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Articles 4201 - 4230 of 4292
Full-Text Articles in Operations Research, Systems Engineering and Industrial Engineering
Decentralized Resource Allocation And Scheduling Via Walrasian Auctions With Negotiable Agents, Huaxing Chen, Hoong Chuin Lau
Decentralized Resource Allocation And Scheduling Via Walrasian Auctions With Negotiable Agents, Huaxing Chen, Hoong Chuin Lau
Research Collection School Of Computing and Information Systems
This paper is concerned with solving decentralized resource allocation and scheduling problems via auctions with negotiable agents by allowing agents to switch their bid generation strategies within the auction process, such that a better system wide performance is achieved on average as compared to the conventional walrasian auction running with agents of fixed bid generation strategy. We propose a negotiation mechanism embedded in auctioneer to solicit bidders’ change of strategies in the process of auction. Finally we benchmark our approach against conventional auctions subject to the real-time large-scale dynamic resource coordination problem to demonstrate the effectiveness of our approach.
Distributed Route Planning And Scheduling Via Hybrid Conflict Resolution, Ramesh Thangarajoo, Hoong Chuin Lau
Distributed Route Planning And Scheduling Via Hybrid Conflict Resolution, Ramesh Thangarajoo, Hoong Chuin Lau
Research Collection School Of Computing and Information Systems
In this paper, we discuss the problem of route planning and scheduling by a group of agents. Each agent is responsible for designing a route plan and schedule over a geographical network, and the goal is to obtain a conflict-free plan/schedule that optimizes a global objective. We present a hybrid conflict resolution method that involves coalition formation and distributed constraint satisfaction depending on the level of coupling between agents. We show how this approach can be effectively applied to solve a distributed convoy movement planning problem.
The Bi-Objective Master Physician Scheduling Problem, Aldy Gunawan, Hoong Chuin Lau
The Bi-Objective Master Physician Scheduling Problem, Aldy Gunawan, Hoong Chuin Lau
Research Collection School Of Computing and Information Systems
Physician scheduling is the assignment of physicians to perform different duties in the hospital timetable. In this paper, the goals are to satisfy as many physicians’ preferences and duty requirements as possible while ensuring optimum usage of available resources. We present a mathematical programming model to represent the problem as a bi-objective optimization problem. Three different methods based on ε–Constraint Method, Weighted-Sum Method and HillClimbing algorithm are proposed. These methods were tested on a real case from the Surgery Department of a large local government hospital, as well as on randomly generated problem instances. The strengths and weaknesses of the …
A Decision Theoretic Approach To Data Leakage Prevention, Janusz Marecki, Mudhakar Srivastava, Pradeep Reddy Varakantham
A Decision Theoretic Approach To Data Leakage Prevention, Janusz Marecki, Mudhakar Srivastava, Pradeep Reddy Varakantham
Research Collection School Of Computing and Information Systems
In both the commercial and defense sectors a compelling need is emerging for rapid, yet secure, dissemination of information. In this paper we address the threat of information leakage that often accompanies such information flows. We focus on domains with one information source (sender) and many information sinks (recipients) where: (i) sharing is mutually beneficial for the sender and the recipients, (ii) leaking a shared information is beneficial to the recipients but undesirable to the sender, and (iii) information sharing decisions of the sender are determined using imperfect monitoring of the (un)intended information leakage by the recipients.We make two key …
Effect Of Human Biases On Human-Agent Teams, Praveen Paruchuri, Pradeep Reddy Varakantham, Katia Sycara, Paul Scerri
Effect Of Human Biases On Human-Agent Teams, Praveen Paruchuri, Pradeep Reddy Varakantham, Katia Sycara, Paul Scerri
Research Collection School Of Computing and Information Systems
As human-agent teams get increasingly deployed in the real-world, agent designers need to take into account that humans and agents have different abilities to specify preferences. In this paper, we focus on how human biases in specifying preferences for resources impacts the performance of large, heterogeneous teams. In particular, we model the inclination of humans to simplify their preference functions and to exaggerate their utility for desired resources, and show the effect of these biases on the team performance. We demonstrate this on two different problems, which are representative of many resource allocation problems addressed in literature. In both these …
Effective Heuristic Methods For Finding Non-Optimal Solutions Of Interest In Constrained Optimization Models, Steven O. Kimbrough, Ann Kuo, Hoong Chuin Lau
