1 Introduction

The concept of shortage emerged as a research object in 1974, when it was offered by Van der Meiden (1974). Since then, related to the topic studies have been shifting towards medicine, e.g. the study made by Seppala and Pulkkinen (1982). However, the topic of shortage remains open to debates among economists that are endlessly searching for the best practices of avoiding shortage. All techniques, methods that improve forecasting, distribution, and handling; may prolong products’ durability and lead-time-demand bring the reduction of shortage in many industries (i.e. Lacko 1989).

The object of this study is to examine the models of inventory management. The article covers the literature regarding inventory planning and avoiding shortage; investigates the causes of operational shortage and formulats the suggestions of possible imprvements. The study provides valuable insights on the topic.

The study has its limitation as the focus area is narrowed to the shortage concept and its reflection on the inventory management models. Nevertheless, the results are very valuable for the experts occupied with the topics of shortage and costs reduction improvement. The originality of the research comes from the author’s analysis of the inventory management models and the shortage concept, while a historical approach to the analysis reveals the development of the concepts, methods, and models that are concerned herein.

2 The Shortage Concept

The shortage might be a result emerging from many activities: stock planning, its distribution, handling, and stock keeping. The author distinguishes the four main shortage causes: (1) Inaccurate demand forecasting; (2) Too long distribution time; (3) Insufficient handling accuracy; and (4) Loses emerging after product’s expiration term comes to affect. The shortage is usually associated with demand forecasting and supply uncertainty. Among all the four shortage reasons only the first one is dedicated to demand forecasting uncertainty, while the other three are oriented to supply uncertainty. In details all these shortage causes are showed in the Fig. 1.

Fig. 1
figure 1

Identifying shortage causes (constructed by the author)

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Each shortage reason is described in detail in the picture. Let’s start from demand forecasting.

  1. 1.

    Forecasting depends heavily on information gathered about previous demand and applicable lead times to meet future needs. Therefore, logistics system should have inventory buffer to cover demand shocks, i.e. “safe” inventory for periods of unexpectedly high demand.

    1. 1.1

      Population forecasting is not an easy issue. Demographics dynamics are analyzed by Keyfitz (1981); ageing population is emphasized by Zabawa et al. (2017). These authors pointed out to the reasons of supply shortage that forecasting is due to deal with the forecasting of demand of medicine products concerns population dynamics in the first place.

    2. 1.2

      Unfulfilled demand could be classified as planned and unplanned. Planned shortage is managed by advanced orders referred to as backorders. A company uses the backorders for new (not yet produced) products as a rule of thumb. In cases of planned shortage, production process starts as customer’s order is received (the concept is referred to as “Make-to-order”). The production process takes additional time that means some wait time for a consumer to receive an item. The unplanned shortage is related with operations management (stock distribution and handling issues; damages that occur in storage, etc.). For example, statistically 95% of products arrive on time, 99.5% of products are picked to customers in full and 98% is a stock accuracy level.

    3. 1.3

      As the number of locations grows in storage, higher inventory level is required. This means that during forecasting the company must consider the number of locations at internal and external facilities. Usually demand is forecasted based on historical data. The demand forecasting may employ two approaches: top-down approach and bottom-up approach. The first one is developed for the market and then it is broken down by region, district, territory, etc. The second one is used to estimate purchases or forecast demand for the specified future time in selling spots. The number of stocking locations is important for demand planning.

  2. 2.

    Lead-time is used as constrain in many inventory models: stochastic and deterministic ones. The lead-time is treated as delivery requisition to order in days.

    1. 2.1

      In studies, the lead-time is presented as having different characteristics: as fixed—by Hoque and Goyal (2004) and as variable—by Moon et al. (2014) and Ouyang et al. (1996); as fuzzy—by Rong et al. (2008) and as stochastic—by Das and Hanaoka (2014).

    2. 2.2

      In addition, time is classified as controllable by Pan and Yang (2006) or having time uncertainty, such is emphasised by Lee et al. (2017), or Cheng (1986).

      Most of the inventory models address only one type of uncertainty that results to shortage in the production process. However, there are much more types of uncertainty.

