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.
[1]
William Feller,et al.
An Introduction to Probability Theory and Its Applications
,
1951
.
[2]
H. Scarf.
THE OPTIMALITY OF (S,S) POLICIES IN THE DYNAMIC INVENTORY PROBLEM
,
1959
.
[3]
E M Beale.
THE USE OF QUADRATIC PROGRAMMING IN STOCHASTIC LINEAR PROGRAMMING
,
1962
.
[4]
Merton H. Miller,et al.
A Model of the Demand for Money by Firms
,
1966
.
[5]
Jr. Arthur F. Veinott.
On the Opimality of $( {s,S} )$ Inventory Policies: New Conditions and a New Proof
,
1966
.
[6]
William D. Whisler.
A Stochastic Inventory Model for Rented Equipment
,
1967
.
[7]
William Feller,et al.
An Introduction to Probability Theory and Its Applications
,
1967
.
[8]
N. M. Girgis.
Optimal Cash Balance Levels
,
1968
.
[9]
Merton H. Miller,et al.
THE DEMAND FOR MONEY BY FIRMS: EXTENSIONS OF ANALYTIC RESULTS
,
1968
.
[10]
E. Fama,et al.
CASH BALANCE AND SIMPLE DYNAMIC PORTFOLIO PROBLEMS WITH PROPORTIONAL COSTS
,
1969
.