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Lecture

Markov Chain: Long-Run Average Costs

Description

This lecture covers the Markov Law of Large Numbers (LLN) applied to an irreducible Markov chain with transition matrix P and invariant distribution π. It discusses the first return time to a state, reward functions, and the long-run average costs in an (S, S) inventory model. The proof of the Markov LLN implies that the expectation of the reward converges to a limit. The lecture also explores the dynamics of the inventory level based on demand, the cost of lost customers, and the unique invariant distribution. It concludes with the implications of the Markov LLN on the long-run average holding cost and the cost of lost customers.

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Related concepts (32)

Inventory

Inventory (American English) or stock (British English) refers to the goods and materials that a business holds for the ultimate goal of resale, production or utilisation. Inventory management is a discipline primarily about specifying the shape and placement of stocked goods. It is required at different locations within a facility or within many locations of a supply network to precede the regular and planned course of production and stock of materials.

Inventory theory

Material theory (or more formally the mathematical theory of inventory and production) is the sub-specialty within operations research and operations management that is concerned with the design of production/inventory systems to minimize costs: it studies the decisions faced by firms and the military in connection with manufacturing, warehousing, supply chains, spare part allocation and so on and provides the mathematical foundation for logistics.

Inventory control

Inventory control or stock control can be broadly defined as "the activity of checking a shop's stock". It is the process of ensuring that the right amount of supply is available within a business. However, a more focused definition takes into account the more science-based, methodical practice of not only verifying a business's inventory but also maximising the amount of profit from the least amount of inventory investment without affecting customer satisfaction.

Vendor-managed inventory

Vendor-managed inventory (VMI) is an inventory management practice in which a supplier of goods, usually the manufacturer, is responsible for optimizing the inventory held by a distributor. Under VMI, the retailer shares their inventory data with a vendor (sometimes called supplier) such that the vendor is the decision-maker who determines the order size, whereas in traditional inventory management, the retailer (sometimes called distributor or buyer) makes his or her own decisions regarding the order size.

Inventory management software

Inventory management software is a software system for tracking inventory levels, orders, sales and deliveries. It can also be used in the manufacturing industry to create a work order, bill of materials and other production-related documents. Companies use inventory management software to avoid product overstock and outages. It is a tool for organizing inventory data that before was generally stored in hard-copy form or in spreadsheets.

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