# American Institute of Mathematical Sciences

ISSN:
1547-5816

eISSN:
1553-166X

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## Journal of Industrial & Management Optimization

2010 , Volume 6 , Issue 4

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2010, 6(4): 709-727 doi: 10.3934/jimo.2010.6.709 +[Abstract](239) +[PDF](367.6KB)
Abstract:
Routine immunization is the most effective public health strategy to prevent the occurrence and spread of infectious diseases. An important factor impacting such effectiveness is the availability of a stable vaccine supply. Vaccine supply interruptions are likely and can prevent children from receiving a full course of vaccinations and hence, increase the risk of disease outbreaks. In the United States, public health officials have established the Pediatric Vaccine Stockpile Program (PVSP) as the best strategy to mitigate the impact of vaccine supply interruptions. The PVSP aims to maintain a six-month national supply of routinely administered pediatric vaccines. When deciding how many vaccine doses to order for the next fiscal year, public health decision-makers must not only minimize the impact of potential vaccine shortages, but also, maintain or increase vaccine coverage rates while minimizing costs. This paper uses the relative importance of each pediatric vaccine to define a multi-attribute utility function that models the conflicting interests of PVSP public health decision-makers when ordering vaccines for the next fiscal year. The aim of the resulting framework is to assist decision-makers in managing the PVSP. As an illustration, this paper explores the implications of optimizing a proposed expected utility function under budget constraints for hypothetical (albeit likely) vaccine supply scenarios, and the potential behavior of public health decision-makers.
2010, 6(4): 729-750 doi: 10.3934/jimo.2010.6.729 +[Abstract](196) +[PDF](277.5KB)
Abstract:
We study the quality ranking of different information structures. In a unified setting, we present eight representations of essentially the same notion that may arise in distinct literatures, and also establish the intricate relationships among them. The forms that are less familiar to researchers turn out to be more relevant to the operations management context. We also relate the information structure ranking to concepts often invoked in operations management literature, such as advance demand information and decision postponement. Using some of the established relations, we argue for building a cross-docking facility as close as possible to the distribution centers it serves.
2010, 6(4): 751-760 doi: 10.3934/jimo.2010.6.751 +[Abstract](185) +[PDF](158.5KB)
Abstract:
We consider uncertainty Euclidean facility location problems. Using the existing robust optimization methodology, we certainly obtain robust optimal solution of the Euclidean facility location problem with unknown-but-bounded uncertainty or with an ellipsoidal uncertainty by solving an SOCP or an SDP. In addition, we show that the robust counterpart of the Euclidean facility location problem with $\cap$-ellipsoidal uncertainty is NP-hard. We give an explicit SDP to approximate the NP-hard problem and estimate the quality of the approximation via the level of conservativeness.
2010, 6(4): 761-777 doi: 10.3934/jimo.2010.6.761 +[Abstract](274) +[PDF](211.0KB)
Abstract:
We consider the optimal control problem with dividend payments and issuance of equity in a dual risk model. Such a model might be appropriate for a company that specializes in inventions and discoveries, which pays costs continuously and has occasional profits. Assuming proportional transaction costs, we aim at finding optimal strategy which maximizes the expected present value of the dividends payout minus the discounted costs of issuing new equity before bankruptcy. By adopting some of the techniques and methodologies in L$\phi$kka and Zervos (2008), we construct two categories of suboptimal models, one is the ordinary dual model without issuance of equity, the other one assumes that, by issuing new equity, the company never goes bankrupt. We identify the value functions and the optimal strategies corresponding to the suboptimal models in two different cases. For exponentially distributed jump sizes, closed-form solutions are obtained.
2010, 6(4): 779-793 doi: 10.3934/jimo.2010.6.779 +[Abstract](170) +[PDF](182.4KB)
Abstract:
An extended canonical dual approach for solving 0-1 quadratic programming problems is introduced. We derive the relationship between the optimal solutions to the extended canonical dual problem and the original problem and prove that there exists no duality gap in-between. The extended canonical dual approach leads to a sufficient condition for global optimality, which is more general than known results of this kind. To solve the extended canonical dual problem, we construct corresponding conic programming problems and study their relationship to the extended canonical dual problem. Using this relationship, we design an algorithm for solving the extended canonical dual problem. Our work extends the known solvable sub-class of 0-1 quadratic programming problems.