  3. 3.

    In the literature, there are distinguished several issues of process accuracies related to the material handling: receiving, in-house handling and shipping. During the material handling inaccurate stock (inventory) measurement appears. The results of accuracy are seen at inventory audit, from client’s claims, from lost sales, from costs associated with errors corrections, and from products availability. When a picker indicates a shortage situation or products are out-of-stock, the system should do three things:

    1. (1)

      to move product to picking;

    2. (2)

      to order a new pick task to be picked (usually with next picking route);

    3. (3)

      to direct product’s stock count to discover, why the discrepancy occurred.

In this circumstance, a buyer wants to purchase extra quantity of, for example, medical products, then available quantity, and material allocation mechanism (such as “first come, first served”) determines which buyer is served. The shortage of inventory is time-dependent and place-dependent. In addition, there are stockage losses, which effect shortage. Several stock losses types are presented below:

  1. 4.1

    Ageing stock—the stock, which fails to move on to the next point beyond its shelf-life, because of expiring term of the products. The actual sales, resulting in unsold stocks and ultimately expired stocks. A company must compensate directly or indirectly the affected parties in the distribution system and ultimately incur the loss itself (Sengupta 2017).

  2. 4.2

    Damages—a term that includes stocks damaged in transit or stocking. The transit requires controlling reverse flow of damaged products. Because the factor of damage occurrence is identified after the shift to next location.

  3. 4.3

    Shrink is the difference between recorded inventory on a company’s balance sheet and its actual inventory. The term “shrinkage” is also used by production in general; it refers to the loss of products or their sub-parts during transactions.

  4. 4.4

    Quality—another emphasis on improving the production process to guarantee products quality reducing number of rework-able items and scrap items.

3 The Inventory Model Encountering Shortage

Since 1915, authors developed different types of inventory models. One type of these models minimizes total costs and optimizes a shortage period. Shortage costs (lost sales costs) are both made up of fixed costs and variable costs which depend on the length of the shortage time. The shortage costs are lost sales because of unserved present customers, which also do not return in future.

The most of inventory models involve only planned shortage (backorders) leaving other shortage causes not considered. Shortage appears when replenishment order does not arrive at or before the inventory position drops to zero. Therein annual shortage costs are equal to penalty for incurring the shortage of a unit, which is multiplied to times shorted throughout a year. Usually average shortage level is used during modelling. Nevertheless, unplanned shortage can still occur if the demand rate and deliveries do not stay on schedule. In addition, defective items could affect shortage and handling errors. Sulak et al. (2015) suggested that inventory model should be oriented to defective items and shortage. The detailed overview of shortage concept in inventory models is presented in Table 1.

Table 1 The employment of historical shortage concept in inventory management models
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The studies presented in Table 1 in the period of 1970–1996 focus on time delays and variable demand. In the second part of presented period, i.e. in 1991, studies involve the understanding that shortage is affected by supply uncertainty. In the pick of popularity of the topic, in studies linked with the period 1997–2010, shortage is analyzed in the line with logistics costs. Extra studies arise in 2010, involving into inventory models batch inspection errors and imperfect quality of products.

The most of inventory models do not involve shortage planning. Inventory model, in fuzzy sense, is developed for inventory planning without backordering by Lee and Yao (1999). The model is used for calculating the appropriate reorder point and the optimal reorder quantity to ensure the instantaneous replenishment of inventory encountering no shortage.

The purpose of inventory model encountering shortage is to reduce fixed costs, minimize inventory holding (carrying) costs, and improve overestimating/underestimating demand costs. The model minimizes the total costs of inventory—such as holding (carrying) costs, ordering and production set-up costs, shortage costs and excess costs. During stock out period, customers are ready to wait for some certain time until their orders are carried out. However, if delay time is too long, some customers will be lost, and a company lose sales and profit, while other customers will ask to provide salvage (discount or time-weighted penalty costs).

Each of inventory models is based on assumptions: Economic production order (EPQ) assumes that a producer will handle its own production quantity, and products will be available after production process. Economic order quantity (EOQ) assumes that quantity will arrive when the order is placed. Following these assumptions author is working with EPQ inventory model formula.