Effective Heuristic Methods For Finding Non-Optimal Solutions Of Interest In Constrained Optimization Models, Steven O. Kimbrough, Ann Kuo, Hoong Chuin Lau
Research Collection School Of Computing and Information Systems
This paper introduces the SoI problem, that of finding nonoptimal solutions of interest for constrained optimization models. SoI problems subsume finding FoIs (feasible solutions of interest), and IoIs (infeasible solutions of interest). In all cases, the interest addressed is post-solution analysis in one form or another. Post-solution analysis of a constrained optimization model occurs after the model has been solved and a good or optimal solution for it has been found. At this point, sensitivity analysis and other questions of import for decision making (discussed in the paper) come into play and for this purpose the SoIs can be of …
Anytime Planning For Decentralized Pomdps Using Expectation Maximization, Akshat Kumar, Shlomo Zilberstein
Anytime Planning For Decentralized Pomdps Using Expectation Maximization, Akshat Kumar, Shlomo Zilberstein
Research Collection School Of Computing and Information Systems
Decentralized POMDPs provide an expressive framework for multi-agent sequential decision making. While finite-horizon DECPOMDPs have enjoyed signifcant success, progress remains slow for the infinite-horizon case mainly due to the inherent complexity of optimizing stochastic controllers representing agent policies. We present a promising new class of algorithms for the infinite-horizon case, which recasts the optimization problem as inference in a mixture of DBNs. An attractive feature of this approach is the straightforward adoption of existing inference techniques in DBNs for solving DEC-POMDPs and supporting richer representations such as factored or continuous states and actions. We also derive the Expectation Maximization (EM) …
Towards Finding Robust Execution Strategies For Rcpsp/Max With Durational Uncertainty, Na Fu, Pradeep Varakantham, Hoong Chuin Lau
Towards Finding Robust Execution Strategies For Rcpsp/Max With Durational Uncertainty, Na Fu, Pradeep Varakantham, Hoong Chuin Lau
Research Collection School Of Computing and Information Systems
Resource Constrained Project Scheduling Problems with minimum and maximum time lags (RCPSP/max) have been studied extensively in the literature. However, the more realistic RCPSP/max problems — ones where durations of activities are not known with certainty – have received scant interest and hence are the main focus of the paper. Towards addressing the significant computational complexity involved in tackling RCPSP/max with durational uncertainty, we employ a local search mechanism to generate robust schedules. In this regard, we make two key contributions: (a) Introducing and studying the key properties of a new decision rule to specify start times of activities with …
Point-Based Backup For Decentralized Pompds: Complexity And New Algorithms, Akshat Kumar, Shlomo Zilberstein
Point-Based Backup For Decentralized Pompds: Complexity And New Algorithms, Akshat Kumar, Shlomo Zilberstein
Research Collection School Of Computing and Information Systems
Decentralized POMDPs provide an expressive framework for sequential multi-agent decision making. Despite their high complexity, there has been significant progress in scaling up existing algorithms, largely due to the use of point-based methods. Performing point-based backup is a fundamental operation in state-of-the-art algorithms. We show that even a single backup step in the multi-agent setting is NP-Complete. Despite this negative worst-case result, we present an efficient and scalable optimal algorithm as well as a principled approximation scheme. The optimal algorithm exploits recent advances in the weighted CSP literature to overcome the complexity of the backup operation. The polytime approximation scheme …
An Analysis Of Extreme Price Shocks And Illiquidity Among Trend Followers, Bernard Lee, Shih-Fen Cheng, Annie Koh
An Analysis Of Extreme Price Shocks And Illiquidity Among Trend Followers, Bernard Lee, Shih-Fen Cheng, Annie Koh
Research Collection School Of Computing and Information Systems
We construct an agent-based model to study the interplay between extreme price shocks and illiquidity in the presence of systematic traders known as trend followers. The agent-based approach is particularly attractive in modeling commodity markets because the approach allows for the explicit modeling of production, capacities, and storage constraints. Our study begins by using the price stream from a market simulation involving human participants and studies the behavior of various trend-following strategies, assuming initially that their participation will not impact the market. We notice an incremental deterioration in strategy performance as and when strategies deviate further and further from the …
Periodic Resource Reallocation In Two-Echelon Repairable Item Inventory Systems, Hoong Chuin Lau, Jie Pan, Huawei Song
Periodic Resource Reallocation In Two-Echelon Repairable Item Inventory Systems, Hoong Chuin Lau, Jie Pan, Huawei Song