2010, 6(4): 795-810 doi: 10.3934/jimo.2010.6.795 +[Abstract](209) +[PDF](191.1KB)
Abstract:
This paper addresses the impact of different contracts on supply chains with different suppliers and a common retailer. Especially, we consider a supply chain system consists of a retailer and two suppliers (supplier1 and supplier 2 and the supply chain consisting of supplier $i$ ($i = 1,2)$and the retailer is named supply chain $i$ ($i = 1,2)$for simplicity) in which supplier 1 and the retailer agree to simple wholesale price contract and supplier 2 and the retailer agree to revenue sharing contract. The products provided by these two suppliers are complement or substitution. We study the external effect of these two contracts. The main findings of this study are as follows: When the two products are not independent, the design of a coordinative contract depends the stocking quantity of another product; Under the same external conditions, both simple wholesale price contract and revenue sharing contract perform the same when the products are complement, and revenue sharing contract performs better than simple wholesale price contract when the products are substitutive or independent; we can also design a wholesale price contract and a revenue sharing contract which coordinate these two supply chains simultaneously when the products are complement. Moreover, we consider the case that the demand depends on costly retailer's effort. We show that wholesale price contract always distorts the retailer's decision on effort in supply chain 1 and we can design a revenue sharing contract coordinating supply chain 2. Whether wholesale price contract or revenue sharing contract is more incentive to encourage retailer's promotion effort depends on the problem and contract parameters. These findings are different from the well known properties of these two contracts in a supply chain consisting of 1-supplier and 1-retailer. As retailers always sell multi-items in reality, we believe that our findings could help managers manage channels efficiently.
2010, 6(4): 811-823 doi: 10.3934/jimo.2010.6.811 +[Abstract](242) +[PDF](238.5KB)
Abstract:
In this paper, we are interested in optimizing a linear function on the set of efficient solutions of a Multiple Objective Integer Linear Programming problem ($MOILP$). We propose an exact algorithm for maximizing a linear function denoted $\phi$ on the set of efficient solutions of a $MOILP$ problem without having to enumerate explicitly all the elements of this set. Two techniques are used: the first is to reduce progressively the admissible domain by adding more constraints eliminating all the dominated points by the current solution; the second, when the new solution obtained by maximizing the function $\phi$ in the reduced area is not efficient, an exploration procedure is applied over the edges incident to this solution in order to find new alternative efficient solutions if they exist. The algorithm produces not only an optimal value of the linear function but also a subset of non-dominated solutions in the direction of $\phi$ that can be helpful in the practice.
2010, 6(4): 825-846 doi: 10.3934/jimo.2010.6.825 +[Abstract](232) +[PDF](226.2KB)
Abstract:
In this paper, we present a geometrical exterior climbing method based on inclusive normal cones for solving general linear programming problems in canonical form. The method iteratively updates the inclusive cone by climbing within its associated inclusive region (also called a ladder), and eventually reaches an optimal solution. This method allows the development of a class of 'ladder algorithms' by using different climbing schemes. Some aspects of the current method are intrinsically related to the dual simplex method. However, it originates from different ideas and provides a new angle to look at the linear programming problem. It can be shown that the dual simplex algorithms are special ladder algorithms in this context. We present two climbing schemes leading to two finitely convergent ladder algorithms. The algorithms are tested by solving a number of linear programming examples. Some numerical results are provided.
2010, 6(4): 847-860 doi: 10.3934/jimo.2010.6.847 +[Abstract](245) +[PDF](185.6KB)
Abstract:
In this paper, we study a two-person knapsack game. Two investors, each with an individual budget, bid on a common pool of potential projects. To undertake a project, investors have their own cost estimation to be charged against their budgets. Associated with each project, there is a potential market profit that can be taken by the only bidder or shared proportionally by both bidders. The objective function of each investor is assumed to be a linear combination of the two bidders' profits. Both investors act in a selfish manner with best-response to optimize their own objective functions by choosing portfolios under the budget restriction. We show that pure Nash equilibrium exists under certain conditions. In this case, no investor can improve the objective by changing individual bid unilaterally. A pseudo polynomial-time algorithm is presented for generating a pure Nash equilibrium. We also investigate the price of anarchy (the ratio of the worst Nash equilibrium to the social optimum) associated with a simplified two-person knapsack game.
