A combined forecast—inventory control procedure for spare parts

This paper examines the performance of two different (s, Q) inventory models, namely a simple and an advanced model, for spare parts in a production plant of a confectionery producer in the Netherlands. The simple approach is more or less standard: the undershoot of the reorder level is not taken into account and the normal distribution is used as the distribution of demand during lead-time. The advanced model takes undershoots into account, differentiates between zero and nonzero demands during lead-time, and utilises the gamma distribution for the demand distribution. Both models are fed with parameters estimated by a procedure that forecasts demand sizes and time between demand occurrences separately (intermittent demand). The results show that the advanced approach yields a service level close to the desired one under many circumstances, while the simple approach is not consistent, in that it leads to much larger inventories in meeting the desired service level for all spare parts.     

On the tradeoff between optimal order-base-stock levels and demand lead-times

We investigate the tradeoff between finished-goods inventory and advance demand information for a model of a single-stage make-to-stock supplier who uses an order-base-stock replenishment policy to meet customer orders that arrive a fixed demand lead-time in advance of their due-dates. We show that if the replenishment orders arrive in the order that they are placed, then the tradeoff between the optimal order-base-stock level and the demand lead-time is “exhaustive”, in the sense that the optimal order-base-stock level drops all the way to zero if the demand lead-time is sufficiently long. We then provide a sufficient condition under which this tradeoff is linear. We verify that this condition is satisfied for the case where the supply process is modeled as an M/M/1 queue. We also show that the tradeoff between the optimal order-base-stock level and the demand lead-time is linear for the case where the supply process is modeled as an M/D/1 queue. More specifically, for this case, we show that the optimal order-base-stock level decreases by one unit if the demand lead-time increases by an amount equal to the supplier’s constant processing time. Finally, we show that the tradeoff between the optimal order-base-stock level and the demand lead-time is exhaustive but not linear in the case where the supply process is modeled as an M/D/∞ queue. We illustrate these results with a numerical example.     

Optimal and heuristic lead-time quotation for an integrated steel mill with a minimum batch size

This paper presents a model of lead-time policies for a production system, such as an integrated steel mill, in which the bottleneck process requires a minimum batch size. An accurate understanding of internal lead-time quotations is necessary for making good customer delivery-date promises, which must take into account processing time, queueing time and time for arrival of the requisite volume of orders to complete the minimum batch size requirement. The problem is modeled as a stochastic dynamic program with a large state space. A computational study demonstrates that lead time for an arriving order should generally be a decreasing function of the amount of that product already on order (and waiting for minimum batch size to accumulate), which leads to a very fast and accurate heuristic. The computational study also provides insights into the relationship between lead-time quotation, arrival rate, and the sensitivity of customers to the length of delivery promises.     

Analysis of Multi-Product Kanban Systems with State-Dependent Setups and Lost Sales

We propose a decomposition-based approximation method that generates fairly accurate estimates for steady-state performance measures of a kanban-controlled production system. The manufacturing facility of this system can process items of several different products. Setup and processing times are assumed to be exponentially distributed. Customers arrive according to mutually independent Poisson processes. A customer whose demand cannot be met from stock leaves the system and satisfies his demand elsewhere (lost sales). The manufacturing facility processes items of a product until a target inventory level given by the number of kanbans has been reached. Then the manufacturing facility is set up for the next product according to a fixed setup sequence if the next product's inventory level is below target. Otherwise, this product is skipped (cyclic-exhaustive processing with state-dependent setups). The manufacturing facility idles when the inventory levels of all products are at their target levels.     

Stochastic decomposition in production inventory with service time

We study an (s, S) production inventory system where the processing of inventory requires a positive random amount of time. As a consequence a queue of demands is formed. Demand process is assumed to be Poisson, duration of each service and time required to add an item to the inventory when the production is on, are independent, non-identically distributed exponential random variables. We assume that no customer joins the queue when the inventory level is zero. This assumption leads to an explicit product form solution for the steady state probability vector, using a simple approach. This is despite the fact that there is a strong correlation between the lead-time (the time required to add an item into the inventory) and the number of customers waiting in the system. The technique is: combine the steady state vector of the classical M/M/1 queue and the steady state vector of a production inventory system where the service is instantaneous and no backlogs are allowed. Using a similar technique, the expected length of a production cycle is also obtained explicitly. The optimal values of S and the production switching on level s have been studied for a cost function involving the steady state system performance measures. Since we have obtained explicit expressions for the performance measures, analytic expressions have been derived for calculating the optimal values of S and s.     

