Inventory theory

<h2 id="issues">Issues</h2>

<p>One issue is infrequent large orders vs. frequent small orders.  Large orders will increase the amount of inventory on hand, which is costly, but may benefit from volume discounts.  Frequent orders are costly to process, and the resulting small inventory levels may increase the probability of <a href="/facts/Stockout/I4nYAyQA">stockouts</a>, leading to loss of <a href="/facts/Customer/oGqcqmhf">customers</a>. In principle all these factors can be calculated mathematically and the <a href="/facts/Optimum/oRn8Iv5I">optimum</a> found.
</p><p>A second issue is related to changes in demand (predictable or random) for the product.  For example, having the needed merchandise on hand in order to make sales during the appropriate buying season(s).  A classic example is a <a href="/facts/Toy_store/U9mr7sIw">toy store</a> before <a href="/facts/Christmas/EsuI0Gds">Christmas</a>: if the items are not on the shelves, they cannot be sold.  And the wholesale market is not perfect; there can be considerable delays, particularly with the most popular toys.  So, the entrepreneur or business manager will buy speculatively.  Another example is a <a href="/facts/Furniture/J7JrsrlA">furniture</a> store.  If there is a six-week, or more, delay for customers to receive merchandise, some sales will be lost.  A further example is a restaurant, where a considerable percentage of the sales are the <a href="/facts/Value-added/TCDUk1bH">value-added</a> aspects of food preparation and presentation, and so it is rational to buy and store somewhat more to reduce the chances of running out of key ingredients.  The situation often comes down to two key questions: confidence in the merchandise selling, and the benefits accruing if it does?
</p><p>A third issue comes from the view that inventory also serves the function of decoupling two separate operations.  For example, <a href="/facts/Work_in_process/e3oUTaxL">work in process</a> inventory often accumulates between two departments because the consuming and the producing department do not coordinate their work.  With improved coordination this buffer inventory could be eliminated. This leads to the whole philosophy of <a href="/facts/Just_In_Time_(business)/h4dNAj42">Just In Time</a>, which argues that the costs of carrying inventory have typically been underestimated, including both the direct, obvious costs of storage space and insurance, but also the harder-to-measure costs of increased variables and complexity, and thus decreased flexibility, for the business enterprise.
</p>
<h2 id="inventory-models">Inventory models</h2>
<p>The mathematical approach is typically formulated as follows:
a store has, at time 
  
    
      
        k
      
    
    {\displaystyle k}
  
, 
  
    
      
        
          x
          
            k
          
        
      
    
    {\displaystyle x_{k}}
  
 items in stock. It then orders (and receives) 
  
    
      
        
          u
          
            k
          
        
      
    
    {\displaystyle u_{k}}
  
 items, and sells 
  
    
      
        
          w
          
            k
          
        
      
    
    {\displaystyle w_{k}}
  
 items, where 
  
    
      
        w
      
    
    {\displaystyle w}
  
 follows a given probability distribution. Thus:
</p>

x
          
            k
            +
            1
          
        
        =
        
          x
          
            k
          
        
        +
        
          u
          
            k
          
        
        −
        
          w
          
            k
          
        
      
    
    {\displaystyle x_{k+1}=x_{k}+u_{k}-w_{k}}

u
          
            k
          
        
        ≥
        0
      
    
    {\displaystyle u_{k}\geq 0}

<p>Whether 
  
    
      
        
          x
          
            k
          
        
      
    
    {\displaystyle x_{k}}
  
 is allowed to go negative, corresponding to back-ordered items, will depend on the specific situation; if allowed there will usually be a penalty for back orders. The store has costs that are related to the number of items in store and the number of items ordered: 
</p>

c
          
            k
          
        
        =
        c
        (
        
          x
          
            k
          
        
        ,
        
          u
          
            k
          
        
        )
      
    
    {\displaystyle c_{k}=c(x_{k},u_{k})}
  
. Often this will be in additive form: 
  
    
      
        
          c
          
            k
          
        
        =
        p
        (
        
          x
          
            k
          
        
        )
        +
        h
        (
        
          u
          
            k
          
        
        )
      
    
    {\displaystyle c_{k}=p(x_{k})+h(u_{k})}

<p>The store wants to select 
  
    
      
        
          u
          
            k
          
        
      
    
    {\displaystyle u_{k}}
  
 in an optimal way, i.e. to minimize
</p>

∑
          
            k
            =
            0
          
          
            T
          
        
        
          c
          
            k
          
        
        .
      
