A robust optimization model for multi-product two-stage capacitated production planning under uncertainty

Production planning (PP) is one of the most important issues carried out in manufacturing environments which seeks efficient planning, scheduling and coordination of all production activities that optimizes the company’s objectives. In this paper, we studied a two-stage real world capacitated production system with lead time and setup decisions in which some parameters such as production costs and customer demand are uncertain. A robust optimization model is developed to formulate the problem in which minimization of the total costs including the setup costs, production costs, labor costs, inventory costs, and workforce changing costs is considered as performance measure. The robust approach is used to reduce the effects of fluctuations of the uncertain parameters with regards to all the possible future scenarios. A mixed-integer programming (MIP) model is developed to formulate the related robust production planning problem. In fact the robust proposed model is presented to generate an initial robust schedule. The performance of this schedule could be improved against of any possible occurrences of uncertain parameters. A case from an Iran refrigerator factory is studied and the characteristics of factory and its products are discussed. The computational results display the robustness and effectiveness of the model and highlight the importance of using robust optimization approach in generating more robust production plans in the uncertain environments. The tradeoff between solution robustness and model robustness is also analyzed.     

Supplier–manufacturer coordination in capacitated two-stage supply chains

Manufacture-to-order is an increasingly popular strategy in commodity electronics and other similar markets where many different product configurations can be produced from common components. To succeed in this environment, manufacturers need to keep both cost and order fulfillment time low. In this article, we compare three different mechanisms that a manufacturer, whose revenues depend on order delays, may use to affect its component supplier’s inventory decisions. These mechanisms are specifying components inventory level, offering a share of the earned revenues to the supplier (called simple revenue sharing), and offering a two-part revenue-sharing scheme. We show that whereas the first two approaches do not lead to supply chain coordination, the two-part scheme does. We demonstrate with numerical experiments that up to a point, the component supplier benefits from having a high utilization of its production facility, whereas the manufacturer benefits from having excess production capacity.     

Manufacturing costs in serial production systems with rework

In this paper, we present a Markovian modelling framework that can describe any serial production system with rework. Each production stage is represented by a state in the Markov chain. Absorbing states indicate the events of scrapping a product at a production stage or the completion of the finished product. Generalizable formulae for the final absorption probabilities are derived that represent: (1) the probability that an unfinished product is scrapped at a certain production stage and (2) the yield of the system. We also derive various expected costs and quantities associated with all products ending in any absorbing state, as well as the equivalent costs and quantities for finished products. The applicability of our modelling framework is demonstrated in a real-life manufacturing environment in the food-packing industry.     

Coordination in a two-stage capacitated supply chain with multiple suppliers

The aim of this paper is to coordinate the inventory policies in a decentralized supply chain with stochastic demand by means of contracts. The system considered is a decentralized two-stage supply chain consisting of multiple independent suppliers and a manufacturer with limited production capacities. The suppliers operate on a make-to-stock basis and apply base stock policy to manage their inventories. On the other hand, the manufacturer employs a make-to-order strategy. Under the necessary assumptions, each supplier is modeled as an M/M/1 make-to-stock queue; and the manufacturer is modeled as a GI/M/1 queue after deriving an approximate distribution for the interarrival times of the manufacturer. Once the supply chain is modeled as a queuing system, centralized and decentralized models are developed. Comparison of the optimal solutions to these models reveals that the supply chain needs coordination. Three different transfer payment contracts are examined in this paper. These are the backorder and holding cost subsidy contracts, the transfer payment contract based on Pareto improvement, and the cost sharing contract. Each contract is evaluated according to its coordination ability and whether it is Pareto improving or not. The results indicate that all three contracts can coordinate the supply chain. However, when the Pareto improvement is taken into account, the cost sharing contract seems to be the one that will be preferred by all parties.     