Managing variances in manufacturing system design

The goal of this paper is to investigate how uncertainties in demand and production should be incorporated into manufacturing system design problems. We examine two problems in manufacturing system design: the resource allocation problem and the product grouping problem. In the resource allocation problem, we consider the issue of how to cope with uncertainties when we utilize two types of resources: actual processing capacity and stored capacity (inventory). A closed form solution of the optimal allocation scheme for each type of capacity is developed, and its performance is compared to that of the conventional scheme where capacity allocation and inventory control decisions are made sequentially. In the product grouping problem, we consider the issue of how we design production lines when each line is dedicated to a certain set of products. We formulate a mathematical program in which we simultaneously determine the number of production lines and the composition of each line. Two heuristics are developed for the problem.     

Production scheduling with supply and delivery considerations to minimize the makespan

In this paper we study a scheduling model that simultaneously considers production scheduling, material supply, and product delivery. One vehicle with limited loading capacity transports unprocessed jobs from the supplier’s warehouse to the factory in a fixed travelling time. Another capacitated vehicle travels between the factory and the customer to deliver finished jobs to the customer. The objective is to minimize the arrival time of the last delivered job to the customer. We show that the problem is NP-hard in the strong sense, and propose an O(n) time heuristic with a tight performance bound of 2. We identify some polynomially solvable cases of the problem, and develop heuristics with better performance bounds for some special cases of the problem. Computational results show that all the heuristics are effective in producing optimal or near-optimal solutions quickly.     

Manufacturing lead-time rules: Customer retention versus tardiness costs

Inaccurate production backlog information is a major cause of late deliveries, which can result in penalty fees and loss of reputation. We identify conditions when it is particularly worthwhile to improve an information system to provide good lead-time information. We first analyze a sequential decision process model of lead-time decisions at a firm which manufactures standard products to order, and has complete backlog information. There are Poisson arrivals, stochastic processing times, customers may balk in response to quoted delivery dates, and revenues are offset by tardiness penalties. We characterize an optimal policy and show how to accelerate computations. The second part of the paper is a computational comparison of this optimum (with full backlog information) with a lead-time quotation rule that is optimal with statistical shop-status information. This reveals when the partial-information method does well and when it is worth implementing measures to improve information transfer between operations and sales.     

Timing order fulfillment of capital goods under a constrained capacity

This paper studies the order-fulfillment process of a supplier producing multiple customized capital goods. The times when orders are confirmed by customers are random. The supplier can only work on one product at any time due to capacity constraints. The supplier must determine the optimal time to start the process for each order so that the total expected cost of having the goods ready before or after their orders are confirmed is minimized. We formulate this problem as a discrete time Markov decision process. The optimal policy is complex in general. It has a threshold-type structure and can be fully characterized only for some special cases. Based on our formulation, we compute the optimal policy and quantify the value of jointly managing the order fulfillment processes of multiple orders and the value of taking into account demand arrival time uncertainty.     

Mathematical model for scheduling operations in cascaded continuous processing units

The scenario under consideration involves n cascaded continuous processing units responsible for processing m product lines. Each product line needs to be processed by all the units in the same sequence, and has dedicated finite capacity storage tanks before and after every processing unit. A unit can process only one product line at a time. Inputs for all the product lines arrive continuously and simultaneously on the input side of the first unit in the sequence. There are multiple intermediate due dates for the final products. An optimal schedule for the units calls for a trade-off among spillage costs, upliftment failure penalties and changeover costs. A mathematical model is developed for the purpose and the resulting MINLP is linearized using standard techniques. The MILP has been tested using GAMS for three units and three product lines as encountered in a refinery situation. The model could output optimal schedules for a ten day scheduling horizon within reasonable time.     

Minimizing customer order lead-time in a two-stage assembly supply chain

Coordination across different process stages of the supply chain is becoming more common as the information needed for this coordination is easier to obtain and share. With the availability of this information, managers are beginning to recognize that there can be benefits to scheduling processes in a coordinated fashion. Thus, finding good schedules for the entire supply chain has added importance to today’s managers. Coordination of the material as it moves from one stage to the next should lead to improved customer order lead-time performance for the whole chain and thus better customer service overall. We look at a two-stage assembly supply chain with the objective of minimizing the average customer order lead-time. Minimizing lead-time is becoming increasingly important as customers demand quicker response. But beyond this better customer service objective, minimizing lead-time is consistent with keeping inventory costs low. We introduce a number of properties of optimal solutions, results for special problem cases, and a series of lower bounds. We also provide a number of intuitive heuristics for coordinated supply chain scheduling and test them to determine their effectiveness.     

