Infinite Planning Horizon Research Articles

Considering human error in optimizing production and corrective and preventive maintenance policies for manufacturing systems

This article extends previous work on co-optimization of production and corrective and preventive maintenance including lockout/tagout. We study the impact of human error on repairable manufacturing systems subject to random failure over an infinite planning horizon and its implications for system capacity and inventory policies, and we derive an optimal policy for minimizing production cost based on machinery maintenance and inventory management while meeting market demand over an infinite horizon. A numerical example is provided to demonstrate the usefulness of the proposed approach and a sensitivity analysis is presented to confirm the efficiency of the control policy.

Production capacity buildup and double marginalization mitigation in a dynamic supply chain

The paper investigates the extent to which capacity investment considerations interact with the double marginalization effect in a simple supply chain governed by a wholesale price contract. To do so, a non-cooperative differential game model is formulated to study the pricing and capacity investment decisions in a supply chain, which consists of a supplier and a manufacturer. In such a game, there are different decision rules—open-loop, closed-loop, feedback—that are available to the supply chain participants, depending on the observability of the current state of the supply chain. While closed-loop and feedback equilibrium strategies involve the observability of other chain member’s production capacity, open-loop equilibrium strategies do not have such requirement. We examine how the supplier and the manufacturer determine, with the different decision rules, their production capacities and pricing policies to maximize their profits over an infinite planning horizon, and determine how the observability of other supply chain’s members’ production capacity affects the magnitude of the double marginalization effect. Our study suggests that the observability of other chain member’s current production capacity entails a lower production efficiency that results in a greater double marginalization effect. This allows us to conclude that observability of other chain member’s current production capacity is associated with a greater double marginalization effect.

Studying a modified ramsey model with variable production flexibility

A one-dimensional nonlinear problem of optimal control on an infinite planning horizon is considered that is a modification of Ramsey‘s model for endogenous economic growth with the Cobb–Douglas production function. The model is innovative in its consideration of variable production flexibility, a parameter of the Cobb–Douglas function. As the first step in the study, the problem is considered for piecewise-smooth and piecewise-constant flexibility functions. Optimal solutions are constructed on the basis of a special integral representation of the functional and are unique for each type of flexibility function. The optimal solutions include singular modes. The final results can be used to estimate the effect of uncertainty in a Ramsey problem with constant parameters.

Technical Note—Managing Nonperishable Inventories with Learning About Demand Arrival Rate Through Stockout Times

We study a periodic review inventory model with a nonperishable product over an infinite planning horizon. The demand for the nonperishable product arrives according to a Poisson process. Lost sales are unobservable but the stockout times are observable. We formulate the problem as a dynamic programming model with learning on arrival rate according to stockout times and further simplify it by using unnormalized probabilities. We then compare the system performance with those under other two information scenarios where lost sales are observable or both lost sales and stockout times are unobservable. We show that the optimal inventory order-up-to level with observable stockout times is larger than the one with observable lost sales. We also show that more information improves the system performance.

Robust Optimal Control Design Using a Differential Game Approach for Open-Loop Linear Quadratic Descriptor Systems

This paper studies the robust optimal control problem for descriptor systems. We applied differential game theory to solve the disturbance attenuation problem. The robust control problem was converted into a reduced ordinary zero-sum game. Within a linear quadratic setting, we solved the problem for finite and infinite planning horizons.

Pulsation in a competitive model of advertising-firm's cost interaction

The literature reveals contradiction between theoretical results (superiority of uniform policy under a concave advertising response function) and empirical results (concavity of the advertising response function and the superiority of a pulsation policy). To reconcile the above difference, this paper offers a resolution based on (1) the concavity of the advertising response function; (2) the convexity of the firm's cost function; and (3) over-advertising. The resolution is reached upon maximizing the net profit per unit time over the infinite planning horizon subject to an exogenous advertising budget constraint. Theoretical results for monopolistic markets are found mostly generalized to competitive markets. A numerical example is introduced to gain more insight into the theoretical findings and an approach is introduced and implemented to empirically assess the shape of a firm's cost function and the advertising policy to be employed.

A dynamic advertising model with reference price effect

This paper develops an advertising model in which goodwill affected by advertising effort has a positive effect on reference price and market demand. In a finite planning horizon, the optimal advertising strategy is provided by solving the optimization problem on the basis of Pontryagin’s maximum principle, then the optimal sales price is obtained through one time pricing strategy. Furthermore, we extend this problem to an infinite planning horizon and present the corresponding optimal strategies. In addition, the relationships between system parameters and optimal solutions are analyzed. Numerical examples are employed to illustrate the effectiveness of the theoretical results, and to assess the sensitivity analysis of system parameters on the optimal strategies.