Research Collection School Of Computing and Information Systems
Given an existing stock allocation in an inventory system, it is often necessary to perform reallocation over multiple time points to address inventory imbalance and maximize availability. In this paper, we focus on the situation where there are two opportunities to perform reallocation within a replenishment cycle. We derive a mathematical model to determine when and how to perform reallocation. Furthermore, we consider the extension of this model to the situation allowing an arbitrary number of reallocations. Experimental results show that the two-reallocation approach achieves better performance compared with the single-reallocation approach found in the literature. We also illustrate how …
Introducing Communication In Dis-Pomdps With Locality Of Interaction, Makoto Tasaki, Yuichi Yabu, Yuki Iwanari, Makoto Yokoo, Janusz Marecki, Pradeep Reddy Varakantham, Milind Tambe
Introducing Communication In Dis-Pomdps With Locality Of Interaction, Makoto Tasaki, Yuichi Yabu, Yuki Iwanari, Makoto Yokoo, Janusz Marecki, Pradeep Reddy Varakantham, Milind Tambe
Research Collection School Of Computing and Information Systems
The Networked Distributed POMDPs (ND-POMDPs) can model multiagent systems in uncertain domains and has begun to scale-up the number of agents. However, prior work in ND-POMDPs has failed to address communication. Without communication, the size of a local policy at each agent within the ND-POMDPs grows exponentially in the time horizon. To overcome this problem, we extend existing algorithms so that agents periodically communicate their observation and action histories with each other. After communication, agents can start from new synchronized belief state. Thus, we can avoid the exponential growth in the size of local policies at agents. Furthermore, we introduce …
A Service Choice Model For Optimizing Taxi Service Delivery, Shih-Fen Cheng, Xin Qu
A Service Choice Model For Optimizing Taxi Service Delivery, Shih-Fen Cheng, Xin Qu
Research Collection School Of Computing and Information Systems
Taxi service has undergone radical revamp in recent years. In particular, significant investments in communication system and GPS devices have improved quality of taxi services through better dispatches. In this paper, we propose to leverage on such infrastructure and build a service choice model that helps individual drivers in deciding whether to serve a specific taxi stand or not. We demonstrate the value of our model by applying it to a real-world scenario. We also highlight interesting new potential approaches that could significantly improve the quality of taxi services.
Sharing Mobile Multimedia Annotations To Support Inquiry-Based Learning Using Mobitop, Khasfariyati Razikin, Dion Hoe-Lian Goh, Yin-Leng Theng, Quang Minh Nguyen, Thi Nhu Quynh Kim, Ee Peng Lim, Chew-Hung Chang, Kalyani Chatterjea, Aixin Sun
Sharing Mobile Multimedia Annotations To Support Inquiry-Based Learning Using Mobitop, Khasfariyati Razikin, Dion Hoe-Lian Goh, Yin-Leng Theng, Quang Minh Nguyen, Thi Nhu Quynh Kim, Ee Peng Lim, Chew-Hung Chang, Kalyani Chatterjea, Aixin Sun
Research Collection School Of Computing and Information Systems
Mobile devices used in educational settings are usually employed within a collaborative learning activity in which learning takes place in the form of social interactions between team members while performing a shared task. We introduce MobiTOP (Mobile Tagging of Objects and People), a geospatial digital library system which allows users to contribute and share multimedia annotations via mobile devices. A key feature of MobiTOP that is well suited for collaborative learning is that annotations are hierarchical, allowing annotations to be annotated by other users to an arbitrary depth. A group of student-teachers involved in an inquiry-based learning activity in geography …
Font Size: Make Font Size Smaller Make Font Size Default Make Font Size Larger Exploiting Coordination Locales In Distributed Pomdps Via Social Model Shaping, Pradeep Varakantham, Jun Young Kwak, Matthew Taylor, Janusz Marecki, Paul Scerri, Milind Tambe
Font Size: Make Font Size Smaller Make Font Size Default Make Font Size Larger Exploiting Coordination Locales In Distributed Pomdps Via Social Model Shaping, Pradeep Varakantham, Jun Young Kwak, Matthew Taylor, Janusz Marecki, Paul Scerri, Milind Tambe
Research Collection School Of Computing and Information Systems
Distributed POMDPs provide an expressive framework for modeling multiagent collaboration problems, but NEXPComplete complexity hinders their scalability and application in real-world domains. This paper introduces a subclass of distributed POMDPs, and TREMOR, an algorithm to solve such distributed POMDPs. The primary novelty of TREMOR is that agents plan individually with a single agent POMDP solver and use social model shaping to implicitly coordinate with other agents. Experiments demonstrate that TREMOR can provide solutions orders of magnitude faster than existing algorithms while achieving comparable, or even superior, solution quality.