The following nomenclature for inventory model encountering shortage is assumed:

Q* :

Optimal order quantity for single item;

Co:

Costs for ordering and set-up production per order/batch;

C h :

Holding costs per unit per year;

C s :

Shortage costs (lost sales including lost profit costs);

C e :

Excess costs;

D :

Annual demand;

p :

Production rate (Production a day);

u :

Consumption rate (Demand a day);

k :

Service level factor.

Inventory Model Encountering Shortage

$$ Q^{*} = \sqrt {\left( {\frac{{2C_{0} D + C_{s} }}{{C_{h} }}} \right)\left( {\frac{p}{p - u}} \right)} k $$
(1)

The equation is helping to calculate optimal order quantity. The last part of equation is dedicated to stock service level factor k, which is calculated using two components shortage costs divided from the sum of shortage costs and excess costs.; k is subject to

$$ k = NORMSINV\left( {\frac{{C_{s} }}{{C_{s} + C_{e} }}} \right) $$
(2)

The Costs Mentioned in Inventory Model Encountering Shortage

During the production run, there are two activities: demand reduces the inventory and production adds more units to the inventory. Ordering/producing costs Co are average costs for specified activities. The primary driver to hold inventories is the presence of a fixed production costs A and ordering costs O that are incurred every time a positive number of units are produced and ordered. To determine the number of units to produce and order every time the fixed production and ordering costs are incurred. Producing or ordering in large quantities reduces the average fixed production/ordering costs.

$$ C_{0} = O + A $$
(3)

The production set-up costs A are presented below. Herein production costs per unit Cp are multiplied from batch size y.

$$ A = C_{p} y $$
(4)

Holding costs Ch represent annual unit holding costs, i.e. holding costs of product in warehouse(s) in period, where product inventory is subject to the average number of units in inventory throughout a year. Holding costs are derived as percentage of item price. Since the item is being storage as inventory, it losses some value by the rate of inflation. So, the holding costs of each item are calculated by multiplying inflation rate i with unit price, which below is represented by unit costs U and sales profit P.

$$ C_{h} = i\left( {U + P} \right) $$
(5)

The order is lost if stock shortage appears. Cs are shortage costs per unit per time i.e. loss of profit. For the equation of shortage (stock-out) costs sales price per unit S and unit costs (or unit purchase price) U are taken.

$$ C_{s} = S - U $$
(6)

Planned shortage is permitted, i.e. backordered demand units are withdrawn from a replenishment order when backorder is delivered. Backordering is as well positively affecting holding costs per unit, per unit time and the variable part of these costs could be taken into the equation of shortage costs.

Excess costs Ce are the losses given from overestimated demand D, i.e. is difference between sales price per unit and salvage price per unit W. These costs are related with ageing stock losses (represented in shortage concept Fig. 1 point 4.1); they are negatively affecting holding costs per unit, per time ChT

$$ C_{e} = S - W + ChT $$
(7)

The Shortage Concept in Inventory Management Model

Author try to find the best way to introduce shortage concept into inventory model. Demand planning is quite complex task. Some ideas on forecasting drug demand are proposed by Clark (2013), where demand is forecasted using probability analysis. Below is quite universal equation for demand forecasting. Population changes could be estimated by trend. The equation is linked with shortage concept (Fig. 1) point 1.1.

$$ D = \left( {underlying\,value + trend} \right)*seasonal\,index $$
(8)

The shortage is planned and unplanned. Probability to have shortage is opposite to probability Prob to arrive to Demand. The next equation is linked with shortage concept (Fig. 1) point 1.2.

$$ 1 - Prob\left( {C_{s} } \right) = Prob\left( D \right) $$
(9)

To serve demand in different regions stock is present in multiple facilities. All locations carry the same amount of inventory, but in multi-locations case, total holding costs can be stated as holding costs in single-location Ch0 plus additional costs CE for each unit (pallet) of the item multiplied from the amount of space E occupied by one unit (pallet) of the same item. The equation is linked with shortage concept (Fig. 1) point 1.3.

$$ C_{h} = C_{h0} + C_{E} *E $$
(10)

In case of Lead-time uncertainty reorder level ROL is calculated as lead-time LT which is multiplied from consumption rate u plus extra inventory Is which is held more than expected demand. The equation is linked with shortage concept (Fig. 1) point 2.

$$ ROL = LT*u + I_{s} $$
(11)

The increase of excess inventory increases holding costs but reduces shortage costs. If excess inventory reduces the opposite actions are taken.

