Optimal and Heuristic Facility Phase-out Strategies

This paper presents an efficient branch and bound algorithm and near optimal heuristic algorithms for solving the problem of withdrawing inventory and/or service facilities for a good or service whose overall demand is declining over time. In particular, this paper models the problem faced by a manager who must consider closing up to M initially open and operating support facilities as demand shifts and declines over a T period planning horizon. The criterion is minimization of total estimated discounted costs. The costs considered are the variable operating cost at each facility, the transportation costs between facilities and demand centers, and the costs to operate and close each facility. Computational results are presented for both the optimum finding and heuristic algorithms.     

An investigation of multiproduct,multimachine production scheduling and inventory control of a flow-shop system

Unlike most of the previous studies of the multiproduct, multimachine systems, hero an attempt is made to consider the man-machine overtime and idleness costs together with the usual inventory-related costs (the set-up costs, carrying costs, and back-logging costs). The major assumptions of the model include: (a) demand for each product is captive and constant; (b) single or multiple facility work centres may be employed: (c) the total manufacturing operations of any single product must be completed before another product can be started; (d) all the manufacturing facilities are set up simultaneously for each of the products and its production can be started only after all the machines are ready. The annual total variable cost function is found to be very complex. Therefore, a recursive algorithm is required to solve this function for the values of optimal produetion-cycle-thnes of the individual products, T_j ^*,s . As the facilities are assumed to produce only one product at a time, these cycle time values need to be fitted into a production schedule so that there is no conflict between any two products on any of the machines and so that any modification in the values of T_j ^*,s result in the least increase of the system's total annual variable costs. Such total annual variable costs, computed with the help of this model and its alogrithm for 15 different situations, are compared with those provided by two other known models. Each time this model performs significantly better than the other two.     

Setting planned orders in master production scheduling under demand uncertainty

For single end-product master production scheduling with time-varying demand uncertainty and supply capacity, we study approaches to set replenishment quantities over the planning horizon. We present a stochastic programming model along with a simulation-based optimisation and two traditional approaches for setting order quantities. We compare these approaches to two new methods: gamma approximation and safety stock search. Computational experiments show that the gamma approximation and safety stock search perform well in terms of holding and shortage costs, with expected total cost on average, respectively, within 0.06% and 0.66% of the optimal from the stochastic program. On average, the two traditional approaches incur 12% and 45% higher cost than optimal. We provide managerial insights on the effects of parameters such as demand coefficient of variation (cv), utilisation, and target service level on the optimal total cost, the corresponding fill rate, and the relative performance of the approaches. We find that, for finite-normal demand, on average, the impact of target service level on cost is larger than that of demand cv, whose impact is larger than utilisation, except at high utilisation. We illustrate that, when demand is not normal, the gamma approximation significantly outperforms the existing normal approximation from Bollapragada and Rao (2006 Bollapragada, R and Rao, US. 2006. Replenishment planning in discrete-time, capacitated, non-stationary, stochastic inventory systems. IIE Transactions, 38(7): 583–595.  [Google Scholar]).     

Impact on inventory costs with consolidation of distribution centers

The consolidation of Distribution Centers (DCs) is a new trend in global logistics management, with a reduction in inventory costs often being cited as one of the main benefits. This paper uses an analytical modeling approach to study the impact on facility investment and inventory costs when several DCs are consolidated into a central DC. In particular, our model suggests that consolidation leads to lower total facility investment and inventory costs if the demands are identically and independently distributed, or when they follow independent but possibly nonidentical Poisson processes. This agrees with the conclusion of the classical EOQ and newsvendor models. However, we show by an example that, for general stochastic demand processes, the total facility investment and inventory costs of a consolidated system can be infinitely worse off than that of a decentralized system. This arises mainly because the order replenishment fixed cost yields a cost component proportional to the square root of the mean value of the demand, while the demand uncertainty yields a cost component proportional to the standard deviation of the demand. Whether consolidation is cost effective or not depends on the trade-off between these two components, as indicated by an extensive numerical study. We also propose an algorithm that solves for a distribution system with the total facility investment and inventory costs within √2 of the optimal.     