Setting Discrete Bid Levels Adaptively In Repeated Auctions, Jilian Zhang, Hoong Chuin Lau, Jialie Shen
Setting Discrete Bid Levels Adaptively In Repeated Auctions, Jilian Zhang, Hoong Chuin Lau, Jialie Shen
Research Collection School Of Computing and Information Systems
The success of an auction design often hinges on its ability to set parameters such as reserve price and bid levels that will maximize an objective function such as the auctioneer revenue. Works on designing adaptive auction mechanisms have emerged recently, and the challenge is in learning different auction parameters by observing the bidding in previous auctions. In this paper, we propose a non-parametric method for determining discrete bid levels dynamically so as to maximize the auctioneer revenue. First, we propose a non-parametric kernel method for estimating the probabilities of closing price with past auction data. Then a greedy strategy …
Multilayer Image Inpainting Approach Based On Neural Networks, Quan Wang, Zhaoxia Wang, Che Sau Chang, Ting Yang
Multilayer Image Inpainting Approach Based On Neural Networks, Quan Wang, Zhaoxia Wang, Che Sau Chang, Ting Yang
Research Collection School Of Computing and Information Systems
This paper describes an image inpainting approach based on the self-organizing map for dividing an image into several layers, assigning each damaged pixel to one layer, and then restoring these damaged pixels by the information of their respective layer. These inpainted layers are then fused together to provide the final inpainting results. This approach takes advantage of the neural network's ability of imitating human's brain to separate objects of an image into different layers for inpainting. The approach is promising as clearly demonstrated by the results in this paper.
An Agent-Based Commodity Trading Simulation, Shih-Fen Cheng, Yee Pin Lim
An Agent-Based Commodity Trading Simulation, Shih-Fen Cheng, Yee Pin Lim
Research Collection School Of Computing and Information Systems
In this paper, an event-centric commodity trading simulation powered by the multiagent framework is presented. The purpose of this simulation platform is for training novice traders. The simulation is progressed by announcing news events that affect various aspects of the commodity supply chain. Upon receiving these events, market agents that play the roles of producers, consumers, and speculators would adjust their views on the market and act accordingly. Their actions would be based on their roles and also their private information, and collectively they shape the market dynamics. This simulation has been effectively deployed for several training sessions. We will …
Distributed Constraint Optimization With Structured Resource Constraints, Akshat Kumar, Boi Faltings, Adrian Petcu
Distributed Constraint Optimization With Structured Resource Constraints, Akshat Kumar, Boi Faltings, Adrian Petcu
Research Collection School Of Computing and Information Systems
Distributed constraint optimization (DCOP) provides a framework for coordinated decision making by a team of agents. Often, during the decision making, capacity constraints on agents' resource consumption must be taken into account. To address such scenarios, an extension of DCOP-Resource Constrained DCOP - has been proposed. However, certain type of resources have an additional structure associated with them and exploiting it can result in more efficient algorithms than possible with a general framework. An example of these are distribution networks, where the flow of a commodity from sources to sinks is limited by the flow capacity of edges. We present …
An Agent-Based Commodity Trading Simulation, Shih-Fen Cheng, Yee Pin Lim, Chao-Chi Liu
An Agent-Based Commodity Trading Simulation, Shih-Fen Cheng, Yee Pin Lim, Chao-Chi Liu
Research Collection School Of Computing and Information Systems
In recent years, the study of trading in electronic markets has received significant amount of attention, particularly in the areas of artificial intelligence and electronic commerce. With increasingly sophisticated technologies being applied in analyzing information and making decisions, fully autonomous software agents are expected to take up significant roles in many important fields. This trend is most obvious in the financial domain, where speed of reaction is highly valued and significant investments have been made in information and communication technologies.Despite the successes of automated trading in many important classes of financial markets, commodity trading has lagged behind, mainly because of …
Constraint-Based Dynamic Programming For Decentralized Pomdps With Structured Interactions, Akshat Kumar, Shlomo Zilberstein
Constraint-Based Dynamic Programming For Decentralized Pomdps With Structured Interactions, Akshat Kumar, Shlomo Zilberstein
