A successive convex approximation method for multistage workforce capacity planning problem with turnover

Workforce capacity planning in human resource management is a critical and essential component of the services supply chain management. In this paper, we consider the planning problem of transferring, hiring, or firing employees among different departments or branches of an organization under an environment of uncertain workforce demands and turnover, with the objective of minimizing the expected cost over a finite planning horizon. We model the problem as a multistage stochastic program and propose a successive convex approximation method which solves the problem in stages and iteratively. An advantage of the method is that it can handle problems of large size where normally solving the problems by equivalent deterministic linear programs is considered to be computationally infeasible. Numerical experiments indicate that solutions obtained by the proposed method have expected costs near optimal.     

Finite-horizon optimal investment with transaction costs: A parabolic double obstacle problem

This paper concerns optimal investment problem of a CRRA investor who faces proportional transaction costs and finite time horizon. From the angle of stochastic control, it is a singular control problem, whose value function is governed by a time-dependent HJB equation with gradient constraints. We reveal that the problem is equivalent to a parabolic double obstacle problem involving two free boundaries that correspond to the optimal buying and selling policies. This enables us to make use of the well-developed theory of obstacle problem to attack the problem. The C2,1 regularity of the value function is proven and the behaviors of the free boundaries are completely characterized.     

Stochastic optimal control of internal hierarchical labor markets

This paper develops an optimal control model for a graded manpower system where the demand for manpower is uncertain. The organization's objective is to minimize the discounted costs of operating the manpower system, including excess demand costs. The stock of workers in various grades can be adjusted in two ways. The first method is outside hiring flows, which is the usual control variable used in previous research. The second method is to control the transition rates between grades of the hierarchy, an instrument not previously studied. Incorporating the transition rates into the control variables creates time lags in the control process. The resulting problem is solved numerically using an approximation for the time-lagged control variables. The numerical example is based on the Air Force officer hierarchy. The model is used to examine such issues as the desirability of granting tenure to workers who are not promoted to the highest grade and the effects of length-of-service and demand uncertainty on manpower policy.     

Optimal portfolio, consumption and retirement decision under a preference change

We investigate an optimal portfolio, consumption and retirement decision problem in which an economic agent can determine the discretionary stopping time as a retirement time with constant labor wage and disutility. We allow the preference of the agent to be changed before and after retirement. It is assumed that the agent's coefficient of relative risk aversion becomes higher after retirement. Under a constant relative risk aversion (CRRA) utility function, we obtain the optimal policies in closed-forms using martingale methods and variational inequality methods. We give some numerical results of the optimal policies. We also consider the relation between the level of disutility and the labor wage with the optimal retirement wealth level.     

Staffing decisions for heterogeneous workers with turnover

In this paper we consider a firm that employs heterogeneous workers to meet demand for its product or service. Workers differ in their skills, speed, and/or quality, and they randomly leave, or turn over. Each period the firm must decide how many workers of each type to hire or fire in order to meet randomly changing demand forecasts at minimal expense. When the number of workers of each type can by continuously varied, the operational cost is jointly convex in the number of workers of each type, hiring and firing costs are linear, and a random fraction of workers of each type leave in each period, the optimal policy has a simple hire- up-to/fire-down-to structure. However, under the more realistic assumption that the number of workers of each type is discrete, the optimal policy is much more difficult to characterize, and depends on the particular notion of discrete convexity used for the cost function. We explore several different notions of discrete convexity and their impact on structural results for the optimal policy.     

Single-person controlled diffusions with discounted costs

This paper describes a class of single-person controlled one-dimensional diffusion processes where the control is a vector-valued function on the state space, which is a compact interval of the real line. These processes generate costs, and the optimal control problem is to choose an admissible control that minimizes expected discounted costs. The major results are necessary and sufficient conditions for a control to be optimal as well as characterizations of the expected costs corresponding to an optimal control. This paper also established sufficient conditions for the existence of piecewise-continuous optimal controls.The author is grateful to M. L. Puterman and R. W. Rosenthal for their helpful comments during the course of this research. The author is especially indebted to A. F. Veinott, Jr., for his guidance of the author's doctoral dissertation, on part of which this paper is based. This research was supported by the National Science Foundation, Grant No. GK-18339, as well as the Office of Naval Research, Contract No. N00014-67-A-0112-0050.     