    
    {\displaystyle \sum _{k=0}^{T}c_{k}.}

<p>Many other features can be added to the model, including multiple products (denoted 
  
    
      
        
          x
          
            i
            k
          
        
      
    
    {\displaystyle x_{ik}}
  
), upper bounds on inventory and so on. Inventory models can be based on different assumptions:<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a><a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>
</p>
<ul><li>Nature of <a href="/facts/Demand/mtb1m6F0">demand</a>: constant, <a href="/facts/Deterministic_system/oT1NvyHD">deterministically</a> time-varying or <a href="/facts/Stochastic/uMbgL94r">stochastic</a></li>
<li><a href="/facts/Cost/V1GX4vhw">Costs</a>: <a href="/facts/Variable_cost/FRwUSFp5">variable</a> versus <a href="/facts/Fixed_cost/BfGf8r1Q">fixed</a></li>
<li>Flow of <a href="/facts/Time/HceJKUGB">time</a>: <a href="/facts/Discrete_time/RR8wpS9e">discrete</a> versus <a href="/facts/Continuity_(mathematics)/n8CGmpos">continuous</a></li>
<li><a href="/facts/Lead_time/GDl9TkTW">Lead time</a>: deterministic or stochastic</li>
<li><a href="/facts/Time_horizon/AeaxrzFw">Time horizon</a>: finite versus infinite (T=+∞)</li>
<li>Presence or absence of <a href="/facts/Scrum_(development)/Ih3ufnVB">back-ordering</a></li>
<li><a href="/facts/Production_rate/pzN4FDct">Production rate</a>: infinite, deterministic or random</li>
<li>Presence or absence of quantity <a href="/facts/Discounts_and_allowances/02F6d4cc">discounts</a></li>
<li>Imperfect <a href="/facts/Quality_(business)/V3IzcDc3">quality</a></li>
<li><a href="/facts/Productive_capacity/FhyA894V">Capacity</a>: infinite or limited</li>
<li><a href="/facts/Product_(business)/l9P9fUJa">Products</a>: one or many</li>
<li><a href="/facts/Location_(geography)/fXux5RaJ">Location</a>: one or many</li>
<li>Echelons: one or many</li></ul>
<h3>Classic models</h3>
<p>Although the number of models described in the literature is immense, the following is a list of classics:
</p>
<ul><li>Infinite fill rate for the part being produced: <a href="/facts/Economic_order_quantity/kYN1N4MK">Economic order quantity</a> model, a.k.a. Wilson EOQ Model</li>
<li>Constant fill rate for the part being produced: <a href="/facts/Economic_production_quantity/S81pm10W">Economic production quantity</a> model</li>
<li>Orders placed at regular intervals: fixed time period model</li>
<li>Demand is random, only one replenishment: classical <a href="/facts/Newsvendor_model/tNlT1Jy5">Newsvendor model</a></li>
<li>Demand is random, continuous replenishment: <a href="/facts/Base_stock_model/Vgly0GBE">Base stock model</a></li>
<li>Continuous replenishment with backorders: <a href="/facts/(Q,r)_model/QfkMKgWB">(Q,r) model</a></li>
<li>Demand varies deterministically over time: <a href="/facts/Dynamic_lot_size_model/12UdzWM4">Dynamic lot size model</a> or Wagner-Whitin model</li>
<li>Demand varies deterministically over time: <a href="/facts/Silver%E2%80%93Meal_heuristic/tqvYEXpu">Silver–Meal heuristic</a></li>
<li>Several products produced on the same machine: <a href="/facts/Economic_lot_scheduling_problem/cxWjsRas">Economic lot scheduling problem</a></li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Inventory_control/170cokUp">Inventory control</a></li>
<li>Strategic inventory</li>
<li><a href="/facts/Safety_stock/Vxchl2l3">Safety stock</a></li>
<li><a href="/facts/Inventory_optimization/vYqTcbnu">Inventory optimization</a></li>
<li><a href="/facts/Inventory_management_software/uBjwd6Sn">Inventory management software</a></li>
<li><a href="/facts/Supply_chain_management/3bEP00vC">Supply chain management</a></li>
<li><a href="/facts/Warehouse_management_system/CHYtEAVx">Warehouse management system</a></li></ul>