Optimal production scheduling for the dairy industry

The increasing variety of products offered by the food industry has helped the industry to respond to market trends, but at the same time has resulted in a more complex production process, which requires flexibility and an efficient coordination of existing resources. Especially in industrial yogurt production, there is a wide variety of products that differ in features like fat content, the whey used to produce the mixture, the flavor, the size of the container or the language on the label. The great diversification and the special features that characterize yogurt production lines (satisfaction of multiple due dates, variable processing times, sequence-dependent setup times and costs and monitoring of inventory levels), render generic scheduling methodologies impractical for real-world applications. In this work we present a customized Mixed Integer Linear Programming (MILP) model for optimizing yogurt packaging lines that consist of multiple parallel machines. The model is characterized by parsimony in the utilization of binary variables and necessitates the use of only a small pre-determined number of time periods. The efficiency of the proposed model is illustrated through its application to the yogurt production plant of a leading dairy product manufacturing company in Greece.     

Managing an integrated production inventory system with information on the production and demand status and multiple non-unitary demand classes

In this paper, we study a system consisting of a manufacturer or supplier serving several retailers or clients. The manufacturer produces a standard product in a make-to-stock fashion in anticipation of orders emanating from n retailers with different contractual agreements hence ranked/prioritized according to their importance. Orders from the retailers are non-unitary and have sizes that follow a discrete distribution. The total production time is assumed to follow a k₀-Erlang distribution. Order inter-arrival time for class l demand is assumed to follow a k_l-Erlang distribution. Work-in-process as well as the finished product incur a, per unit per unit of time, carrying cost. Unsatisfied units from an order from a particular demand class are assumed lost and incur a class specific lost sale cost. The objective is to determine the optimal production and inventory allocation policies so as to minimize the expected total (discounted or average) cost. We formulate the problem as a Markov decision process and show that the optimal production policy is of the base-stock type with base-stock levels non-decreasing in the demand stages. We also show that the optimal inventory allocation policy is a rationing policy with rationing levels non-decreasing in the demand stages. We also study several important special cases and provide, through numerical experiments, managerial insights including the effect of the different sources of variability on the operating cost and the benefits of such contracts as Vendor Managed Inventory or Collaborative Planning, Forecasting, and Replenishment. Also, we show that a heuristic that ignores the dependence of the base-stock and rationing levels on the demands stages can perform very poorly compared to the optimal policy.     

Supply chain scheduling in a collaborative manufacturing mode: model construction and algorithm design

In collaborative manufacturing, the supply chain scheduling problem becomes more complex according to both multiple product demands and multiple production modes. Aiming to obtain a reasonable solution to this complexity, we analyze the characteristics of collaborative manufacturing and design some elements, including production parameters, order parameters, and network parameters. We propose four general types of collaborative manufacturing networks and then construct a supply chain scheduling model composed of the processing costs, inventory costs, and two penalty costs of the early completion costs and tardiness costs. In our model, by considering the urgency of different orders, we design a delivery time window based on the least production time and slack time. Additionally, due to the merit of continuously processing orders belonging to the same product type, we design a production cost function by using a piecewise function. To solve our model efficiently, we present a hybrid ant colony optimization (HACO) algorithm. More specifically, the Monte Carlo algorithm is incorporated into our HACO algorithm to improve the solution quality. We also design a moving window award mechanism and dynamic pheromone update strategy to improve the search efficiency and solution performance. Computational tests are conducted to evaluate the performance of the proposed method.

     

Measure of bullwhip effect in supply chains: The case of high order autoregressive demand process

Bullwhip effect – the phenomenon in which variance of demand is amplified when moving upstream – has attracted the attention of many researchers for the last few decades. Although the main sources that cause bullwhip effect have been identified, quantifying the bullwhip effect still remains a challenge. In the past, measuring the bullwhip effect for supply chains with autoregressive demand process has been conducted by some researchers. However, most past researches focused mainly on the simple AR(1) model. In many practical situations, autoregressive models with higher order should be employed because they might better represent the demand process. Up to now, very little effort has been spent on this matter. Therefore, this research is conducted to fill this gap by first dealing with AR(2) demand process and investigating the behavior of the developed measure with respect to autoregressive coefficients and order lead-time. Extension to the general AR(p) demand process is then considered.     