A combinatorial approach to a class of parallel-machine,continuous-time scheduling problems

The paper analyzes a manufacturing system with N non-identical, parallel machines continuously producing one product type in response to its demand. Inventory and backlog costs are incurred when tracking the demand results in inventory surpluses and shortages respectively. In addition, the production cost of a machine is incurred when the machine is not idle. The objective is to determine machine production rates so that the inventory, backlog, and production costs are minimized. For problems with demand defined as an arbitrary function of time, numerical methods are suggested to approximate an optimal solution. The complexity of the approximation methods is polynomial, while finding an exact optimal solution requires exponential time. In a case when production is to cope with a special form of a single-mode, K-level piece-wise constant demand, we prove, with the aid of the maximum principle, that the exact optimal solution can be found as a combination of analytical and combinatorial tools in O(KN ²( max {K,2N})²) time.     

Capacitated dynamic lot sizing with capacity acquisition

One of the fundamental problems in operations management is determining the optimal investment in capacity. Capacity investment consumes resources and the decision, once made, is often irreversible. Moreover, the available capacity level affects the action space for production and inventory planning decisions directly. In this article, we address the joint capacitated lot-sizing and capacity-acquisition problems. The firm can produce goods in each of the finite periods into which the production season is partitioned. Fixed as well as variable production costs are incurred for each production batch, along with inventory carrying costs. The production per period is limited by a capacity restriction. The underlying capacity must be purchased up front for the upcoming season and remains constant over the entire season. We assume that the capacity acquisition cost is smooth and convex. For this situation, we develop a model which combines the complexity of time-varying demand and cost functions and of scale economies arising from dynamic lot-sizing costs with the purchase cost of capacity. We propose a heuristic algorithm that runs in polynomial time to determine a good capacity level and corresponding lot-sizing plan simultaneously. Numerical experiments show that our method is a good trade-off between solution quality and running time.     

Two-critical-number control policy for a stochastic production inventory system with partial backlogging

This paper deals with a production–inventory control model with partial backlogging, in which a reflected Brownian motion governs the inventory level variation. We consider a single storage facility with infinite capacity and assume that shortages are allowed and the total amount of stock-out is a mixture of backordering and lost sales. In addition, the production facility is controlled by a two-parameter (m, M) policy, which switches the production rate when the inventory level reaches the threshold values. The aim is to determine the optimal control parameters m and M by minimising the long-run total expected cost of the system. Some results are illustrated using numerical examples. A sensitivity analysis of the optimal solution with respect to major parameters is also carried out.     

Fuzzy approach to the robust facility layout in uncertain production environments

The proposed method approaches the problem of the optimal facility layout using fuzzy theory. The optimal layout is a robust layout that minimizes the total material handling cost, when the product market demands are uncertain variables, which are defined as fuzzy numbers. Since each department has a limited production capacity, not all possible combinations, deriving from each product's market demand, are taken into account because some combination could exceed the overall department's productivity. Therefore, the optimal solution results by solving a 'constrained' fuzzy optimization problem, in which the fuzzy material handling costs corresponding to the layouts are evaluated, and a ranking method, which considers the grade of pessimism of the decision maker, is established to determine the optimal layout.     

OPTIMAL PRODUCTION PLANNING AND INVENTORY CONTROL*

This paper is concerned with the optimal production planning and inventory control. The first problem is a multiperiod production scheduling problem in which the objective is to minimize the operating cost for a planning period. This cost is composed principally of the sum of the production cost and inventory carrying cost. The second problem considers an inventory system with two decision variables in each planning period. These are the production schedule and work force which are to be determined so as to minimize the operating cost which includes the costs of changing the production rate, of changing the work force and of carrying the inventory. The maximum principle in the discrete form is used to reduce both the first problem which has N decision variables and the second problem which has 2N decision variables respectively to a series of two decision variables problems. The so-called sequential simplex pattern search technique is used to determine the optimal values of these two decision variables. Numerical examples are given to demonstrate the method.     

Capacity requirements planning—optimal work station capacities

The How of work through a work station is analysed in terms of planned input from previous stations, planned output to subsequent stations, planned work-in-process waiting for processing and the planned queue of unprocessed work delayed for processing during sonic subsequent period. The planner is assumed to have some leeway in determining the maximum throughput during each period7mdash; the planned capacity—and is thus interested in determining the optimal capacity, to be maintained at some constant value throughout the planning horizon. This paper presents a methodology for finding the optimal planned capacity of a work station, which minimizes total costs over the planning horizon. These costs are of three types: costs of establishing and maintaining the capacity level; costs of work processing; and costs of hoi ding up work in the queue owing to insufficient capacity at any period.