An LP-based heuristic for two-stage capacitated facility location problems

In this paper, a linear programming based heuristic is considered for a two-stage capacitated facility location problem with single source constraints. The problem is to find the optimal locations of depots from a set of possible depot sites in order to serve customers with a given demand, the optimal assignments of customers to depots and the optimal product flow from plants to depots. Good lower and upper bounds can be obtained for this problem in short computation times by adopting a linear programming approach. To this end, the LP formulation is iteratively refined using valid inequalities and facets which have been described in the literature for various relaxations of the problem. After each reoptimisation step, that is the recalculation of the LP solution after the addition of valid inequalities, feasible solutions are obtained from the current LP solution by applying simple heuristics. The results of extensive computational experiments are given.     

Minimax production planning in failure-prone manufacturing systems

In this paper, we consider a minimax production planning model of a flexible manufacturing system with machines that are subject to random breakdown and repair. The objective is to choose the rate of production that minimizes the related minimax cost of production and inventory/shortage. The value function is shown to be the unique viscosity solution to the associated Hamilton-Jacobi-Isaacs equation. Under certain conditions, it is shown that the value function is continuously differentiable. A verification theorem is given to provide a sufficient condition for optimal control. Finally, two examples are solved explicitly.This research was supported by the Natural Sciences and Engineering Research Council of Canada under Grants OGP0036444 and A4169.     

Multi-period efficiency and Malmquist productivity index in two-stage production systems

Conventional two-stage data envelopment analysis (DEA) models measure the overall performance of a production system composed of two stages (processes) in a specified period of time, where variations in different periods are ignored. This paper takes the operations of individual periods into account to develop a multi-period two-stage DEA model, which is able to measure the overall and period efficiencies at the same time, with the former expressed as a weighted average of the latter. Since the efficiency of a two-stage system in a period is the product of the two process efficiencies, the overall efficiency of a decision making unit (DMU) in the specified period of time can be decomposed into the process efficiency of each period. Based on this decomposition, the sources of inefficiency in a DMU can be identified. The efficiencies measured from the model can also be used to calculate a common-weight global Malmquist productivity index (MPI) between two periods, in that the overall MPI is the product of the two process MPIs. The non-life insurance industry in Taiwan is used to verify the proposed model, and to explain why some companies performed unsatisfactorily in the specified period of time.     

Decomposition of technical and scale efficiencies in two-stage production systems

Conventional data envelopment analysis (DEA) models are used to measure the technical and scale efficiencies of a system when it is considered as a whole unit. This paper extends the efficiency measurement to two-stage systems where each stage has one process and all the outputs from the first process become the inputs of the second. An input-oriented DEA model for the first process is developed to separate the process efficiency into the input technical and scale efficiencies, and an output-oriented model is developed for the second process to separate the process efficiency into the output technical and scale efficiencies. Combining the two models, the system efficiency is expressed as the product of the overall technical and scale efficiencies, where the overall technical and scale efficiencies are the products of the corresponding efficiencies of the two processes, respectively. The detailed decomposition allows the sources of inefficiency to be identified.     

Variational approach to serial multilevel production/inventory systems

We discuss stochastic, continuous-time models of serial multilevel production/inventory systems using quasi-variational inequalities. The value functions are characterized as maximum solutions of suitable equivalent problems. Then, we show how to construct optimal centralized policies and, if possible, decentralized policies. Finally, we present some numerical examples for the deterministic case.The work of the second author was supported by the National Science Foundation under Grant No. DMS-87-02236.     

Look-back policies for two-stage,pull-type production/inventory systems

We consider a two-stage, pull-type production/inventory system with a known service mechanism at the first stage. Set-ups and start-ups are involved in the operation of the second stage. We develop a production control policy for the second stage, within the class of (R, r) continuous-review policies, that minimizes the long run average total cost. We use a semi-Markov decision model to obtain an optimal policy for the operation of the second stage. The structure of the optimal policy suggests the use of a suboptimal look-back policy that delays the set-up at the second stage if the buffer lacks sufficient raw material. The performance of the system and the average total cost under the suboptimal policy can be obtained approximately using a decomposition algorithm. We show examples justifying the use of this suboptimal policy.This research is supported by the NSF Grant No. NSF-NCR-9110105, NSF Grant No. NSF-DDM-9014868 and by the North Atlantic Treaty Organization Grant No. NATO-CRG-900580.     

