where dedicated technology should be adopted for a larger investment cost range, and thus, is the best response to the tighter capital budget, whereas flexible technology is the best response to lower financial flexibility. We identify the key roles that the capacity intensity (the ratio of unit capacity cost to total unit capacity and production cost) of each technology and the pooling value of operating budget with dedicated technology, which brings this technology closer to flexible technology in terms of the resource network’s flexibility, play in a budget-constrained environment. Managerially, our results underline that in the presence of financial constraints, firms should manage technology adoption together with plant location, which shapes capacity intensity, or product portfolio, which shapes financial flexibility. This paper was accepted by Serguei Netessine, operations management.

Ad exchanges are emerging Internet markets where advertisers may purchase display ad placements, in real time and based on specific viewer information, directly from publishers via a simple auction mechanism. Advertisers join these markets with a prespecified budget and participate in multiple second-price auctions over the length of a campaign. This paper studies the competitive landscape that arises in ad exchanges and the implications for publishers’ decisions. The presence of budgets introduces dynamic interactions among advertisers that need to be taken into account when attempting to characterize the bidding landscape or the impact of changes in the auction design. To this end, we introduce the notion of a fluid mean-field equilibrium (FMFE) that is behaviorally appealing and computationally tractable, and in some important cases, it yields a closed-form characterization. We establish that an FMFE approximates well the rational behavior of advertisers in these markets. We then show how this framework may be used to provide sharp prescriptions for key auction design decisions that publishers face in these markets. In particular, we show that ignoring budgets, a common practice in this literature, can result in significant profit losses for the publisher when setting the reserve price. This paper was accepted by Dimitris Bertsimas, optimization.

This paper analyzes the impact of endogenous credit terms under capital market imperfections in a capacity investment setting. We model a monopolist firm that decides on its technology choice (flexible versus dedicated) and capacity level under demand uncertainty. Differing from the majority of the stochastic capacity investment literature, we assume that the firm is budget constrained and can relax its budget constraint by borrowing from a creditor. The creditor offers technology-specific loan contracts to the firm, after which the firm makes its technology choice and subsequent decisions. Capital market imperfections impose financing frictions on the firm. Our analysis contributes to the capacity investment literature by extending the theory of stochastic capacity investment and flexible versus dedicated technology choice to understand the impact of capital market imperfections, and by analyzing the impact of demand uncertainty (variability and correlation) on the operational decisions and the performance of the firm under different capital market conditions. We demonstrate that the endogenous nature of credit terms in imperfect capital markets may modify or reverse conclusions concerning capacity investment and technology choice obtained under the perfect market assumption and we explain why. The theory developed in this paper suggests some rules of thumb for the strategic management of the capacity and technology choice in imperfect capital markets. This paper was accepted by John Birge, focused issue editor.

Public health budget constraints force policy makers to prioritize resources toward those interventions that yield the highest perceived benefits. Intuitively, it appears optimal to focus resources on affordable interventions against prevalent diseases. However, due to the dynamics of infectious disease eradication, policies focusing on a static perception of priorities may lead to economically suboptimal outcomes. Using a hypothetical two-disease dynamic transmission model, we explore several different decision rules with respect to vaccination policy for eradicable diseases. The simulations show that cost-effectiveness decreases as the extent of priority shifting increases. This model suggests the need for a longer-term dynamic perspective to appropriately recognize costs and benefits of different policies for eradicable diseases.

Selecting program portfolios within a budget constraint is an important challenge in the management of new product development (NPD). Optimal portfolios are difficult to define because of the combinatorial complexity of project combinations. However, at the aggregate level of the strategic allocation of resources across product lines, investment in a program is not an all-or-nothing decision, but can be adjusted, resulting in a higher or lower program benefit (e.g., higher or lower quality). In some cases, resources can be adjusted even for individual projects. With this insight, one can use marginal analysis to optimally allocate the scarce budget. This article develops a dynamic model of resource allocation, taking into account multiple interacting factors, such as independent or correlated, uncertain market payoffs that change over time, increasing or decreasing returns from the NPD investment, carry-over of the investment benefit over multiple periods, and interactions across market segments. We characterize optimal policies in closed form and derive qualitative decision rules for managers.

