A Scenario-Based Objective Function for an M/M/K Queuing Model with Priority (A Case Study in the Gear Box Production Factory) H. R. Rezaei, A. Asgari, H. Mogouei, Amirhossein Amiri Abstract— In most optimization problems of an M/M/K queuing model with priority, the scenarios are defined and optimized separately. However, sometimes, the scenarios are dependent and should be optimized simultaneously. In this paper, an efficient approach is developed for this purpose. The performance of the proposed approach is evaluated through a real case study in a ware house of a Gear Box production factory. Key words— Queuing model, Optimization, M/M/K with priority I. INTRODUCTION HE case study investigated in this paper, is related to Gear Box production company which has a high market share of Gear Box in Iran. In this company, the queuing system for the final product warehouse is studied to determine the optimum number of docks. Now, there exists three docks available and each dock requires two operators for loading trucks. This company has two types of customers. The main customer makes up 40% of the cash flow and is considered as the most important customer among the others. The other 60% cash flow is distributed among four less priority customers. The cash flow shares and the priority of all customers are summarized in TABLE 1. If the customers are waited in the queues, the company should pay a penalty cost to customers. The top manager of the company has decided to decrease the waiting time as much as possible. As faster as the company can send products to the customer, the cash flow would increase. Hence, less wait in queue time is required even if the top Manuscript received March 7, 2011; revised April 09, 2011 H. R. Rezaei is with Industrial Engineering Department, Shahed University, Tehran, Iran (Corresponding author; phone: (+9821) 66794959; e-mail:hamidreza.rezaei21@gmail.com) A. Asgari is with Industrial Engineering Department, Shahed University, Tehran, Iran (e-mail: aliasgari_1987@yahoo.com). H. Mogouei is with Industrial Engineering Department, Shahed University, Tehran, Iran (e-mail: mogouie@ shahed.ac.ir) Amirhossein Amiri is with Industrial Engineering Department, Shahed University, Tehran, Iran (e-mail: amiri@shahed.ac.ir) manager has to omit a low priority customer in future contracts. TABLE 1 Customers Information Costumer No. Demand Share Priority 1 40% 1 2 21% 2 3 16% 3 4 12% 4 5 11% 5 As mentioned above, top manager policy will accept to eliminate one of the customers among customers number 2 to 5 from future contract list. The way that queuing theory models have been used in practice is categorized in two major groups. The first group contains the cases where queuing theory models have been used to obtain the value of queuing system evaluation indices such as  ௤ and ݓ ௤ for a few decision scenarios and choosing the best optimal solution. In more advanced cases an objective function is considered in which decision variables are  ௤ or ݓ ௤ and the value of this function is calculated under any of scenarios S q l and S q w and finally, the best alternative among available scenarios are selected. The application of the first category was optimization of Berth Crane operations in a maritime terminal [1]. Also a few other researches have used this approach in various cases to improve queuing systems. This approach has shown a good efficiency in optimization of complex queuing networks [2]. By combining queuing theories with scheduling knowledge, optimizing the loading and unloading operations in container terminal is considered in [3]. Another example is the usage of this approach for selecting the best configuration for a manufacturing system in which queuing networks existed [4]. The other category that can be called optimization based application of queuing theory models focuses on defining an objective function in which the decision variables are  ௤ or ݓ ௤ and the cost or benefit function are computed based on  ௤ or ݓ ௤ . For the second category, an optimization procedure for finding the optimal value of N in an N policy M/ ܧ ௄ /1 queuing system with a removable service station is considered in [5]. A recursive method to control an M/G/1queuing system is proposed by [6]. The optimal policy of an M/ k E /1 while the service station was removable is given in [7]. An M/G/1 queuing model was studied in [8] to determine the optimal control of the queuing model while the capacity of the queue was definite. T Proceedings of the World Congress on Engineering 2011 Vol I WCE 2011, July 6 - 8, 2011, London, U.K. ISBN: 978-988-18210-6-5 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online) WCE 2011