Portfolio Rules with Log Consumption Utility and Cox–Ingersoll–Ross Interest Rate

We consider the portfolio management problem with the objective of consumption optimization. Dynamic programming is applied to find the portfolio and consumption management strategy. Lower and upper solutions of an ordinary differential equation are applied to prove that the proposed strategy is optimal for an infinite planning horizon with a log utility function and interest rate specified by the generalized Cox---Ingersoll---Ross model. The optimal control is determined analytically for the Vasicek model.

Joint Production and Replacement Strategy for a Quality Deteriorating Failure-Prone Manufacturing System

This paper deals with the simultaneous production planning and quality control problem in manufacturing systems with unreliable single machine responding to a single product type demand. The machine is subject to random failures and repairs, and deteriorates with its age while in operation. This deterioration manifests by imperfect repairs and is characterized to increase the failures rates and decrease the quality of the parts produced. Because the availability and the quality of the parts of the machine decrease with repair activities, the introduction of the option to replace the machine will be taken to completely restore the age of the machine. When the replacement action is selected, the machine is replaced by a new identical one. The objective of this research is to find the simultaneous production and replacement policy under quality and failures deteriorations. The optimal solution is established in order to minimize the discounted total cost including the inventory holding, backlog, production, repair and replacement costs over an infinite planning horizon and satisfy the customer demand. Due to imperfect repairs, the semi-Markov processes are more appropriate to describe the systems subject to deteriorations caused by the machine aging. Optimality conditions are developed using the optimal control theory based on stochastic dynamic programming. Numerical methods are used to obtain the optimal control policies (production rate and replacement policies). Finally, an illustrative example and a sensitivity analysis are provided to confirm the structure of the obtained optimal control policies.

Dynamic Pricing and Production Control of an Inventory System with Remanufacturing

We consider optimal pricing and manufacturing control of a continuous-review inventory system with remanufacturing. Customer demand and product return follow independent Poisson processes. Customer demand is filled by serviceable product, which can be either manufactured or remanufactured from the returned product. The lead times for both manufacturing and remanufacturing are exponentially distributed. The objective is to maximize the expected total discounted profit over an infinite planning horizon. We characterize the structural properties of the optimal policy through the optimality equation. Specifically, the optimal manufacturing policy is a base-stock policy with the base-stock level nonincreasing in the return inventory level. The optimal pricing policy is also a threshold policy, where the threshold level is nonincreasing in the return inventory level.

Optimality test in fuzzy inventory model for restricted budget and space: Move forward to a non-linear programming approach

In this paper, the concept of fuzzy Non-Linear Programming Technique is applied to solve an economic order quantity (EOQ) model for restricted budget and space. Since various types of uncertainties and imprecision are inherent in real inventory problems, they are classically modeled using the approaches from the probability theory. However, there are uncertainties that cannot be appropriately treated by the usual probabilistic models. The questions are how to define inventory optimization tasks in such environment and how to interpret the optimal solutions. This paper allow the modification of the Single item EOQ model in presence of fuzzy decision making process where demand is related to the unit price, and the setup cost varies with the quantity produced/Purchased. The modification of objective function, budget, and storage area in the presence of imprecisely estimated parameters are considered. The model is developed by employing different approaches over an infinite planning horizon. It incorporates all the concepts of a fuzzy arithmetic approach and comparative analysis with other non linear models. Investigation of the properties of an optimal solution allows developing an algorithm whose validity is illustrated by an example problem, and two and three dimensional diagrams are represented to this application through MATL(R2009a) software. Sensitivity analysis of the optimal solution is studied with respect to the changes of different parameter values for obtaining managerial insights of the decision problem.

Optimal Pricing and Order-Up-To S Inventory Policy with Expected Utility of the Present Value Criterion

The article determines pricing and order-up-to level S inventory decisions over an infinite planning horizon from the point of view of a risk-averse decision maker. The demand is assumed to be stochastic but influenced by the selling price which is a decision variable. Shortages are allowed and backordered partially. We calculate the present value of the cash flow over the entire planning horizon and incorporate the notion of risk aversion into the model using a concave utility function. We numerically demonstrate the model and investigate the impact of different model-parameters on the optimal decisions. It is observed that the optimal selling price for a risk-averse decision maker is not less than the optimal selling price of a risk-neutral decision maker while the optimal order level for the risk-averse decision maker is always less than that of the risk-neutral decision maker.

