Problems & Solutions in Inventory Management

In this chapter, we discuss the meaning of the term inventory. Different types of inventory are used by firms depending on the nature of their business. We discuss the types of inventory as well as the purpose of holding those. Inventory managers are expected to ensure availability of an item in the right quantity when a demand arises. To achieve this objective, they need to consider the characteristics of each of the items they are managing, such as demand, replenishment lead time, the timing of review, and item lifetime. We examine these characteristics in detail and describe how the element of uncertainty increases the complexity of making reliable inventory decisions.

Toward the end of this chapter, we present a classification of inventory models that give the reader an idea of the breadth of available solutions.

In this chapter, we discuss the key design factors – frequency of review, timing of order placement, and order size – that need to be kept in mind while adopting an inventory control system for a given business. While several inventory control systems and their variants are available in literature, we review three popular inventory control systems in this chapter:

Cost is often the basis for optimization in most of the problems that businesses encounter. An inventory control system that minimizes the total inventory costs would be the one an organization needs to adopt. In this chapter, we discuss in detail three relevant inventory costs – carrying cost, ordering cost, and shortage cost – as well as methods to estimate those.

Toward the end of this chapter, we present a running example that we would use across all chapters. This would help get a common basis for learning inventory management concepts.

In this chapter, we discuss mathematical models to manage inventory of a single item whose demand is known and is constant. We start our discussion with the most fundamental of inventory models – the Economic Order Quantity (EOQ) model – which assumes that the demand for the item is constant, the order is filled instantaneously, and there are no shortages. We derive an expression for the total annual inventory cost (TIC) comprising the ordering cost, carrying cost, and the cost of procurement, and determine the optimal order quantity that minimizes the TIC. We also discuss computation of the reorder level when lead time is assumed to be constant and known. Later in this chapter, we discuss extensions of the EOQ model to other situations where supply is not instantaneous (i.e., supply is gradual), and another where planned shortages are allowed. Toward the end of this chapter, we also discuss a periodic review model under certainty.

It is usual for a seller to offer discounted prices to buyers that procure goods from them in large volumes. A buyer wanting to take advantage of the discount being offered will have to balance the reduced purchase cost with the increase in carrying cost. In this chapter, we discuss continuous review (EOQ) models with quantity discounts, also known as price-break models. We start our discussions describing the concept of discount with a single price-break model. We then extend our discussions to models with multiple price-breaks. Two types of multiple price-break models have been discussed in this chapter:

Toward the end of this chapter, we also discuss another type of discount model – special discount model, also known as a one-off discount or promotional discount model.

In this chapter, we discuss lot-sizing heuristics that can be used to manage inventory of a single item whose demand varies from period to period. Because the demand is not constant the classical EOQ formula cannot be applied. Several heuristics are found in literature. The popular ones that have been considered in this chapter are (1) lot-for-lot method, (2) part-period balancing method, (3) least unit cost method, (4) silver-meal method, and (5) Wagner-Whitin method. The primary goal of these heuristics is to determine the lot size that would minimize the total ordering and carrying costs. Depending on the situation, one of these heuristics can be implemented to obtain least cost inventory management solution.

In the previous chapters, we discussed models that may be used when demand and lead time are constant. In this chapter, we explore the uncertainty of demand (and lead time) and its effect on order size and replenishment strategies. When demand for an item is uncertain, it becomes difficult for inventory managers to decide the order quantity and the timing of replenishment orders. In some situations, demand could exceed expectations or orders may arrive late. This could result in a stockout – a situation when inventory does not exist on hand to meet the demand. Besides how much and when to order, an additional question becomes important – that of how much to stock in order to offset the uncertainties of demand and lead time. This inventory, held in excess of regular usage quantities, is referred to as safety stock. Carrying safety stock is one of the most popular methods of reducing the effects of demand and lead time uncertainty. Reliable, historical data of demand and lead time is an important factor in the calculation of safety stock. In this chapter, we use historical data and fit it into a probability distribution (discrete as well as continuous) to determine the size of safety stock. (s, Q) and (T, S) models have been discussed.

The inventory models discussed in the previous chapters assumed a single item being managed. Inventory managers at warehouses and retail shops manage a large number of items in their inventory. Inventory decisions – such as timing of replenishment orders, determining order quantities – are not made for each item independently. Decisions are often made jointly for several items managed at one location. Such decisions may be constrained by either the value of inventory they can hold in stock, or the availability of space to stock the items in their warehouse, or some other similar scarce resource. In this chapter, we discuss multi-item inventory models that are subject to one or more resource constraints such as budget, space, or number of orders. We also discuss methods of treating inventory problems that are subject to more than one constraint. Later in this chapter, we discuss another point of interest to inventory managers – that of optimizing the inventory costs by replenishing jointly to derive the benefits of economies of scale.

Inventory control models reviewed till now require that each item held in the inventory system be given equal importance irrespective of their usage, criticality, cost, or any similar attribute. This would be extremely difficult in organizations that manage several thousands of items. During the time when computing resources were scarce, inventory managers devised methods to selectively control inventory items – focusing their effort more on items that are key to the organization, and applying lesser control on other items. In this chapter, some of these popular selective inventory control methods such as ABC analysis, VED analysis, and FSN analysis have been discussed with numerical examples. Each of these methods classifies inventory items into three groups (Some selective inventory control methods group inventory items into more than 3 groups.) and apply inventory management strategies around those groups. A hybrid method – that of combining the ABC and the VED techniques – has also been discussed. We also discuss an important multi-item inventory management technique that has EOQs as its basis – the exchange curve. This curve can help inventory managers identify an appropriate ratio of ordering cost to inventory carrying rate that can achieve a management-desired level (constraint) of inventory value.

In this chapter, we discuss the management of maintenance items – items that do not go into the final product but those that support the production of a product. Maintenance items such as spare parts, coolants, lubricants, sensors etc. have been traditionally classified into three groups – fast-moving items, slow-moving items, and rotables. Fast-moving items may be managed using one of the deterministic or stochastic techniques learned in the earlier chapters. However, the same techniques may not be applicable to manage slow-moving items because their consumption (demand) is random, and reliability of historical records is low. Despite the demand being low because the time between failures is far and in between, organizations would still need to maintain an inventory of slow-moving parts. This is because they may not be available a few years later as original equipment manufacturers would no longer be producing those parts, or would have changed their design. Rotables are repairable items that are reconditioned and made available for service. The characteristics of this type of item as well as its demand pattern make it unsuitable to apply inventory management techniques we learned earlier in this book to manage this class of spares. Analytical as well as graphical techniques, suitable for management of slow-moving inventories as well as rotables, have been discussed in detail in this chapter.

In the previous chapters, we dealt with the problem of inventory items stocked in one location. In this chapter, we discuss mathematical models to manage inventory items that are maintained at different stocking points (or locations). These models are referred to as multi-echelon models. These models, while being closer to reality, are mathematically quite complex, and therefore only a high-level view of this topic is presented. We start our discussions with a simple two-stage (warehouse and retailer) model with deterministic demand and derive expressions to compute the order quantities for both stages. We then extend the model to probabilistic demand to compute the order quantities assuming demand to be normally distributed.