Variance minimization for constrained discounted continuous-time MDPs with exponentially distributed stopping times

This paper deals with minimization of the variances of the total discounted costs for constrained Continuous-Time Markov Decision Processes (CTMDPs). The costs consist of cumulative costs incurred between jumps and instant costs incurred at jump epochs. We interpret discounting as an exponentially distributed stopping time. According to existing theory, for the expected total discounted costs optimal policies exist in the forms of randomized stationary and switching stationary policies. While the former is typically unique, the latter forms a finite set whose number of elements grows exponentially with the number of constraints. This paper investigates the problem when the process stops immediately after the first jump. For costs up to the first jump we provide an index for selection of actions by switching stationary policies and show that the indexed switching policy achieves a smaller variance than the randomized stationary policy. For problems without instant costs, the indexed switching policy achieves the minimum variance of costs up to the first jump among all the equivalent switching policies.     

Optimal empty vehicle repositioning and fleet-sizing for two-depot service systems

This paper considers the problem of determining optimal control policies for empty vehicle repositioning and fleet-sizing in a two-depot service system with uncertainties in loaded vehicle arrival at depots and repositioning times for empty vehicles in the fleet. The objective is to minimise the sum of the costs incurred by vehicle maintenance, empty vehicle repositioning and vehicle leasing. A novel integrated model is presented. The optimal empty repositioning policy for a particular fleet size is shown to be of the threshold control type. The explicit form of the cost function under such threshold controls is obtained. The optimal threshold values and fleet-size are then derived. Numerical examples are given to demonstrate the results.     

Optimal production,purchasing and pricing: A differential game approach

This paper deals with a problem of determining optimal production and pricing policies of a manufacturing firm which is supplying a retailer, The latter faces a price-dependent demand function towards the final consumers and wishes to determine optimal purchasing and pricing policies. Both firms carry inventories and backlogging is permitted. The problem is modelled as a two-player nonzero-sum differential game with the inventory levels as the state variables. The controls are the rates of production and purchasing as well as the prices. Assuming linear production costs, but convex ordering and holding costs, open-loop Nash controls are characterized by using switching point analysis as well as phase diagrams.     

Reduction of total-cost and average-cost MDPs with weakly continuous transition probabilities to discounted MDPs

This note describes sufficient conditions under which total-cost and average-cost Markov decision processes (MDPs) with general state and action spaces, and with weakly continuous transition probabilities, can be reduced to discounted MDPs. For undiscounted problems, these reductions imply the validity of optimality equations and the existence of stationary optimal policies. The reductions also provide methods for computing optimal policies. The results are applied to a capacitated inventory control problem with fixed costs and lost sales.     

Simple heuristics for push and pull remanufacturing policies

Inventory policies for joint remanufacturing and manufacturing have recently received much attention. Most efforts, though, were related to (optimal) policy structures and numerical optimization, rather than closed form expressions for calculating near optimal policy parameters. The focus of this paper is on the latter. We analyze an inventory system with unit product returns and demands where remanufacturing is the cheaper alternative for manufacturing. Manufacturing is also needed, however, since there are less returns than demands. The cost structure consists of setup costs, holding costs, and backorder costs. Manufacturing and remanufacturing orders have non-zero lead times. To control the system we use certain extensions of the familiar (s, Q) policy, called push and pull remanufacturing policies. For all policies we present simple, closed form formulae for approximating the optimal policy parameters under a cost minimization objective. In an extensive numerical study we show that the proposed formulae lead to near-optimal policy parameters.     

Optimal investment with transaction costs based on exponential utility function: a parabolic double obstacle problem

This paper concerns optimal investment problem with proportional transaction costs and finite time horizon based on exponential utility function. Using a partial differential equation approach, we reveal that the problem is equivalent to a parabolic double obstacle problem involving two free boundaries that correspond to the optimal buying and selling policies. Numerical examples are obtained by the binomial method.     

An efficient heuristic for inventory control when the customer is using a (s,S) policy

We consider a capacitated supply chain in which the supplier has the information of the (s,S) policy used by the retailer as well as the end-customer demand distribution. For the resulting inventory control problem at the supplier, optimal policies and structural properties were presented by Gavirneni et al. (Management Sci. 45(1) (1999) 16). They detailed an efficient solution procedure for the uncapacitated problem and resorted to computationally expensive infinitesimal perturbation analysis (IPA) for the capacitated situation. In this paper, we study a heuristic, based on the uncapacitated solution, for the capacitated situation. A detailed computational study showed that this heuristic is very efficient in that the costs increased by only 3.3% on the average. The heuristic was especially effective at higher capacities, lower holding costs, and extreme values of demand variance.     

An O(T log T) Algorithm for the Dynamic Lot Size Problem with Limited Storage and Linear Costs

In this paper the dynamic lot size problem with time varying storage capacities and linear costs is addressed. Like in the uncapacitated version, this problem can be formulated as a network flow problem. Considering the properties of the underlying network, we devise an O(T log T) greedy algorithm to obtain optimal policies and we report computational results for randomly generated problems.     

