Intuitionistic fuzzy optimization technique for Pareto optimal solution of manufacturing inventory models with shortages

This paper discusses a manufacturing inventory model with shortages where carrying cost, shortage cost, setup cost and demand quantity are considered as fuzzy numbers. The fuzzy parameters are transformed into corresponding interval numbers and then the interval objective function has been transformed into a classical multi-objective EPQ (economic production quantity) problem. To minimize the interval objective function, the order relation that represents the decision maker’s preference between interval objective functions has been defined by the right limit, left limit, center and half width of an interval. Finally, the transformed problem has been solved by intuitionistic fuzzy programming technique. The proposed method is illustrated with a numerical example and Pareto optimality test has been applied as well.     

Cooperation and game-theoretic cost allocation in stochastic inventory models with continuous review

We study cooperation strategies for companies that continuously review their inventories and face Poisson demand. Our main goal is to analyze stable cost allocations of the joint costs. These are such that any group of companies has lower costs than the individual companies. If such allocations exist they provide an incentive for the companies to cooperate.     

Linear programming based decomposition methods for inventory distribution systems

We consider an inventory distribution system consisting of one warehouse and multiple retailers. The retailers face random demand and are supplied by the warehouse. The warehouse replenishes its stock from an external supplier. The objective is to minimize the total expected replenishment, holding and backlogging cost over a finite planning horizon. The problem can be formulated as a dynamic program, but this dynamic program is difficult to solve due to its high dimensional state variable. It has been observed in the earlier literature that if the warehouse is allowed to ship negative quantities to the retailers, then the problem decomposes by the locations. One way to exploit this observation is to relax the constraints that ensure the nonnegativity of the shipments to the retailers by associating Lagrange multipliers with them, which naturally raises the question of how to choose a good set of Lagrange multipliers. In this paper, we propose efficient methods that choose a good set of Lagrange multipliers by solving linear programming approximations to the inventory distribution problem. Computational experiments indicate that the inventory replenishment policies obtained by our approach can outperform several standard benchmarks by significant margins.     

Managing stochastic inventory systems with free shipping option

In many industries, customers are offered free shipping whenever an order placed exceeds a minimum quantity specified by suppliers. This allows the suppliers to achieve economies of scale in terms of production and distribution by encouraging customers to place large orders. In this paper, we consider the optimal policy of a retailer who operates a single-product inventory system under periodic review. The ordering cost of the retailer is a linear function of the ordering quantity, and the shipping cost is a fixed constant K whenever the order size is less than a given quantity – the free shipping quantity (FSQ), and it is zero whenever the order size is at least as much as the FSQ. Demands in different time periods are i.i.d. random variables. We provide the optimal inventory control policy and characterize its structural properties for the single-period model. For multi-period inventory systems, we propose and analyze a heuristic policy that has a simple structure, the (s, t, S) policy. Optimal parameters of the proposed heuristic policy are then computed. Through an extensive numerical study, we demonstrate that the heuristic policy is sufficiently accurate and close to optimal.     

A stochastic program based lower bound for assemble-to-order inventory systems

In this paper we introduce a multi-stage stochastic program that provides a lower bound on the long-run average inventory cost of a general class of assemble-to-order (ATO) inventory systems. The stochastic program also motivates a replenishment policy for these systems. Our lower bound generalizes a previous result of Do?ru et al. (2010) [3] for systems with identical component replenishment lead times to those with general deterministic lead times. We provide a set of sufficient conditions under which our replenishment policy, coupled with an allocation policy, attains the lower bound (and is hence optimal). We show that these sufficient conditions hold for two examples, a single product system and a special case of the generalized W model.     

On a stochastic demand jump inventory model

This paper considers a Quasi-Variational Inequality (QVI) arising from a stochastic demand jump inventory model in a continuous review setting with a fixed ordering cost and where demand is made up of a deterministic part (which is a function of the stock level) punctuated by random jumps. Under some restrictions on the parameters, a solution to the QVI is found which corresponds to an (s,S) policy.     

On a stochastic inventory model with a generalized holding costs

This paper is concerned with finding the optimal replenishment policy for an inventory model that minimizes the total expected discounted costs over an infinite planning horizon. The demand is assumed to be driven by a Brownian motion with drift and the holding costs (inventory and shortages) are assumed to take some general form. This generalizes the earlier work where holding costs were assumed linear. It turns out that problem of finding the optimal replenishment schedule reduces to the problem of solving a Quasi-Variational Inequality Problem (QVI). This QVI is then shown to lead to an (s, S) policy, where s and S are determined uniquely as a solution of some algebraic equations.     