There joint calculation of order quantity with shortage where holding costs are replaced to C *h due to many conditions influencing these costs.

$$ Q^{*} = k\sqrt {\left( {\frac{2D}{{C_{h}^{*} }}} \right)\left( {\frac{p}{p - u}} \right)\left[ {C_{0} + C_{s} \mathop \sum \limits_{D = ROL}^{\infty } \left( {D - ROL} \right)*Prob\left( D \right)} \right]} $$
(12)

Production activities are always linked with screening and reworking. The Author is going to present equations, which are linked with shortage concept (Fig. 1) point 4.4. Screening and reworking costs fall under Co, which includes production set-up costs A. Total screening costs C are calculated accordingly: screening costs per unit Cq are multiplied from batch size y and present the costs of inspection.

$$ C = C_{q} y $$
(13)

In addition, rework costs R are equal to rework costs per unit for defective items Cr multiplied from rate r of rework-able defective items and the size of batch y:

$$ R = C_{r} ry $$
(14)

After the products screening and reworking activities are finished products are moved to storage. The Holding costs, which integrate the the level of non-rework-able imperfect items t and scrap items k, are stated below. Holding costs per unit, per unit time is marked as ChT and screening rate x

$$ C_{h} = C_{hT} \left[ {\left( {t + k} \right)\frac{{y^{2} }}{x} + \frac{{y^{2} (1 - \left( {t + k} \right))^{2} }}{2D} - \frac{{y^{2} }}{2p}} \right] $$
(15)

Handling Accuracy and Holding Costs

Nevertheless, handling costs fall under holding (carrying) costs, the handling accuracy, which relates to handling activity, are not covered in inventory models. The most critical factors in the process are human mistakes, which must be solved seeking to avoid unplanned shortage. Author delivers simulation model and tests the impact of handling accuracy to holding (carrying) costs. Results are presented in Table 2 and Fig. 2.

Table 2 Effect of handling accuracy on holding (carrying) costs
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Fig. 2
figure 2

Handling accuracy generates extra holding (carrying) costs (author’s simulation case, study results)

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The impact of handling accuracy y to the revised holding (carrying) costs C *h could be calculated according the next equation. This equation is linked with shortage concept (Fig. 1) point 3.

$$ C_{h}^{*} = C_{h} *y $$
(16)

The inventory model encountering shortage is developed with the assumption that all products are delivered to the customers prior to their expiry date, although the practice is different. The final equation of order quantity encountering shortage covers also damages and shrink rate h (represented in shortage concept Fig. 1 points 4.2 and 4.3).

$$ Q_{f}^{*} = Q^{*} \left( {1 + h} \right) $$
(17)

The suggested equation summarizes all components the author has involved into inventory model encountering shortage.

4 Conclusions

The classical inventory models are meant for cases when 100% of environment and products are perfect. However, this assumption may not be valid for the most cases evident in real environments. The analysis of historical shortage concept shows that shortage is quite popular in inventory management models. But more often authors provide single studies in the area, i.e. these studies do not combine all shortage causes together.

Author is trying to map causes, which influence shortage, especially which is dependent on supply uncertainty and in most cases is not planned, to present the shortage concept. The components of shortage concept are covered by inventory models encountering shortage, considering demand and supply uncertainties. Herein, the components of costs are analyzed in separate seeking to present the links between activities.

Aiming to specify holding costs, author provides simulation model, where relationship between handling accuracy and holding (carrying) costs is presented. Finally, author delivers the inventory model encountering shortage, which widens the range of inventory models and their application. This study could be expanded to various directions, including the practical applications of suggested model.