Structuring production planning systems for computer applications

Activities in a job shop type mechanical company can be split in production, product cycle and production cycle. The corresponding flow in each of these are materials, manufacturing specifications and product requirements. Production planning will plan and control these flows. Basic data structures are a product model and a corresponding hierarchy linked to production resources. A planning system may be designed by combining a set of operation or building blocks. These are three types: user communication, data base management and basic operations. The basic operations cover all processing necessary in production planning and represent the fundamental building blocks. A list of basic operations is suggested.     

Operations planning test bed under rolling horizons,multiproduct, multiechelon,multiprocess for capacitated production planning modelling with strokes

One of the problems when conducting research in mathematical programming models for operations planning is having an adequate database of experiments that can be used to verify advances and developments with enough factors to understand different consequences. This paper presents a test bed generator and instances database for a rolling horizons analysis for multiechelon planning, multiproduct with alternatives processes, multistroke, multicapacity with different stochastic demand patterns to be used with a stroke-like bill of materials considering production costs, setup, storage and delays for operations management. From the analysis of the operations planning obtained from this test bed, it is concluded that a product structure with an alternative process obtains the lowest total cost and the highest service level. In addition, decreasing seasonal demand could present a lower total cost than constant demand, but would generate a worse service level. This test bed will allow researchers further investigation so as to verify improvements in forecast methods, rolling horizons parameters, employed software, etc.

     

Plant location and procurement planning in knockdown production systems

We consider an integrated problem of plant location and capacity planning for components procurement in knockdown production systems. The problem is that of determining the schedule of opening components manufacturing plants, plans for acquisition of capacities in opened components manufacturing plants, and plans for components procurement in final assembly plants with the objective of minimizing the sum of fixed costs for opening plants, acquisition and operation costs of facilities, and delivery and subcontracting costs of components. The problem is formulated as a mixed integer linear program and solved by a two-stage solution procedure. In the solution procedure, the problem is decomposed into two tractable subproblems and these subproblems are solved sequentially. In the first stage, a dynamic plant location problem is solved using a cut and branch algorithm based on Gomory cuts, while a multiperiod capacity planning problem is solved in the second stage by a heuristic algorithm that uses a cut and branch algorithm and a variable reduction scheme. The solution procedure is tested on problems of a practical size and results show that the procedure gives reasonably good solutions.     

Multi-location transshipment problem with capacitated production

We consider coordination among stocking locations through replenishment strategies that explicitly take into account lateral transshipments, i.e., transfer of a product among locations at the same echelon level. Our basic contribution is the incorporation of supply capacity into the traditional transshipment model. Our goal is to analyze the impact on system behavior and on stocking locations’ performance of the fact that the supplier may fail to fulfill all the replenishment orders. We therefore formulate the capacitated supply scenario as a network flow problem embedded in a stochastic optimization problem, which is solved through a sample average approximation method. We find that, depending on the production capacity, system behavior can vary drastically. Moreover, in a production-inventory system, we find evidence that either capacity flexibility (i.e., extra production) or transshipment flexibility (i.e., pooling) is required to maintain a desired level of service.     

Decision architectures for production planning in multi-stage multi-product manufacturing systems

Recent trends in automated manufacturing call for hierarchical decision architectures for production planning, suitable for integration with part flow controls. Different design approaches are currently adopted for implementing production planning architectures, depending either on the objective of defining a centralized production plan for the whole manufacturing system (as in the case of MRP and OPT), or on the desire of coordinating local plans for the component work cells (as for JIT). The paper analyzes such approaches by use of a unifying mathematical formulation of the production plan optimization problem, to recognize the main features of the existing planning approaches, and compare their usefulness in different manufacturing processes.     