<h2 id="further-reading">Further reading</h2>
<ul><li>International Journal of Inventory Research is an <a href="/facts/Academic_journal/SBI7Ue0q">academic journal</a> on inventory theory publishing current research.</li></ul>
<p>Classic books that established the field are:
</p>
<ul><li>Kenneth J. Arrow, Samuel Karlin, and Herbert E. Scarf: Studies in the Mathematical Theory of Inventory and Production, Stanford University Press, 1958</li>
<li>Thomson M. Whitin, G. Hadley, Analysis of Inventory Systems, Englewood Cliffs: Prentice-Hall 1963</li></ul>
<p>Many university courses in inventory theory use one or more of the following current textbooks:
</p>
<ul><li>Silver, Edward A., David F. Pyke, and Rein Peterson. Inventory Management and Production Planning and Scheduling, 3rd ed. Hoboken, NJ: Wiley, 1998. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-471-11947-4</li>
<li>Zipkin, Paul H. Foundations of Inventory Management. Boston: McGraw Hill, 2000. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-256-11379-3</li>
<li>Axsaeter, Sven. Inventory Control. Norwell, MA: Kluwer, 2000. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-33250-2</li>
<li>Porteus, Evan L. Foundations of Stochastic Inventory Theory. Stanford, CA: Stanford University Press, 2002. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8047-4399-1</li>
<li>Simchi-Levi, David, Xin Chen, and Julien Bramel. The Logic of Logistics: Theory, Algorithms, and Applications for Logistics Management, 2nd ed. New York: Springer Verlag, 2004. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-22199-9</li>
<li>Sethi, S.P., Yan, H., and Zhang, H., <a href="https://www.springer.com/us/book/9781402081231"><i>Inventory and Supply Chain Management with Forecast Updates</i></a>, in series International Series in Operations Research & Management Science, Springer, NY, NY, 2005.(310 pages - <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 1-4020-8123-5)</li>
<li>Beyer, D., Cheng, F., Sethi, S.P., and Taksar, M.I., <a href="https://www.springer.com/us/book/9780387716039"><i>Markovian Demand Inventory Models</i></a>, in series: International Series in Operations Research and Management Science, Springer, New York, NY, 2010. (253 pages - <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-71603-9)</li>
<li>Tempelmeier, Horst. Inventory Management in Supply Networks, 3rd. Edition, Norderstedt (Books on Demand) 2011, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-8423-4677-8</li>
<li>Snyder, Lawrence V. Fundamentals of Supply Chain Theory, 2nd ed. Hoboken, NJ: John Wiley & Sons, Inc, 2019. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-119-02484-2</li>
<li>Rossi, Roberto. Inventory Analytics. Cambridge, UK: Open Book Publishers, 2021. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-800-64176-1</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Zipkin Paul H., Foundations of Inventory Management, Boston: McGraw Hill, 2000, ISBN 0-256-11379-3 <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>W. Hopp, M. Spearman, Factory Physics, 3rd ed. Waveland Press, 2011 <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
</ol>

Inventory theory open-in-new

Inventory theory