Convergence guarantees for generalized adaptive stochastic search methods for continuous global optimization

This paper presents some simple technical conditions that guarantee the convergence of a general class of adaptive stochastic global optimization algorithms. By imposing some conditions on the probability distributions that generate the iterates, these stochastic algorithms can be shown to converge to the global optimum in a probabilistic sense. These results also apply to global optimization algorithms that combine local and global stochastic search strategies and also those algorithms that combine deterministic and stochastic search strategies. This makes the results applicable to a wide range of global optimization algorithms that are useful in practice. Moreover, this paper provides convergence conditions involving the conditional densities of the random vector iterates that are easy to verify in practice. It also provides some convergence conditions in the special case when the iterates are generated by elliptical distributions such as the multivariate Normal and Cauchy distributions. These results are then used to prove the convergence of some practical stochastic global optimization algorithms, including an evolutionary programming algorithm. In addition, this paper introduces the notion of a stochastic algorithm being probabilistically dense in the domain of the function and shows that, under simple assumptions, this is equivalent to seeing any point in the domain with probability 1. This, in turn, is equivalent to almost sure convergence to the global minimum. Finally, some simple results on convergence rates are also proved.     

A continuous review replenishment–disposal policy for an inventory system with autonomous supply and fixed disposal costs

In this study, we analyze an inventory system facing stochastic external demands and an autonomous supply (independent return flow) in the presence of fixed disposal costs and positive lead times under a continuous review replenishment–disposal policy. We derive the analytical expressions of the operating characteristics of the system; and, construct the objective function to minimize the total expected costs of ordering, holding, purchasing and disposal per unit time subject to a fill rate constraint. An extensive numerical analysis is conducted to study the sensitivity of the policy parameters and the benefit of employing a policy which allows for disposal of excess stock in this setting. We model the net demand process as the superposition of normally distributed external demand and inflows, which is expressed as a Brownian motion process. Our findings indicate that the disposal option results in considerable savings even (i) in the presence of non-zero fixed disposal costs, (ii) large actual demand rates with high return ratios (resulting in small net demands) and (iii) for moderate return ratios with high demand variability.     

Fill rate of single-stage general periodic review inventory systems

This paper develops an exact formula for the fill rate of a single-stage inventory system that uses a general periodic-review base-stock policy. For normal demand, we present a fill-rate expression that uses the standard normal PDF and CDF, and develop two approximations for the fill rate.     

Sensitivity of continuous review stochastic (s,S) inventory systems to ordering delays

This paper examines continuous review stochastic (s, S) inventory systems with ordering delays. That is, systems where the difference between the time the order should be placed and the time the order is actually placed is non-trivial. The traditional inventory ordering, holding and penalty costs are included and the average cost for an (s, S) policy is developed and examined. Computational results are presented for two cases. In the first case, the manager is aware of the delay and uses the policy that minimizes all costs. We present the increase in cost due to having a delay. In the second case the manager is unaware of the delay and uses the (integer) square root formula. We present the increase in cost due to using the square root formula when it is inappropriate and in fact our computational results indicate that there may very well be a large increase in cost due to being unaware of the ordering delay.     

Coordinating inventory control and pricing strategies: The continuous review model

We analyze an infinite horizon, single product, continuous review model in which pricing and inventory decisions are made simultaneously and ordering cost includes a fixed cost. We show that there exists a stationary (s,S) inventory policy maximizing the expected discounted or expected average profit under general conditions.     

Cárnico-ICSPEA2—A metaheuristic co-evolutionary navigator for a complex co-evolutionary farming system

This paper introduces Cárnico-ICSPEA2, a metaheuristic co-evolutionary navigator designed by its end-user as an aid for the analysis and multi-objective optimisation of a beef cattle enterprise running on temperate pastures and fodder crops in Chalco, Mexico State, in the central plateau of Mexico. By combining simulation routines and a multi-objective evolutionary algorithm with a deterministic and stochastic framework, the software imitates the evolutionary behaviour of the system of interest, helping the farm manager to ‘navigate’ through his system’s dynamic phase space. The ultimate goal was to enhance the manager’s decision-making process and co-evolutionary skills, through an increased understanding of his system and the discovery of new, improved heuristics. This paper describes the numerical simulation and optimisation resulting from the application of Cárnico-ICSPEA2 to solve a specific multi-objective optimisation problem, along with implications for the management of the system of interest.     