Solving the robust two-stage capacitated facility location problem with uncertain transportation costs

In this study, we start from a multi-source variant of the two-stage capacitated facility location problem (TSCFLP) and propose a robust optimization model of the problem that involves the uncertainty of transportation costs. Since large dimensions of the robust TSCFLP could not be solved to optimality, we design a memetic algorithm (MA), which represents a combination of an evolutionary algorithm (EA) and a modified simulated annealing heuristic (SA) that uses a short-term memory of undesirable moves from previous iterations. A set of computational experiments is conducted to examine the impact of different protection levels on the deviation of the objective function value. We also investigate the impact of variations of transportation costs that may occur on both transhipment stages on the total cost for a fixed protection level. The obtained results may help in identifying a sustainable and efficient strategy for designing a two stage capacitated transportation network with uncertain transportation costs, and may be applicable in the design and management of similar transportation networks.     

Managing production with flexible capacity deployment for serial multi-stage manufacturing systems

This paper presents a model for serial multi-stage manufacturing systems facing variability from two sources. One source is demand uncertainty; the other is manufacturing uncertainty associated with all manufacturing stages. A production control policy based on the planned lead time and the manufacturing capacity requirement is developed. It is shown that this production control policy has the effect of reducing the variance of production output for all manufacturing stages. Some specific analyses are provided to illustrate the production control policy developed. The model developed provides a vehicle for examining the interrelationships among the production output, the planned lead time and the actual manufacturing flow time. The risk-pooling value over both demand randomness and manufacturing uncertainty, which is achieved through consolidating some manufacturing capacity and deploying flexible capacity among the manufacturing stages, is analyzed. This risk-pooling value can be realized in the form of either reduced manufacturing flow time or increased effective capacity to meet more demand. It is shown that the risk-pooling value increases as the planned lead time decreases.     

A two-stage fuzzy robust integer programming approach for capacity planning of environmental management systems

In this study, a two-stage fuzzy robust integer programming (TFRIP) method has been developed for planning environmental management systems under uncertainty. This approach integrates techniques of robust programming and two-stage stochastic programming within a mixed integer linear programming framework. It can facilitate dynamic analysis of capacity-expansion planning for waste management facilities within a multi-stage context. In the modeling formulation, uncertainties can be presented in terms of both possibilistic and probabilistic distributions, such that robustness of the optimization process could be enhanced. In its solution process, the fuzzy decision space is delimited into a more robust one by specifying the uncertainties through dimensional enlargement of the original fuzzy constraints. The TFRIP method is applied to a case study of long-term waste-management planning under uncertainty. The generated solutions for continuous and binary variables can provide desired waste-flow-allocation and capacity-expansion plans with a minimized system cost and a maximized system feasibility.     

A stochastic production planning problem in hybrid manufacturing and remanufacturing systems with resource capacity planning

DIRECT is derivative-free global-search algorithm has been found to perform robustly across a wide variety of low-dimensional test problems. The reason DIRECT’s robustness is its lack of algorithmic parameters that need be “tuned” to make the algorithm perform well. In particular, there is no parameter that determines the relative emphasis on global versus local search. Unfortunately, the same algorithmic features that enable DIRECT to perform so robustly have a negative side effect. In particular, DIRECT is usually quick to get close to the global minimum, but very slow to refine the solution to high accuracy. This is what we call DIRECT’s “eventually inefficient behavior.” In this paper, we outline two root causes for this undesirable behavior and propose modifications to eliminate it. The paper builds upon our previously published “MrDIRECT” algorithm, which we can now show only addressed the first root cause of the “eventually inefficient behavior.” The key contribution of the current paper is a further enhancement that allows MrDIRECT to address the second root cause as well. To demonstrate the effectiveness of the enhanced MrDIRECT, we have identified a set of test functions that highlight different situations in which DIRECT has convergence issues. Extensive numerical work with this test suite demonstrates that the enhanced version of MrDIRECT does indeed improve the convergence rate of DIRECT.