Various constraints are considered when characterizing the operating environment of the work station: minimum and maximum permissible queue lengths, delays, work-in-process amounts, lead times, underloading and backlogging. It is shown that these restrictions may be transformed into minimum and maximum bounds on allowable capacity levels, so that the problem becomes one of constrained optimization in a single decision variable—the planned capacity. For complex cost functions or when underloading is permissible, complete enumeration of all feasible solutions provides the simplest approach. For linear or quadratic cost functions and no underloading, the shape and smoothness of the total cost curve indicates that set of feasible capacities within which the optimal capacity lies.     

The use of rush deliveries in periodic review assemble-to-order systems

We calculate optimal safety stock in a periodic review (T,S) assemble-to-order system having multiple components and multiple finished goods (FGs). Customer orders for FGs arrive according to independent Poisson processes, and cannot be neither backlogged nor lost. In case of potential component stock-out, the studied system uses rush deliveries from suppliers. For this setting, approximate expressions of the optimal safety stock that minimise the sum of inventory holding and rush ordering costs are developed. Exact optimal safety stocks are calculated using Discrete Event Simulation, and compared numerically to the approximate expressions. The model is applied to a first-tier automotive supplier and yields to a significant reduction in terms of inventory holding and rush ordering costs. A sensitivity analysis on relevant system parameters such as components demand, assembly coefficients and unit rush ordering cost is conducted.     

Optimal Acceptance Test Plans with Grouping

This paper considers the problem of selecting an optimal acceptance test, plan. A cost model is developed which permits comparison of single, dollble, multiple, and sequential test plans. Cost elements include fixed costs to initiate testing, costs to purchase and ship components, and variable costs for testing and return or disposal of untested items.

In the case of seqllential testing, consideration of these cost factors leads to an optimization model having as decision variables the number of samples to ship to the test facility for each possible round of tests. Solution of this optimization problenl enables explicit comparison of test costs with non-sequential designs. A numerical example is provided for illtlstration.     

The determination of capacity expansion programmes with economies of scale

For many production processes there are economies of scale with plant investments and certain operating costs. With long-term increases in demand this affects the timing and sizing of capacity expansion projects. This paper is concerned with the development of optimal expansion programmes. It applies dynamic programming models to minimize the present value of capita! costs, operating expenses and capacity shortage losses.

The paper considers homogeneous production capacity and heterogeneous plant where constituent units have different economies of scale arid working lives. The algorithm is applied to the analysis of an expansion programme for an integrated pulp mill.

The planning implications of such models are considered. Using sensitivity analysis, a number of variables such as discount rates, inflation effects and demand patterns are related to the economy scale factor. This indicates that the optimum capacity expansion schedule is sensitive to discount and inflation rates.     

Investigating reliability improvement of second-hand production equipment considering warranty and preventive maintenance strategies

A cost model for optimal reliability improvement of warranted second-hand production equipment is developed. The second-hand production equipment of age x is subjected to an upgrade action of a certain level u before it is sold with a Free Repair Warranty. We look at determining the optimal upgrade level when not performing and when performing periodic preventive maintenance (PM) during the warranty period. Two different PM strategies are considered: (a) periodic PM actions having the same efficiency level; (b) periodic multi-phase PM actions with a maintenance efficiency level which varies according to the phase. The proposed model aims at helping the dealer to find the optimal upgrade level to perform before selling the second-hand equipment, and to assess whether performing PM actions during the warranty period, according to a specific maintenance strategy, is worthwhile in terms of cost reduction. Numerical experimentations considering each PM scenario are performed in order to investigate how each PM strategy impacts the improvement level to be performed and the associated total expected cost. The obtained results showed that the expected total cost incurred by the dealer is governed by a sensitive trade-off between the warranty servicing cost and the costs associated with the reliability improvement, and with the PM performed during the warranty period. It is also found that the proposed new periodic multi-phase PM policy with an increasing maintenance efficiency level yields lower upgrade levels, inducing lower costs for the dealer.     