Research Collection School Of Computing and Information Systems
Decentralized partially observable MDPs (DEC-POMDPs) provide a rich framework for modeling decision making by a team of agents. Despite rapid progress in this area, the limited scalability of solution techniques has restricted the applicability of the model. To overcome this computational barrier, research has focused on restricted classes of DEC-POMDPs, which are easier to solve yet rich enough to capture many practical problems. We present CBDP, an efficient and scalable point-based dynamic programming algorithm for one such model called ND-POMDP (Network Distributed POMDP). Specifically, CBDP provides magnitudes of speedup in the policy computation and generates better quality solution for all …
Dynamic Programming Approximations For Partially Observable Stochastic Games, Akshat Kumar, Shlomo Zilberstein
Dynamic Programming Approximations For Partially Observable Stochastic Games, Akshat Kumar, Shlomo Zilberstein
Research Collection School Of Computing and Information Systems
Partially observable stochastic games (POSGs) provide a rich mathematical framework for planning under uncertainty by a group of agents. However, this modeling advantage comes with a price, namely a high computational cost. Solving POSGs optimally quickly becomes intractable after a few decision cycles. Our main contribution is to provide bounded approximation techniques, which enable us to scale POSG algorithms by several orders of magnitude. We study both the POSG model and its cooperative counterpart, DEC-POMDP. Experiments on a number of problems confirm the scalability of our approach while still providing useful policies.
Optimizing Service Systems Based On Application-Level Qos, Qianhui Liang, Xindong Wu, Hoong Chuin Lau
Optimizing Service Systems Based On Application-Level Qos, Qianhui Liang, Xindong Wu, Hoong Chuin Lau
Research Collection School Of Computing and Information Systems
Making software systems service-oriented is becoming the practice, and an increasingly large number of service systems play important roles in today's business and industry. Currently, not enough attention has been paid to the issue of optimization of service systems. In this paper, we argue that the key elements to be considered in optimizing service systems are robustness, system orientation, and being dynamic and transparent. We present our solution to optimizing service systems based on application-level QoS management. Our solution incorporates three capabilities, i.e., 1) the ability to cater to the varying rigidities on Web service QoS in distinct application domains …
Ontology-Based Business Process Customization For Composite Web Services, Qianhui (Althea) Liang, Xindong Wu, E. K. Park, T. Khoshgoftaar, C. Chi
Ontology-Based Business Process Customization For Composite Web Services, Qianhui (Althea) Liang, Xindong Wu, E. K. Park, T. Khoshgoftaar, C. Chi
Research Collection School Of Computing and Information Systems
A key goal of the Semantic Web is to shift social interaction patterns from a producer-centric paradigm to a consumer-centric one. Treating customers as the most valuable assets and making the business models work better for them are at the core of building successful consumer-centric business models. It follows that customizing business processes constitutes a major concern in the realm of a knowledge-pull-based human semantic Web. This paper conceptualizes the customization of service-based business processes leveraging the existing knowledge of Web services and business processes. We represent this conceptualization as a new Extensible Markup Language (XML) markup language Web Ontology …
The Price Of Stability In Selfish Scheduling Games, Lucas Agussurja, Hoong Chuin Lau
The Price Of Stability In Selfish Scheduling Games, Lucas Agussurja, Hoong Chuin Lau
Research Collection School Of Computing and Information Systems
Game theory has gained popularity as an approach to analysing and understanding distributed systems with self-interested agents. Central to game theory is the concept of Nash equilibrium as a stable state (solution) of the system, which comes with a price − the loss in efficiency. The quantification of the efficiency loss is one of the main research concerns. In this paper, we study the quality and computational characteristics of the best Nash equilibrium in two selfish scheduling models: the congestion model and the sequencing model. In particular, we present the following results: (1) In the congestion model: first, the best …
Integrated Resource Allocation And Scheduling In Bidirectional Flow Shop With Multi-Machine And Cos Constraints, Hoong Chuin Lau, Zhengyi Zhao, Shuzhi Sam Ge
Integrated Resource Allocation And Scheduling In Bidirectional Flow Shop With Multi-Machine And Cos Constraints, Hoong Chuin Lau, Zhengyi Zhao, Shuzhi Sam Ge
