Supply chain management is becoming an increasingly important issue, especially when in most industries the cost of materials purchased comprises 40--60% of the total sales revenue. Despite the benefits cited for single sourcing in the popular literature, there is enough evidence of industries having two/three sources for most parts. In this paper we address the operational issue of quantity allocation between two uncertain suppliers and its effects on the inventory policies of the buyer. Based on the type of delivery contract a buyer has with the suppliers, we suggest three models for the supply process. Model I is a one-delivery contract with all of the order quantity delivered either in the current period with probability \beta , or in the next period with probability 1 -- \beta . Model II is also a one-delivery contract with a random fraction of the order quantity delivered in the current period; the portion of the order quantity not delivered is cancelled. Model III is similar to Model II with the remaining quantity delivered in the next period. We derive the optimal ordering policies that minimize the total ordering, holding and penalty costs with backlogging. We show that the optimal ordering policy in period n for each of these models is as follows: for x \ge \bar{u} n, order nothing; for v\bar n \le x n , use only one supplier; and for x n , order from both suppliers. For the limiting case in the single period version of Model I, we derive conditions under which one would continue ordering from one or the other or both suppliers. For Model II, we give sufficient conditions for not using the second (more expensive) supplier when the demand and yield distributions have some special form. For the single period version of Models II and III with equal marginal ordering costs we show that the optimal order quantities follow a ratio rule when demand is exponential and yields are either normal or gamma distributed.dual sourcing, supply uncertainty, inventory

Increasing product complexity, manufacturing environment complexity and an increased emphasis on product quality are all factors leading to uncertainties in production processes. These uncertainties are in the form of unplanned machine maintenance, varying production yields and rework, among others. In planning for production, an adequate model must incorporate these uncertainties into the representation of the production process. This paper treats the aggregate planning problem for a single product with random demand and random capacity. In the single-period problem, random capacity does not affect the optimal policy but results in a unimodal, nonconvex cost function. In the multiple-period and infinite-horizon settings order-up-to policies that are dependent on the distribution of capacity are shown to be optimal in spite of a nonconvex cost. In the infinite-horizon setting an intuitive description of the situation leads to the notion of a class of extended myopic policies, requiring the consideration of review periods of uncertain length.

We address the problem of controlling the production rate of a failure prone manufacturing system so as to minimize the discounted inventory cost, where certain cost rates are specified for both positive and negative inventories, and there is a constant demand rate for the commodity produced.The underlying theoretical problem is the optimal control of a continuous time system with jump Markov disturbances, with an infinite horizon discounted cost criterion.We use two complementary approaches. First, proceeding informally, and using a combination of stochastic coupling, linear system. arguments, stable and unstable eigenspaces, renewal theory, parametric optimization etc., we arrive at a conjecture for the optimal policy. Then we address the previously ignored mathematical difficulties associated with differential equations with discontinuous right hand sides, singularity of the optimal control problem, smoothness and validity of the dynamic programming equation etc., to give a rigorous proof of optimality of the conjectured policy.It is hoped that both approaches will find uses in other such problems also.We obtain the complete solution and show that the optimal solution is simply characterized by a certain critical number, which we call the optimal inventory levelo If the current inventory level exceeds the optimal, one should not produce at all, if less, one should produce at the maximum rate, while if exactly equal one should produce exactly enough to meet demand. We also give a simple explicit formula for the optimal inventory level.

We study a single period multiproduct inventory problem with substitution and proportional costs and revenues. We considerNproducts andNdemand classes with full downward substitution, i.e., excess demand for classican be satisfied using productjfori≥j. We first discuss a two-stage profit maximization formulation for the multiproduct substitution problem. We show that a greedy allocation policy is optimal. We use this to write the expected profits and its first partials explicitly. This in turn enables us to prove additional properties of the profit function and several interesting properties of the optimal solution. In a limited computational study using two products, we illustrate the benefits of solving for the optimal quantities when substitution is considered at the ordering stage over similar computations without considering substitution while ordering. Specifically, we show that the benefits are higher with high demand variability, low substitution cost, low profit margins (or low price to cost ratio), high salvage values, and similarity of products in terms of prices and costs.

Abstract-Machine learning classifiers have recently emerged as a way to predict the introduction of bugs in changes made to source code files. The classifier is first trained on software history, and then used to predict if an impending change causes a bug. Drawbacks of existing classifier-based bug prediction techniques are insufficient performance for practical use and slow prediction times due to a large number of machine learned features. This paper investigates multiple feature selection techniques that are generally applicable to classification-based bug prediction methods. The techniques discard less important features until optimal classification performance is reached. The total number of features used for training is substantially reduced, often to less than 10 percent of the original. The performance of Naive Bayes and Support Vector Machine (SVM) classifiers when using this technique is characterized on 11 software projects. Naive Bayes using feature selection provides significant improvement in buggy F-measure (21 percent improvement) over prior change classification bug prediction results (by the second and fourth authors [28]). The SVM's improvement in buggy F-measure is 9 percent. Interestingly, an analysis of performance for varying numbers of features shows that strong performance is achieved at even 1 percent of the original number of features.

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We address the problem of controlling the production rate of a failure prone manufacturing system so as to minimize the discounted inventory cost, where certain cost rates are specified for both positive and negative inventories, and there is a constant demand rate for the commodity produced.The underlying theoretical problem is the optimal control of a continuous time system with jump Markov disturbances, with an infinite horizon discounted cost criterion.We use two complementary approaches. First, proceeding informally, and using a combination of stochastic coupling, linear system. arguments, stable and unstable eigenspaces, renewal theory, parametric optimization etc., we arrive at a conjecture for the optimal policy. Then we address the previously ignored mathematical difficulties associated with differential equations with discontinuous right hand sides, singularity of the optimal control problem, smoothness and validity of the dynamic programming equation etc., to give a rigorous proof of optimality of the conjectured policy.It is hoped that both approaches will find uses in other such problems also.We obtain the complete solution and show that the optimal solution is simply characterized by a certain critical number, which we call the optimal inventory levelo If the current inventory level exceeds the optimal, one should not produce at all, if less, one should produce at the maximum rate, while if exactly equal one should produce exactly enough to meet demand. We also give a simple explicit formula for the optimal inventory level.

No abstract