Optimal Continuous Policies for Repair and Replacement

This paper investigates the problem of finding optimal replacement policies for equipment subject to failures with randomly distributed repair costs, the degree of reliability of the equipment being considered as a state of a Markov process. Algorithms have been devised to find optimal combined policies both for preventive replacement and for replacement in case of failure by using repair-limit strategies.First a simple procedure to obtain an optimal discrete policy is described. Then an algorithm is formulated in order to calculate an optimal continuous policy: it is shown how the optimal repair limit is the solution to an ordinary differential equation, and how the value of the repair limit determines the optimal preventive replacement policy.     

Age-dependent production planning and maintenance strategies in unreliable manufacturing systems with lost sale

In this paper, we formulate an analytical model for the joint determination of an optimal age-dependent buffer inventory and preventive maintenance policy in a production environment that is subject to random machine breakdowns. Traditional preventive maintenance policies, such as age and periodic replacements, are usually studied based on simplified and non-realistic assumptions, as well as on the expected costs criterion. Finished goods inventories and the age-dependent likelihood of machine breakdowns are usually not considered. As a result, these policies could significantly extend beyond the anticipated financial incomes of the system, and lead to crises. In order to solve this problem, a more realistic analysis model is proposed in this paper to consider the effects of both preventive maintenance policies and machine age on optimal safety stock levels. Hence, a unified framework is developed, allowing production and preventive maintenance to be jointly considered. We use an age-dependent optimization model based on the minimization of an overall cost function, including inventory holdings, lost sales, preventive and corrective maintenance costs. We provide optimality conditions for the manufacturing systems considered, and use numerical methods to obtain an optimal preventive maintenance policy and the relevant age-dependent threshold level production policy. In this work, this policy is called the multiple threshold levels hedging point policy. We include numerical examples and sensitivity analyses to illustrate the importance and the effectiveness of the proposed methodology. Compared with other available optimal production and maintenance policies, the numerical solution obtained shows that the proposed age-dependent optimal production and maintenance policies significantly reduce the overall cost incurred.     

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This paper investigates the investment-dividend optimization problem for a corporation with transaction costs and investment constraints. The main feature is that we assume general constraints on investments including the special case of short-sale and borrowing constraints. This results in a regular-impulse stochastic control problem. The nontrivial case is that the investment can't meet the loss of wealth due to discounting. In this case, delicate analysis is carried out on QVI w.r.t. three possible situations, leading to an explicit construction of the value functions together with the optimal policies. We also give explicit conclusion of the trivial case at last.     

Value iteration in average cost Markov control processes on Borel spaces

This paper deals with discrete-time Markov control processes withBorel state and control spaces, with possiblyunbounded costs andnoncompact control constraint sets, and the average cost criterion. Conditions are given for the convergence of the value iteration algorithm to the optimal average cost, and for a sequence of finite-horizon optimal policies to have an accumulation point which is average cost optimal.This research was partially supported by the Consejo Nacional de Ciencia y Tecnología (CONACyT) under grant 1332-E9206.     

On scheduling a multiclass queue with abandonments under general delay costs

We consider a multiclass queueing system with abandonments and general delay costs. A system manager makes dynamic scheduling decisions to minimize long-run average delay and abandonment costs. We consider the three types of delay cost: (i) linear, (ii) convex, and (iii) convex–concave, where the last one corresponds to settings where customers may have a particular deadline in mind but once that deadline passes there is increasingly little difference in the added delay. The dynamic control problem for the queueing system is not tractable analytically. Therefore, we consider the system in the conventional heavy traffic regime and study the approximating Brownian control problem (BCP). We observe that the approximating BCP does not admit a pathwise solution due to abandonments. In particular, the celebrated cμ rule and its extension, the generalized cμ rule, which is asymptotically optimal under convex delay costs with no abandonments, are not optimal in this case. Consequently, we solve the associated Bellman equation, which yields a dynamic index policy (derived from the value function) as the optimal control for the approximating BCP. Interpreting that control in the context of the original queueing system, we propose practical policies for each of the three cases considered and demonstrate their effectiveness through a simulation study.     

On economically based quality control decisions

A new approach to the problem of quality control is investigated. It is based on the theories and models developed for optimal maintenance policies for equipment subject to stochastic failure. Four alternative sampling/inspection procedures are examined. They are easy to implement, do not require the process operator to compute his own operating decisions, and are capable of finding optimal intervals between successive samplings/inspections. Both fixed and variable costs are included in the models. Relatively efficient algorithms have been developed for finding the optimal quality control procedures; this is due, partially, to the simple form of the objective function.