The Stochastic Cash Balance Problem with Fixed Costs for Increases and Decreases
TL;DR: In this article, a convex upper and lower bound on the nonconvex cost functions partially describes the optimal policy for the stochastic cash balance problem, where both fixed and proportional costs are incurred whenever the inventory is changed in either direction.
Abstract: The stochastic cash balance problem is an inventory problem in which the stochastic cash or inventory change can either be positive or nonpositive, and in which decisions to increase or decrease the inventory are permitted at the beginning of each time period. The paper studies problems in which both fixed and proportional costs can be incurred whenever the inventory is changed in either direction. An example is used to demonstrate that when these costs are positive and the loss function is convex, a simple policy analogous to a two-sided s, S policy is not generally optimal. The example is also used to display the relations between the cash balance problem and inventory problems previously studied by Scarf and Veinott. When proportional costs of changing the inventory are zero, the two fixed costs are equal, the loss function is symmetric quasi-convex, and the problem's probability densities are quasi-concave a simple policy is shown to be optimal. For the cases in which simple policies are not optimal, the paper develops a technique which employs convex upper and lower bounds on the nonconvex cost functions partially to describe the optimal policy. It is suggested that this convex bounding technique may provide an approach to studying the cost implications of following simple, nonoptimal policies in inventory problems for which the optimal policy is complex.
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TL;DR: This paper is the first attempt to incorporate asset-based financing into production decisions by model the available cash in each period as a function of assets and liabilities that may be updated periodically according to the dynamics of the production activities.
Abstract: Most of the traditional models in production and inventory control ignore the financial states of an organization and can lead to infeasible practices in real systems. This paper is the first attempt to incorporate asset-based financing into production decisions. Instead of setting a known, exogenously determined budgetary constraint as most existing models suggest, we model the available cash in each period as a function of assets and liabilities that may be updated periodically according to the dynamics of the production activities. Furthermore, our models allow different interest rates on cash balance and outstanding loans, which is an enhancement over most traditional models in that inventory financed by a loan may be more expensive than that by out-of-pocket cash. We demonstrate the importance of joint consideration of production and financing decisions in a start-up setting in which the ability to grow the firm is mainly constrained by its limited capital and dependence on bank financing. We then explain the motivation for asset-based financing by examining the decision making at a bank and a set of retailers in a newsvendor setting.
559 citations
Cites background from "The Stochastic Cash Balance Problem..."
...In particular, dynamic stochastic programming models are established to characterize the structure of optimal cash-management policies (see Girgis 1968, Neave 1970, Miller and Orr 1966, Ziemba and Vickson 1975)....
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TL;DR: A stochastic dynamic programming (DP) model of the fashion buying problem that incorporates the model of demand and an updated Newsboy heuristic that is intuitively appealing and easily implemented are developed.
Abstract: We focus on the problem of buying fashion goods for the “big book” of a catalogue merchandiser. This company also owns outlet stores and thus has the opportunity, as the season evolves, to divert inventory originally purchased for the big book to the outlet store. The obvious questions are: (1) how much to order originally, and (2) how much to divert to the outlet store as actual demand is observed. We develop a model of demand for an individual item. The model is motivated by data from the women's designer fashion department and uses both historical data and buyer judgement. We build a stochastic dynamic programming (DP) model of the fashion buying problem that incorporates the model of demand. The DP model is used to derive the structure of the optimal inventory control policy. We then develop an updated Newsboy heuristic that is intuitively appealing and easily implemented. When this heuristic is compared to the optimal solution for a wide variety of scenarios, we observe that it performs very well. Si...
222 citations
TL;DR: It is proved that there always exists an optimal policy for the inventory model and that this policy is of a simple form.
Abstract: We formulate a continuous-time, infinite-horizon, discounted-cost cash management model with both fixed and proportional transactions costs and with linear holding and penalty costs. We model the cumulative demand for cash by a Wiener process with drift and use the optimal control technique of "impulse control" to find sufficient conditions under which an optimal policy exists. We show that these conditions are always met. Therefore, we prove that there always exists an optimal policy for the cash management problem and that this policy is of a simple form. When the proportional transactions cost of decreasing the cash balance is sufficiently high, it is never optimal to decrease the cash balance. Then the cash management model degenerates to the inventory model. We prove that there always exists an optimal policy for the inventory model and that this policy is of a simple form. Under special cases of the cash management model we obtain analytic expressions for the parameters of the optimal policy.
192 citations
TL;DR: This work considers the empty container allocation problem in a port to be a nonstandard inventory problem with positive and negative demands at the same time under a general holding-penalty cost function and one-time period delay availability for full containers just arriving at the port.
126 citations
Cites background from "The Stochastic Cash Balance Problem..."
...Considering the model with negative demand, some related research concentrated on the cash balance, see [ 16-18 ] and their references, and on the equipment rentals, see [19]....
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...Girgis [16], Eppen and Fama [17], and Neave [ 18 ] consider the stochastic cash balance problem as an inventory model in which the stochastic cash change can either be positive or nonpositive, and in which the decision to increase or decrease the inventory is permitted at the beginning of each time period....
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TL;DR: In this paper, a continuous time model of cash management is formulated with stochastic demand and allowing for positive and negative cash balances, and the form of the optimal policy is assumed to be of a simple form d, D, U, u.
Abstract: A continuous time model of cash management is formulated with stochastic demand and allowing for positive and negative cash balances. The form of the optimal policy is assumed to be of a simple form d, D, U, u. The parameters of the optimal policy are explicitly evaluated and the properties of the system are discussed.
100 citations
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11,456 citations
TL;DR: In this paper, a model of cash flows and the costs of cash management for business firms is presented, with assumptions underlying the model, and optimal values of the policy parameters are derived.
Abstract: I. Introduction, 413. â II. A model of cash flows and the costs of cash management for business firms, 416; assumptions underlying the model, 417; optimal values of the policy parameters, 420; some properties of the solution, 423; implications for the demand for money by firms, 425; extension to allow for non-zero drift, 427.â III. The applicability of the model, 429. â Appendix, 433.
1,288 citations
"The Stochastic Cash Balance Problem..." refers background in this paper
...In other papers dealing with cash balances optimality of a simple policy is assumed; see [8] and [9]....
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