This paper examines dynamic R&D investment policies and the valuation of R&D programs in a contingent claims framework. We incorporate the following characteristics of R&D programs into our model: learning-by-doing, collateral learning between different projects in the program, interaction between project cash flows, periodic reevaluations of the program, different intensities of investment, capital rationing constraints, and competition. We show that a firm may invest in multiple projects even if only one can be implemented after development is complete. Furthermore, the firm may significantly alter its funding policy over time. For example, it may simultaneously develop multiple projects for a period of time, then focus on a lead project, and potentially resume funding of a “backup” project if the lead project fails to deliver on its early promise. We show how a firm can forecast expected R&D spending through time for an optimally executed R&D program. While project volatility plays an important role in determining R&D program value, we find that for high volatility projects the optimal investment policy is not very sensitive to changes in (or misestimation of) volatility. In considering whether to accelerate development of a project, a firm should balance the adverse effects of increased costs and the loss of investment flexibility against the positive effects of rapid uncertainty resolution and accelerated cash flows. In the presence of a budget constraint that prevents the firm from simultaneously accelerating projects and developing projects in parallel, we find that, if one project significantly dominates another early in the development stage, the option to accelerate the lead project is likely to be more valuable than the option to exchange projects. Thus, the backup project would be shelved in order to commit extra resources to development of the lead project. Finally, competition from other firms leads to more parallel investment in the early development stages of projects, less parallel investment in the latter stages of development, and lower overall investment.

We use mathematical modeling in decision analysis to help us obtain a profit lottery that is “better” than the one we can assess directly. The “goodness” of the profit lottery is defined by the concept of authenticity of probabilities. The role of modeling is to simplify our assessment task through decomposition of the profit lottery. However, budgetary constraints force us to make approximations in the modeling process and thereby force us to misstate the profit lottery. When dealing with our dissatisfactions about the modeling in a decision analysis, we regard models as subjective expressions of our uncertainty rather than as objective descriptions of the real-world.

In this paper we describe a multiperiod linear programming model designed to aid the administrator of a school in planning and decision-making for a five-year horizon. The technique of goal programming is employed to accommodate the multiplicity of goals and the marginal trade-offs inherent in such a process. The model includes faculty staffing goals, career constraints, teaching and course loads, tenure constraints and budget constraints. A weighted sum of the deviations from the staffing goals make up the objective function of the prototype model considered. Several methods of estimating the various parameters of the model are discussed, and the results of a small example are also discussed.

This paper considers a multi-contract bidding situation in which the contracts are interrelated because of limited productive facilities and because of budgetary constraints. A dynamic programming model and computational algorithm are presented which derive a bidding strategy which maximizes total expected return from all contracts subject to the constraints. The computational algorithm is efficient in terms of storage capacity required and computation time.

Can city governments take meaningful actions to relieve the growing shortage of rental housing units for low income families without heavy reliance on state or federal support? To answer this question a net present value model is developed from the viewpoint of the existing landlords. Using this model the municipal policies which optimize the objective function of providing the maximum renovated housing units subject to budgetary constraints are identified by a convex programming algorithm. The empirical application of this approach to a Pittsburgh neighborhood indicates that it is possible to increase the quality and quantity of the rental housing stock while actually increasing the municipal net revenue.

In a recent article, in Management Science, W. F. Massy [Massy, W. F. 1976. A dynamic equilibrium model for university budget planning. Management Sci. 23 (3, November) 248–256.] outlines an interesting difference equation model of the Stanford University budget system. He proceeds then to solve for first and second order (i.e., short-term) balanced budget conditions. However, Massy's assertion that the balanced budget requirement cannot be modeled “effectively” for an extended time horizon is inaccurate. This note shows that such long-range solutions are readily available from the eigenstructure of his model. We derive simple sufficient conditions for maintaining balanced university budgets. Related issues of balanced budget attainability, empirical correspondence, and policy feasibility are also discussed.

This study examines the problem faced by a firm which wishes to position a new choice object in an existing product class. It is assumed that both the consumer and the firm are involved in a two-stage decision process. The consumer first decides on his budget for the product class. He then evaluates, within the product class, that subset of competing objects which have prices approximately equal to his budget constraint. This evaluation is performed through a weighted multi-attribute utility model. The product classes considered here are ones for which the consumer has a finite ideal level on each attribute. The consumer is hypothesized to choose, without error, that object which is closest to his ideal. Different individuals are assumed to be heterogeneous in both attribute weights and ideal levels. The firm is assumed to first identify, in the attribute space which contains ideal points and competing objects, promising product positions which would attract a large number of consumers. It then evaluates these positions in terms of costs and resulting profits. The problem of identifying the promising positions, given information on a sample of individuals, is formulated as a Mixed Integer Nonlinear Program. Due to the inability of such a program to solve even small sample problems, the spatial properties of the problem are examined. An exact solution algorithm which is computationally feasible for small samples is developed. It is based on an examination of intersections of indifference hyperellipsoids. For larger sample problems an efficient heuristic which is an extension of the random point search used in nonlinear programming is provided. It involves random line search procedures for our noncontinuous-type problem. The positioning approach and the heuristic are illustrated in a simulated positioning problem in the small car market.