Research Collection School Of Computing and Information Systems
An integer programming (IP) model is proposed for integrated resource allocation and operation scheduling for a multiple job-agents system. Each agent handles a specific job-list in a bidirectional flowshop. For the individual agent scheduling problem, a formulation is proposed in continuous time domain and compared with an IP formulation in discrete time domain. Of particular interest is the formulation of the machine utilization function-- both in continuous time and discrete time. Fast heuristic methods are proposed with the relaxation of the machine capacity. For the integrated resource allocation and scheduling problem, a linear programming relaxation approach is applied to solve …
Event-Detecting Multi-Agent Mdps: Complexity And Constant-Factor Approximation, Akshat Kumar, S. Zilberstein
Event-Detecting Multi-Agent Mdps: Complexity And Constant-Factor Approximation, Akshat Kumar, S. Zilberstein
Research Collection School Of Computing and Information Systems
Planning under uncertainty for multiple agents has grown rapidly with the development of formal models such as multi-agent MDPs and decentralized MDPs. But despite their richness, the applicability of these models remains limited due to their computational complexity. We present the class of event-detecting multi-agent MDPs (eMMDPs), designed to detect multiple mobile targets by a team of sensor agents. We show that eMMDPs are NP-Hard and present a scalable 2-approximation algorithm for solving them using matroid theory and constraint optimization. The complexity of the algorithm is linear in the state-space and number of agents, quadratic in the horizon, and exponential …
Distributing Complementary Resources Across Multiple Periods With Stochastic Demand, Shih-Fen Cheng, John Tajan, Hoong Chuin Lau
Distributing Complementary Resources Across Multiple Periods With Stochastic Demand, Shih-Fen Cheng, John Tajan, Hoong Chuin Lau
Research Collection School Of Computing and Information Systems
In this paper, we evaluate whether the robustness of a market mechanism that allocates complementary resources could be improved through the aggregation of time periods in which resources are consumed. In particular, we study a multi-round combinatorial auction that is built on a general equilibrium framework. We adopt the general equilibrium framework and the particular combinatorial auction design from the literature, and we investigate the benefits and the limitation of time-period aggregation when demand-side uncertainties are introduced. By using simulation experiments, we show that under stochastic conditions the performance variation of the process decreases as the time frame length (time …
Generating Robust Schedules Subject To Resource And Duration Uncertainties, Na Fu, Hoong Chuin Lau, Fei Xiao
Generating Robust Schedules Subject To Resource And Duration Uncertainties, Na Fu, Hoong Chuin Lau, Fei Xiao
Research Collection School Of Computing and Information Systems
We consider the Resource-Constrained Project Scheduling Problem with minimal and maximal time lags under resource and duration uncertainties. To manage resource uncertainties, we build upon the work of Lambrechts et al 2007 and develop a method to analyze the effect of resource breakdowns on activity durations. We then extend the robust local search framework of Lau et al 2007 with additional considerations on the impact of unexpected resource breakdowns to the project makespan, so that partial order schedules (POS) can absorb both resource and duration uncertainties. Experiments show that our proposed model is capable of addressing the uncertainty of resources, …
A Heuristic Method For Job-Shop Scheduling With An Infinite Wait Buffer: From One-Machine To Multi-Machine Problems, Z. J. Zhao, J. Kim, M. Luo, Hoong Chuin Lau, S. S. Ge
A Heuristic Method For Job-Shop Scheduling With An Infinite Wait Buffer: From One-Machine To Multi-Machine Problems, Z. J. Zhao, J. Kim, M. Luo, Hoong Chuin Lau, S. S. Ge
Research Collection School Of Computing and Information Systems
Through empirical comparison of classical job shop problems (JSP) with multi-machine consideration, we find that the objective to minimize the sum of weighted tardiness has a better wait property compared with the objective to minimize the makespan. Further, we test the proposed Iterative Minimization Micro-model (IMM) heuristic method with the mixed integer programming (MIP) solution by CPLEX. For multi-machine problems, the IMM heuristic method is faster and achieves a better solution. Finally, for a large problem instance with 409 jobs and 30 types of machines, IMM-heuristic method is compared with ProModel and we find that the heuristic method is slightly …