Süleyman Özekici

Inventory models with unreliable suppliersin a random environment

We consider infinite‐horizon periodic‐review inventory models with unreliable suppliers where the demand, supply and cost parameters change with respect to a randomly changing environment. Although our analysis will be in the context of an inventory model, it is also appropriate for production systems with unreliable machines where planning is done on a periodic basis. It is assumed that the environmental process follows a Markov chain. The stock‐flow equations of the inventory system subject to environmental fluctuations is represented using a two‐dimensional stochastic process. We show that an environment‐dependentorder‐up‐to level (i.e., base‐stock) policy is optimal when the order cost is linearin order quantity. When there is also a fixed cost of ordering, we show that a two‐parameter environment‐dependent (s, S) policy is optimal under reasonable conditions. We also discuss computational issues and some extensions.

Multiperiod portfolio optimization models in stochastic markets using the mean-variance approach

We consider several multiperiod portfolio optimization models where the market consists of a riskless asset and several risky assets. The returns in any period are random with a mean vector and a covariance matrix that depend on the prevailing economic conditions in the market during that period. An important feature of our model is that the stochastic evolution of the market is described by a Markov chain with perfectly observable states. Various models involving the safety-first approach, coefficient of variation and quadratic utility functions are considered where the objective functions depend only on the mean and the variance of the final wealth. An auxiliary problem that generates the same efficient frontier as our formulations is solved using dynamic programming to identify optimal portfolio management policies for each problem. Illustrative cases are presented to demonstrate the solution procedure with an interpretation of the optimal policies.

Reliability of Complex Devices in Random Environments

Abstract : The lifetimes of the components of a device depend on each other because of their joint dependence on the environmental conditions. The authors introduce intrinsic age processes, one for each component, to handle such dependence. The data required can be obtained by experiments under controlled laboratory conditions. The computations needed for randomly varying conditions are recursive and can be used for making decisions regarding maintenance and replacement. Keywords: multi-component devices; semimarkov processes.

Portfolio optimization in stochastic markets

We consider a multiperiod mean-variance model where the model parameters change according to a stochastic market. The mean vector and covariance matrix of the random returns of risky assets all depend on the state of the market during any period where the market process is assumed to follow a Markov chain. Dynamic programming is used to solve an auxiliary problem which, in turn, gives the efficient frontier of the mean-variance formulation. An explicit expression is obtained for the efficient frontier and an illustrative example is given to demonstrate the application of the procedure.

Inventory models with random yield in a random environment

Abstract We consider a single item inventory model which is observed periodically in a randomly changing environment. All model parameters are specified by the state of the environment which is assumed to be a time-homogeneous Markov chain. Yield is random due to the random capacity of the vendor, i.e., a given order is fully received if the order quantity is less than this capacity. Otherwise, the quantity received is equal to the available capacity. The problem is analyzed in single, multiple and infinite periods and it is shown that in all cases, the optimal policy is the well-known base-stock policy where the optimal order-up-to level depends on the state of the environment. The results are compared with the solutions of the certain yield model when there is infinite capacity. We show that the order-up-to levels are equal in the single period case. However, in multiple and infinite periods, we order the same or more if the yield is random.

Optimal maintenance policies in random environments

Abstract Most devices are designed to function under different environmental conditions which vary in time. The deterioration and failure process depends on the environment so that the failure rate at any time is a function of the prevailing environmental state. This necessitates the use of the intrinsic age of a device, rather than the real age, in reliability and maintenance problems. The measurement of the age of a device with respect to an intrinsic clock which ticks differently in varying environments is a rather new approach with important implications on the optimal replacement and repair problems. We consider an attractive model where the underlying environmental process has a somewhat general semi-Markov structure. Under the usual assumption requiring increasing failure rate distribution functions in all environments and reasonable cost structures, we show that the control-limit type intrinsic age replacement and repair policies are still optimal.

Portfolio selection in stochastic markets with HARA utility functions

In this paper, we consider the optimal portfolio selection problem where the investor maximizes the expected utility of the terminal wealth. The utility function belongs to the HARA family which includes exponential, logarithmic, and power utility functions. The main feature of the model is that returns of the risky assets and the utility function all depend on an external process that represents the stochastic market. The states of the market describe the prevailing economic, financial, social, political and other conditions that affect the deterministic and probabilistic parameters of the model. We suppose that the random changes in the market states are depicted by a Markov chain. Dynamic programming is used to obtain an explicit characterization of the optimal policy. In particular, it is shown that optimal portfolios satisfy the separation property and the composition of the risky portfolio does not depend on the wealth of the investor. We also provide an explicit construction of the optimal wealth process and use it to determine various quantities of interest. The return-risk frontiers of the terminal wealth are shown to have linear forms. Special cases are discussed together with numerical illustrations.

Reliability of software with an operational profile

Abstract This article provides the stochastic and statistical framework to model software reliability in the presence of an operational profile. The software system is used under a randomly changing operational process so that the failure characteristics depend on the specific operation performed. The operational process describes, in a probabilistic sense, how the software is utilized by the users. The time to failure distribution for each fault is exponentially distributed with a rate that depends on the operation. As soon as a failure is experienced, the fault that caused the fault is removed immediately with certainty. We discuss several issues related to software reliability and statistical inference.

Optimal scheduling of inspections: a delayed Markov model with false positives and negatives

A system subject to catastrophic failure deteriorates according to a delayed Markov process and is subjected to a series of binary tests that may yield false negative and false positive outcomes. A corrective action is carried out when a true positive is observed, thereby reducing the chance of system failure. Costs of inspections, false positives, the corrective action, and failure are incurred, and dynamic programming is used to compute the optimal inspection schedule. Two tractable computational methods are developed. The model, which is suited for medical screening, is applied to the problems of post-operative periumbilical pruritis and breast cancer.

Complex Systems in Random Environments

In this paper, we consider various inventory, queueing and reliability models where there is apparent complexity due to interacting components or subsystems. In particular, our analysis focuses on a multi-item inventory model with stochastically dependent demands, a queueing network where there are dependent arrival and service processes, or a reliability model with stochastically dependent component lifetimes. We discuss cases where this dependence is induced only by a random environmental process which the system operates in. This process represents the sources of variation that affect all deterministic and stochastic parameters of the model. Thus, not only the parameters of the model are now stochastic processes, but they are all dependent due to the common environment they are all subject to. Our objective is to provide a convincing argument that, under fairly reasonable conditions, the analysis techniques used in these models as well their solutions are not much more complicated than those where there is no environmental variation.