Warehouse location and two-echelon inventory management with concave operating cost

In this paper, we study a location–inventory network design problem which jointly optimises the warehouse location, the warehouse–retailer assignments, the warehouse–retailer echelon inventory replenishment and the safety stock-level decisions over an infinite planning horizon. The consideration of the facility operating cost, the safety stock cost and the two-echelon inventory cost results in an MIP model with several nonlinear terms. Due to the complex trade-offs among the various costs and multiple nonlinear terms in the model, traditional solution approaches no longer work for this problem. We outline a polymatroid cutting-plane approach based on the submodular property of the cost terms to address this problem. Computational results demonstrate that the cutting-plane method based on polymatroid inequalities can efficiently solve randomly generated instances with moderate sizes.

Optimal lockout/tagout, preventive maintenance, human error and production policies of manufacturing systems with passive redundancy

Purpose – The analysis of the optimal production and preventive maintenance with lockout/tagout planning problem for a manufacturing system is presented in this paper. The considered manufacturing system consists of two non-identical machines in passive redundancy producing one type of part. These machines are subject to random breakdowns and repairs. The purpose of this paper is to minimize production, inventory, backlog and maintenance costs over an infinite planning horizon; in addition, it aims to verify the influence of human reliability on the inventory levels for illustrating the importance of human error during the maintenance and lockout/tagout activities. Design/methodology/approach – This paper is different compared to other research projects on preventive maintenance and lockout/tagout. The influence of human error on lockout/tagout as well as on preventive maintenance activities are presented in this paper. The preventive maintenance policy depends on the machine age. For the considered manufacturing system the optimality conditions are provided, and numerical methods are used to obtain machine age-dependent optimal control policies (production and preventive maintenance rates with lockout/tagout). Numerical examples and sensitivity analysis are presented to illustrate the usefulness of the proposed approach. The system capacity is described by a finite-state Markov chain. Findings – The proposed model taking into account the preventive maintenance activities with lockout/tagout and human error jointly, instead of taking into account separately. It verifies the influence of human error during preventive maintenance and lockout/tagout activities on the optimal safety stock levels using an extension of the hedging point structure. Practical implications – The model proposed in this paper might be extended to manufacturing systems, but a number of conditions must be met to make effective use of it. Originality/value – The originality of this paper is to consider the preventive maintenance activities with lockout/tagout and human error simultaneously. The control policy is obtained in order to find the solution for the considered manufacturing system. This paper also brings a new vision on the importance of human reliability during preventive maintenance and lockout/tagout activities.

Ratchet consumption over finite and infinite planning horizons

Ratchet consumers want their spending to always increase and never decrease. We find an optimal consumption rule for ratchet consumers by maximizing an expected utility that eschews spending declines, yet permits a range of choices for felicity and time preference functions. This solution can be tailored to fit both retirees with finite planning horizons and endowments with infinite planning horizons. We assume complete markets modeled by a pricing kernel generated by a Lévy process. When the kernel is log-normal, we obtain closed-form solutions for both finite and infinite horizons.

A 4-Stated DICE: Quantitatively Addressing Uncertainty Effects in Climate Change

We introduce a version of the DICE-2007 model designed for uncertainty analysis. DICE is a wide-spread deterministic integrated assessment model of climate change. Climate change, long-term economic development, and their interactions are highly uncertain. The quantitative analysis of optimal mitigation policy under uncertainty requires a recursive dynamic programming implementation of integrated assessment models. Such implementations are subject to the curse of dimensionality. Every increase in the dimension of the state space is paid for by a combination of (exponentially) increasing processor time, lower quality of the value or policy function approximations, and reductions of the uncertainty domain. The paper promotes a state-reduced, recursive dynamic programming implementation of the DICE-2007 model. We achieve the reduction by simplifying the carbon cycle and the temperature delay equations. We compare our model’s performance and that of the DICE model to the scientific AOGCM models emulated by MAGICC 6.0 and find that our simplified model performs equally well as the original DICE model. Our implementation solves the infinite planning horizon problem in an arbitrary time step. The paper is the first to carefully analyze the quality of the value function approximation using two different types of basis functions and systematically varying the dimension of the basis. We present the closed form, continuous time approximation to the exogenous (discretely and inductively defined) processes in DICE, and we present a numerically more efficient re-normalized Bellman equation that, in addition, can disentangle risk attitude from the propensity to smooth consumption over time.