The capacitated lot sizing problem with overtime decisions and setup times

The Capacitated Lot Sizing-Problem (CLSP) consists of planning the lot sizes of multiple items over a planning horizon with the objective of minimizing setup and inventory holding costs. In each period that an item is produced a setup cost is incurred. Capacity is limited and homogeneous. Here, the CLSP is extended to include overtime decisions and capacity consuming setups. The objective function consists of minimizing inventory holding and overtime costs. Setups incur costs implicitly via overtime costs, that is, they lead to additional overtime costs when setup times contribute to the use of overtime capacity in a certain period. The resulting problem becomes more complicated than the standard CLSP and requires methods different from the ones proposed for the latter. Consequently, new heuristic approaches are developed to deal with this problem. Among the heuristic approaches are the classical HPP approach and its modifications, an iterative approach omitting binary variables in the model, a Genetic Algorithm approach based on the transportation-like formulation of the single item production planning model with dynamic demand and a Simulated Annealing approach based on shifting family lot sizes among consecutive periods. Computational results demonstrate that the Simulated Annealing approach produces high quality schedules and is computationally most efficient.     

A mixed-integer programming approach to the machine loading and process planning problem in a process layout environment

A mixed-integer programming (MIP) model for the machine loading and process planning problem has been developed for a process layout environment. This model relaxes the most commonly used assumption that each operation can be assigned to only one machine. In this model, the most important production cost components: machining cost, material handling (MH) cost, setup cost and machine idle cost are considered. In addition, by assigning penalty cost for long makespans, the optimal process routes for each part and optimal load for each machine in terms of minimum production cost can be obtained with lower makespan. An example problem is solved, followed by discussions with regard to the impact of batch splitting, setup cost and demand change on optimal solution.     

改进麻雀算法求解带模糊需求的低碳路径优化

目的 针对低碳背景下带模糊需求的低碳多式联运规划问题（Low-carbon Multimodal Transportation Planning Problem with Fuzzy Demand, LCMTPP-FD），以成本最小化构建数学模型。同时，结合现有的强制碳排放、碳税、碳交易和碳补偿等政策对LCMTPP-FD进行模型转换，研究不同低碳政策对物流成本和碳排放量的影响。方法 主要根据模型的特征，设计一种t分布麻雀搜索算法，对不同低碳政策下的模型进行求解，将迭代次数作为t分布的自由度来提高麻雀算法的性能。结果 将改进算法及多个模型应用于实际运输案例中，改进的麻雀算法能在较短时间内获得最优解，并且在强制碳排放下碳排放量最少为9 522.28，在碳交易和碳补偿政策下成本分别降低了11.41%、17.24%。结论 改进的麻雀搜索算法具有较好的收敛性和搜索能力。强制碳排放能有效地降低碳排放量，碳交易和碳补偿能有效降低企业成本，适合于低碳运输的推广。     

A multi-objective mixed-integer programming model for a multi-floor facility layout

We consider a multi-floor facility layout problem in which the overall length and width of the facility, the size and location of each department, the number and the location of elevators and the number of floors in the facility are all modelled as decision variables. We adapt a linear approximation scheme to represent the area of each department. We consider two objective functions in our model, namely minimising material handling and facility building costs, and propose a lexicographic ordering technique to handle multiple objectives. The numerical experiments show that the slack used in the lexicographic ordering approach has a significant impact on the optimal solution. The experiments also show that the material handling cost can be significantly reduced in a multi-floor facility compared with a single-floor facility.     

A multiperiod stochastic production planning and sourcing problem with service level constraints

We study a stochastic multiperiod production planning and sourcing problem of a manufacturer with a number of plants and/or subcontractors. Each source, i.e. each plant and subcontractor, has a different production cost, capacity, and lead time. The manufacturer has to meet the demand for different products according to the service level requirements set by its customers. The demand for each product in each period is random. We present a methodology that a manufacturer can utilize to make its production and sourcing decisions, i.e., to decide how much to produce, when to produce, where to produce, how much inventory to carry, etc. This methodology is based on a mathematical programming approach. The randomness in demand and related probabilistic service level constraints are integrated in a deterministic mathematical program by adding a number of additional linear constraints. Using a rolling horizon approach that solves the deterministic equivalent problem based on the available data at each time period yields an approximate solution to the original dynamic problem. We show that this approach yields the same result as the base stock policy for a single plant with stationary demand. For a system with dual sources, we show that the results obtained from solving the deterministic equivalent model on a rolling horizon gives similar results to a threshold subcontracting policy. Correspondence to: Fikri KaraesmenThe authors are grateful to Yves Dallery for his ideas, comments and suggestions on the earlier versions of this paper.