Effective and simple EOQ-like solutions for stochastic demand periodic review systems

In this paper we examine a periodic review system under stochastic demand with variable stockout costs. The optimal values for cycle length and amount of safety stock are difficult to obtain because one of the First Order Conditions does not have a closed form solution. However, by using a Taylor series expansion to approximate part of the cost function, we produce a simple cost function structure which is similar to that of deterministic models.We argue that this simple structure is also beneficial to promote the solution in other problems where coordination of cycles is required. To illustrate, we use the joint replenishment problem for multiple items under stochastic demand and suggest simple and efficient solution procedures.     

Vendor managed inventory for environments with stochastic product usage

We develop technology to plan delivery routes for the supply of blood products to hospitals by a blood bank. The technology produces low cost, robust plans that hedge against the natural uncertainty associated with blood product usage at hospitals. The technology relies on sampling-based approaches involving integer programming and variable neighborhood search. An extensive computational study shows the efficacy of the two approaches and highlights the impact of product usage uncertainty on the resulting delivery plans.     

Periodic review inventory control with fluctuating purchasing costs

For an inventory control problem in which the purchasing cost changes (e.g. exchange rate fluctuations), we show that an order up to policy is optimal and determine conditions under which the optimal up to levels are monotonically ordered. We also propose a simple method for predicting the effectiveness of myopic heuristics.     

Evolutionary computation of a deterministic switching regressions estimator

Summary  A deterministic switching regressions estimator is evaluated using an evolutionary method based on genetic algorithms. Distinctive aspects of the method include (1) a combination of simple and random chromosomal crossover and (2) extension of the principle of natural selection to internal parameterization. The evolutionary computation duplicates, significantly faster, the results of an existing enumerative method in samples small enough to permit enumeration. It also provides the ability to calculate the estimator in much larger sample sizes than is possible with the enumerative approach. An example problem from the United States gasoline market is given. Funding support provided by the Gerondelis Foundation, Inc. is gratefully acknowledged.     

Interactive evolutionary multi-objective optimization for quasi-concave preference functions

We present a new hybrid approach to interactive evolutionary multi-objective optimization that uses a partial preference order to act as the fitness function in a customized genetic algorithm. We periodically send solutions to the decision maker (DM) for her evaluation and use the resulting preference information to form preference cones consisting of inferior solutions. The cones allow us to implicitly rank solutions that the DM has not considered. This technique avoids assuming an exact form for the preference function, but does assume that the preference function is quasi-concave. This paper describes the genetic algorithm and demonstrates its performance on the multi-objective knapsack problem.     

Relief inventory modelling with stochastic lead-time and demand

The irregular demand and communication network disruption that are characteristics of situations demanding humanitarian logistics, particularly after large-scale earthquakes, present a unique challenge for relief inventory modelling. However, there are few quantitative inventory models in humanitarian logistics, and assumptions inherent in commercial logistics naturally have little applicability to humanitarian logistics. This paper develops a humanitarian disaster relief inventory model that assumes a uniformly distributed function in both lead-time and demand parameters, which is appropriate considering the limited historical data on relief operation. Furthermore, this paper presents different combinations of lead-time and demand scenarios to demonstrate the variability of the model. This is followed by the discussion of a case study wherein the decision variables are evaluated and sensitivity analysis is performed. The results reveal the presence of a unique reorder level in the inventory wherever the order quantity is insensitive to some lead-time demand values, providing valuable direction for humanitarian relief planning efforts and future research.     

An order expediting policy for continuous review systems with manufacturing lead-time

Expedited shipments are often seen in practice. When the inventory level of an item gets dangerously low after an order has been placed, material managers are often willing to expedite the order at extra fixed and/or variable costs. This paper proposes a single-item continuous-review order expediting inventory policy, which can be considered as an extension of ordinary (s,Q)$(s\text{,}Q)$ models. Besides the two usual operational parameters: reorder point s and order quantity Q, it consists of a third parameter called the expedite-up-to level R. If inventory falls below R at the end of the manufacturing lead-time, the buyer can request the upstream supplier to deliver part of an outstanding order via a fast transportation mode. The amount expedited will raise inventory to R, while the remaining order is delivered via a slow (regular) supply mode. Simple procedures are developed to obtain optimal operational parameters. Computational results show that the proposed policy can save large costs for a firm if service level is high, demand variability is large, the extra cost for expediting is small, or the manufacturing lead-time is long.