A 4-Stated Dice: Quantitatively Addressing Uncertainty Effects in Climate Change

We introduce a version of the DICE-2007 model designed for uncertainty analysis. DICE is a wide-spread deterministic integrated assessment model of climate change. Climate change, long-term economic development, and their interactions are highly uncertain. The quantitative analysis of optimal mitigation policy under uncertainty requires a recursive dynamic programming implementation of integrated assessment models. Such implementations are subject to the curse of dimensionality. Every increase in the dimension of the state space is paid for by a combination of (exponentially) increasing processor time, lower quality of the value or policy function approximations, and reductions of the uncertainty domain. The paper promotes a state reduced, recursive dynamic programming implementation of the DICE-2007 model. We achieve the reduction by simplifying the carbon cycle and the temperature delay equations. We compare our model's performance and that of the DICE model to the scientific AOGCM models emulated by MAGICC 6.0 and find that our simplified model performs equally well as the original DICE model. Our implementation solves the infinite planning horizon problem in an arbitrary time step. The paper is the first to carefully analyze the quality of the value function approximation using two different types of basis functions and systematically varying the dimension of the basis. We present the closed form, continuous time approximation to the exogenous (discretely and inductively defined) processes in DICE, and we present a numerically more efficient re-normalized Bellman equation that, in addition, can disentangle risk attitude from the propensity to smooth consumption over time.

Optimal production and pricing policies in a combined make-to-order/make-to-stock system

In this paper, we consider a production system which is capable to produce two types of products. The first type of products is make-to-order, while the second type is make-to-stock whose demand is satisfied by the on-hand inventory. The demand arrival rates of both types of products are price-sensitive. The excess demand that cannot be satisfied immediately is either backlogged or lost. The system costs include the holding costs of product inventories and shortage costs of unsatisfied demand. The objective is to maximise the total discounted profit over an infinite planning horizon by coordinating the production process and pricing decisions. By analysing the properties of objective functions, we characterise the optimal control policy by two switch curves and the optimal price is also given for different ordering and inventory levels. We also explore the monotonicity of both switch curves which will reduce the computation effort. Numerical experiments are conducted to demonstrate the use of the switch curves in managing the production system and illustrate that compared with the static pricing policy, the optimal integrated price and inventory control policy can result in a significant profit improvement in the make-to-order/make-to-stock system that is much higher than in a single-product system.

Joint optimization of pavement design, resurfacing and maintenance strategies with history-dependent deterioration models

The subject of this paper is the joint optimization of pavement design, maintenance and resurfacing (M&R) strategies. This problem is solved for continuous pavement state, continuous time, infinite planning horizon and non-Markovian (history-dependent) pavement deterioration model. This paper presents a mathematical formulation of the joint optimization problem to minimize the total discounted lifecycle costs, using a combination of analytical and numerical tools. The lifecycle costs include both user costs and agency (construction, resurfacing and maintenance) costs. This paper shows that resurfacing schedule converges to a steady state after a few resurfacing cycles. The research results should be of use to developing countries in the process of expanding their highway networks facing multiple constraints.

An adaptive large neighborhood search algorithm for a selective and periodic inventory routing problem

We study a selective and periodic inventory routing problem (SPIRP) and develop an Adaptive Large Neighborhood Search (ALNS) algorithm for its solution. The problem concerns a biodiesel production facility collecting used vegetable oil from sources, such as restaurants, catering companies and hotels that produce waste vegetable oil in considerable amounts. The facility reuses the collected waste oil as raw material to produce biodiesel. It has to meet certain raw material requirements either from daily collection, or from its inventory, or by purchasing virgin oil. SPIRP involves decisions about which of the present source nodes to include in the collection program, and which periodic (weekly) routing schedule to repeat over an infinite planning horizon. The objective is to minimize the total collection, inventory and purchasing costs while meeting the raw material requirements and operational constraints. A single-commodity flow-based mixed integer linear programming (MILP) model was proposed for this problem in an earlier study. The model was solved with 25 source nodes on a 7-day cyclic planning horizon. In order to tackle larger instances, we develop an ALNS algorithm that is based on a rich neighborhood structure with 11 distinct moves tailored to this problem. We demonstrate the performance of the ALNS, and compare it with the MILP model on test instances containing up to 100 source nodes.