Optimal integrated production and inventory control of an assemble-to-order system with multiple non-unitary demand classes

We study a pure assemble-to-order system subject to multiple demand classes where customer orders arrive according to a compound Poisson process. The finished product is assembled from m different components that are produced on m distinct production facilities in a make-to-stock fashion. We show that the optimal production policy of each component is a state-dependent base-stock policy and the optimal inventory allocation policy is a multi-level state-dependent rationing policy. Using numerical experimentation, we first study the system behavior as a function of order size variability and order size. We show that the optimal average cost rate is more sensitive to order size variability than to order size. We also compare the optimal policy to the first-come first-serve policy and show that there is great benefit to inventory rationing. We also propose two simple heuristics and show that these can effectively mimic the optimal policy which is generally much more difficult to determine and, especially, to implement.     

Multiple-buyer multiple-vendor multi-product multi-constraint supply chain problem with stochastic demand and variable lead-time: A harmony search algorithm

In this paper, a multi-buyer multi-vendor supply chain problem is considered in which there are several products, each buyer has limited capacity to purchase products, and each vendor has warehouse limitation to store products. In this chain, the demand of each product is stochastic and follows a uniform distribution. The lead-time of receiving products from a vendor to a buyer is assumed to vary linearly with respect to the order quantity of the buyer and the production rate of the vendor. For each product, a fraction of the shortage is backordered and the rest are lost. The ordered product quantities are placed in multiple of pre-defined packets and there are service rate constraints for the buyers. The goal is to determine the reorder points, the safety stocks, and the numbers of shipments and packets in each shipment of the products such that the total cost of the supply chain is minimized. We show that the model of this problem is of an integer nonlinear programming type and in order to solve it a harmony search algorithm is employed. To validate the solution and to compare the performance of the proposed algorithm, a genetic algorithm is utilized as well. A numerical illustration and sensitivity analysis are given at the end to show the applicability of the proposed methodology in real-world supply chain problems.     

Modelling and scheduling of an asynchronous cyclic production line with multiple parts

In this study, we model and analyse a production line with asynchronous part transfers, processing time variability, and cyclic scheduling in the same framework. We consider a production line with multiple parts and finite interstation buffers. The line produces a batch of n jobs repetitively using the same order of jobs in every batch. The processing time of a job on a station is a random variable and is assumed to have a phase-type distribution. Parts are transferred between the stations in an asynchronous manner. We first present a continuous time Markov chain model to analyse the performance of this system for a given sequence. A state-space representation of the model and the associated rate matrix are generated automatically. The steady state probabilities of the Markov chain are determined by using a recursive method that exploits the special structure of the rate matrix. The cycle time, the production rate, and the expected Work-In-Progress (WIP) inventory are used as the main performance measures. We then present an approximate procedure to determine the cyclic sequence that minimises the cycle time. We then investigate the effects of operating decisions, system structure, processing time variability, and their interaction in the same framework. Numerical results for the performance evaluation and scheduling of cyclic production lines are also presented.     

Optimizing delivery lead time/inventory placement in a two-stage production/distribution system

In this paper we study a system composed of a supplier and buyer(s). We assume that the buyer faces random demand with a known distribution function. The supplier faces a known production lead time. The main objective of this study is to determine the optimal delivery lead time and the resulting location of the system inventory. In a system with a single-supplier and a single-buyer it is shown that system inventory should not be split between a buyer and supplier. Based on system parameters of shortage and holding costs, production lead times, and standard deviations of demand distributions, conditions indicating when the supplier or buyer(s) should keep the system inventory are derived. The impact of changes to these parameters on the location of system inventory is examined. For the case with multiple buyers, it is found that the supplier holds inventory for the buyers with the smallest standard deviations, while the buyers with the largest standard deviations hold their own inventory.     

Capacity optimization and competition with cyclical and lead-time-dependent demands

Capacity acquisition is often capital- and time-consuming for a business, and capacity investment is often partially or fully irreversible and difficult to change in the short term. Moreover, capacity determines the action space for service/production scheduling and lead-time quotation decisions. The quoted lead-time affects the customer’s perceived service quality. Thus, capacity acquisition level and lead-time quotation affect a firm’s revenue/profit directly or indirectly. In this paper, we investigate a joint optimization problem of capacity acquisition, delivery lead-time quotation and service-production scheduling with cyclical and lead-time-dependent demands. We first explore the structural properties of the optimal schedule given any capacity and lead-time. Then, the piecewise concave relationship between the delay penalty cost and the capacity acquisition level is found. Thereby, an efficient and effective polynomial time algorithm is provided to determine the optimal capacity acquisition level, delivery lead-time quotation and service/production schedule simultaneously. Furthermore, a capacity competition game among multiple firms is addressed. The numerical studies show that capacity equilibrium often exists and converges to a unique solution.