2010, 6(4): 861-880 doi: 10.3934/jimo.2010.6.861 +[Abstract](247) +[PDF](1499.7KB)
Abstract:
In this paper we consider the adaptive control problem for a class of systems governed by nonlinear differential equations. Using Takagi-Sugeno approach, we have proposed a fuzzy model, which is linear in nature, the behavior of which is close to that of the unknown (nonlinear) plant. Based on this fuzzy model, we have proposed certain control structure with the help of which the plant output is capable of tracking certain desired trajectory. Using a suitable objective function and variation arguments, we have developed a set of necessary conditions with the help of which the parameters of the proposed fuzzy model and controller can be determined. Based on these necessary conditions, a numerical scheme is presented for computing the unknowns. Further, the question of continuous dependence of the proposed estimator and controller on system parameters (robustness) has been studied. Finally, the proposed adaptive control scheme has been applied to two different examples to illustrate the effectiveness of the proposed adaptive control scheme.
2010, 6(4): 881-893 doi: 10.3934/jimo.2010.6.881 +[Abstract](197) +[PDF](182.1KB)
Abstract:
In this paper, duals for standard semidefinite programming problems from both the primal and dual sides are studied. Explicit expressions of the minimal cones and their dual cones are obtained under closeness assumptions of certain sets. As a result, duality formulations resulting from regularizations for both primal and dual problems can be expressed explicitly in terms of equality and inequality constraints involving three vector and matrix variables under such assumptions. It is proved in this paper that these newly developed duals can be cast as the Extended Lagrange-Slater Dual (ELSD) and the Extended Lagrange-Slater Dual of the Dual (ELSDD) with one reduction step. Therefore, the duals formulated in this paper guarantee strong duality, i.e., a zero duality gap and dual attainment.
2010, 6(4): 895-910 doi: 10.3934/jimo.2010.6.895 +[Abstract](387) +[PDF](211.0KB)
Abstract:
In this paper, a computational approach based on a new exact penalty function method is devised for solving a class of continuous inequality constrained optimization problems. The continuous inequality constraints are first approximated by smooth function in integral form. Then, we construct a new exact penalty function, where the summation of all these approximate smooth functions in integral form, called the constraint violation, is appended to the objective function. In this way, we obtain a sequence of approximate unconstrained optimization problems. It is shown that if the value of the penalty parameter is sufficiently large, then any local minimizer of the corresponding unconstrained optimization problem is a local minimizer of the original problem. For illustration, three examples are solved using the proposed method. From the solutions obtained, we observe that the values of their objective functions are amongst the smallest when compared with those obtained by other existing methods available in the literature. More importantly, our method finds solution which satisfies the continuous inequality constraints.
2010, 6(4): 911-927 doi: 10.3934/jimo.2010.6.911 +[Abstract](235) +[PDF](204.9KB)
Abstract:
This paper analyzes a finite buffer bulk arrival queueing system with a single working vacation and partial batch rejection in which the inter-arrival and service times are, respectively, arbitrarily and exponentially distributed. Using the supplementary variable and the imbedded Markov chain techniques, we obtain the system length distributions at pre-arrival and arbitrary epochs. We also present Laplace-Stiltjes transform of the actual waiting time distribution in the system. Finally, several performance measures and a variety of numerical results in the form of tables and graphs are discussed.
2010, 6(4): 929-944 doi: 10.3934/jimo.2010.6.929 +[Abstract](177) +[PDF](224.8KB)
Abstract:
Queues with Markovian service process ($MSP$) are mainly useful in modeling and performance analysis of telecommunication networks based on asynchronous transfer mode (ATM) environment. This paper analyzes a finite buffer single server batch service ($a, b)$ queue with general input and Markovian service process ($MSP$). The server accesses new arrivals even after service has started on any batch of initial number $a$. This operation continues till the service time of the ongoing batch is completed or the maximum accessible capacity $d ~(a\le d < b)$ of the batch being served is attained whichever occurs first. Using the embedded Markov chain technique and the supplementary variable technique we obtain the steady state queue length distributions at pre-arrival and arbitrary epochs. The primary focus is on the various performance measures of the steady state distribution of the batch service, special cases and also on numerical illustrations.

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