This paper examines the system fill-rate of a one-warehouse N-identical retailer distribution system as a function of warehouse and retailer safety stock. Using the approximation model of Deuermeyer and Schwarz (Deuermeyer, B. L., L. B. Schwarz. 1981. A model for the analysis of system service level in warehouse/retailer distribution systems: the identical retailer case. Chapter 8 in Multi-Level Production/Inventory Control Systems: Theory and Practice. North-Holland, Amsterdam.) we examine the problem of maximizing system fill-rate subject to a constraint on system safety stock. Optimal safety stock policy is characterized to be the intersection of a fill-rate policy line and the safety stock budget line. Properties of fill-rate policy lines are given. These properties may be used to provide managerial insight into system optimization and as the basis for heuristics. One heuristic, the vertical heuristic, based on the corresponding deterministic version of the problem, yields trivially simple and near-optimal policies.

In most policy analysis studies there are a large number of alternative policies and a large number of impacts to be considered. Time and budget constraints make it impossible to calculate all the impacts for all of the alternatives. As a result, such studies include some process for reducing the number of alternatives to be examined in detail. The process is often implicit and nonscientific (e.g., only the decisionmaker's three favorite alternatives are considered). This paper suggests that policy analysis studies explicitly include a screening step, in which the alternatives to be examined in detail and those to be excluded from further consideration are chosen in a scientific and systematic manner. The output from this step is a relatively small set of policy alternatives that are sufficiently attractive that they deserve a more thorough evaluation. Two general screening strategies are described. They are illustrated by describing the screening step in a study to help determine an overall water management policy for the Netherlands.

A model is developed which allocates an available budget to research alternatives subject to budgetary constraints on both organizational and technical entities. The solution algorithm is based on the out-of-kilter minimum cost network flow algorithm and is programmed for use in a time-share, conversational mode. Although the model does not describe the decision processes observed empirically, its output is sufficiently similar to actual allocations that it can be viewed as predictive. Strengths and weaknesses of the model are detailed based on questionnaire and interview data provided by the potential users of the system. The decision information system is used to illustrate that experimentation concerned with better understanding of the adoption process can be used to facilitate model design and application.

Suppose that an investing firm is primarily interested in achieving a specified minimum return critical to its economic survival. Then it seems appropriate to maximize the probability of exceeding the aspiration level. This decision criterion will be applied in the context of several stochastic static investment models with budget constraint. As long as the aspiration level does not exceed the maximum expected return achievable with the given budget, the desired investment strategy must be sought among the efficient solutions in the Markowitz sense. For higher aspiration levels this is no longer true. For the special case of the Markowitz model we show that all investment projects with expected yield not exceeding the aspiration level (of yield) should be rejected. This is a surprising confirmation of a well-known deterministic rule of investment theory and practice in a rather different context.

Dynamic programming algorithms are developed for optimal capital allocation subject to budget constraints. We extend the work of Weingartner [Weingartner, H. M. 1966. Capital budgeting of interrelated projects: Survey and synthesis. Management Sci. 12(7, March) 485–516.] and Weingartner and Ness [Weingartner, H. M., D. N. Ness. 1967. Methods for the solution of the multi-dimensional 0/1 knapsack problem. Oper. Res. 15(1, January–February) 83–108.] by including multilevel projects, reinvesting returns, borrowing and lending, capital deferrals, and project interactions. We are able to handle dynamic programming models with several state variables because the optimal returns are monotone non-decreasing step functions. Computational experience with a variety of problems is reported.

Recently considerable attention has been focused on the inadequacy of presently used methods of compensating school district personnel. The increasing incidence of teacher strikes in many school districts has established the need for an effective alternative to the inherent rigidity of the commonly used fixed step salary schedule. The purpose of this study is to propose one such alternative that is based upon mathematical programming techniques. In this study a salary evaluation model was derived then applied to a school district. Salary schedules were derived which (1) were consistent with both the imposed hierarchical and budgetary constraints, (2) considered ten factors in the salary evaluation, and (3) reflected established school district priorities and objectives. This latter feature was accomplished by solving the model with various objective functions.