Application of queuing theory in inventory systems with substitution flexibility
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Abstract
Considering the competition in today’s business environment, tactical planning of a supply chain becomes more complex than before. In many multiproduct inventory systems, substitution flexibility can improve profits. This paper aims to prepare a comprehensive substitution inventory model, where an inventory system with two substitute products with ignorable lead time has been considered, and effects of simultaneous ordering have been examined. In this paper, demands of customers for both of the products have been regarded as stochastic parameters, and queuing theory has been used to construct a mathematical model. The model has been coded by C++, and it has been analyzed due to a real example, where the results indicate efficiency of proposed model.
Keywords
Inventory management Substitution flexibility Simultaneous ordering Stochastic demand Queuing theoryIntroduction
One of the challenges in supply chain management is to find optimal policy for inventory system, the main objective of inventory management is to balance conflicting goals like optimization of stock costs and shortage costs (Arda and Hennet 2006). Using flexible inventories is one of the ways to reduce inventory costs. Flexibility could be considered in different ways, for example, through using product substitution, postponement (TibbenLembke and Bassok 2005) and lateral transshipments (Herer et al. 2006).
In substitution systems, flexible stock (mostly more expensive) will be used only when regular (cheaper) item stockout (Deflem and van Nieuwenhuyse 2011). For instance, if inventory of regular product cannot satisfy its demand, a higher quality item can be used as a substitute inventory (Liu and Lee 2007).

Demands have been considered as stochastic parameters.

To prepare a comprehensive model, bilevel Markov process has been used.

All the steadystate equations have been solved in terms of one state.
This model can be applied probably for inventory systems where demand is uncertain and two way substitutions can be used. For example, some items of dairy inventories have mostly stochastic demand, and some of them can use substitution with each other.
The reminder of this paper is organized as follows. In “Literature review”, a brief literature review has been presented. In “Model development and analysis”, first we represent a mathematical model for an inventory system with substitute products. The model is validated and some numerical examples are tested in “Solving approach”. We conclude our study in “Numerical results”.
Literature review
There are lots of researchers that worked on inventory models. For instance, AriaNezhad et al. (2013) proposed a two echelon system for perishable items in supply chains, and used a case study to analyze their model.
Generally, in a supply chain, most of the parameters are not deterministic, for this reason, some researchers used queuing theory to construct a mathematical model. In this area, there are some who prepared a model for stochastic demand and some who prepared a model for stochastic lead time. Parlar (1996) presented an inventory model which was combined with queuing theory to consider demand and lead time stochastic parameters. Hosseini et al. (2013) considered stochastic lead time, and developed a multiobjective pricinginventory model for a retailer, where their main objective was to maximize retailer’s profit and service level. Seyedhoseini et al. (2014) considered poison demand for customers in a crossdocking problem, and prepare a stochastic model.
For better modeling of stochastic environment, some researchers used queuing theory, for example, Ha (1997) considered poison demand and exponential production times for a singleitem maketostock production system. He proposed an M/M/1/S queuing system for modeling the system. Arda and Hennet (2006) analyzed inventory control of a multisupplier strategy in a twolevel supply chain. They considered random arrival for customers and random delivery time for suppliers, and represented their system as a queuing network. Isotupa (2006), considered a lost sales (s, Q) inventory system with two customer groups, and illustrated the model by Markov processes.
Babai et al. (2010) considered demand and lead time stochastic and analyzed a singleechelon singleitem inventory system by means of queuing theory. Considering effectiveness of queuing theory in inventory problems, we also used queuing theory to develop our model. ToktasPalut and Ülengin (2011) coordinated the inventory policies in a twostage decentralized supply chain, where each supplier has been considered as an M/M/1 queue and the manufacture has been assumed as a GI/M/1 queue.
Alimardani et al. (2013) applied continuous review (S−1,S) policy for inventory control and supposed a biproduct threeechelon supply chain which is modeled as an (M/M/1) queue model for each type of products offered through the developed network. In addition, to show the performance of the proposed biproduct supply chain, they also considered a network including two (M/M/1) queue for each type of products.
Some researchers studied substitution flexibility. For example, Bayindir et al. (2007) consider a oneway substitution system with two products which uses S−1, S policy. They use a twodimensional Markov process to develop the model, where the objective of their research was to find the optimal order up to levels. Liu and Lee (2007) proposed three different policies to use oneway substitution, and developed an inventory system with backlogs.
Olssen (2010) considered a continuous review inventory system where oneway lateral transshipment is allowed. Nagarajan and Rajagopalan (2008) dealt with a twoproduct problem, where substitution has been assumed for both of the products. Bahri and Tarokh (2012) presented a coordinated seller–buyer supply chain model in two stages, which is called Joint Economic Lot Sizing (JELS) in the literature. They assumed that the delivery lead time is stochastic and follows an exponential distribution and delivery activities consist of a single raw material. Tan and Karabati (2013) proposed an inventory management that incorporates the effects of stockoutbased dynamic substitutions.
Deflem and van Nieuwenhuyse (2011) presented an approach to analyze twoitem periodic inventory system with oneway substitution flexibility, where the objective function was to minimizing the expected purchasing costs, holding costs, shortage costs and adjustment costs. Ye (2014) dealt with the problem of inventory management and simultaneously horizontal (equivalently interbrand) and vertical (equivalently intrabrand) substitution.
Considering great literature of stochastic inventory we prepare a table in which research on substitute flexibility and stochastic inventory have been analyzed.
Classification of papers in stochastic inventory and substitution flexibility
Authors  Shortage  Lead time  Demand  Replenishment policy  Substitute flexibility  

Lost sales  Back order  One way  Two way  
Parlar (1996 )  ✓  Stochastic  Stochastic  (R, Q)  
Isotupa (2006)  ✓  Stochastic  Stochastic  (R, Q)  
(Arda and Hennet 2006)  ✓  Stochastic  Stochastic  (S−1, S)  
Boute et al. (2007 )  ✓  Stochastic  Stochastic  (T, S)  
Olssen (2010 )  ✓  ✓  Deterministic  Stochastic  (R, Q), (S−1, S)  
Hill et al. (2007 )  ✓  Deterministic  Stochastic  (S−1, S)  
Hannet and Arda (2008 )  ✓  Stochastic  Stochastic  (S−1, S)  
Teimoury et al. (2010 )  ✓  Stochastic  Stochastic  (R, Q)  
ToktasPalut and Ülengin (2011)  ✓  Stochastic  Stochastic  (S−1, S)  
Babai et al. (2010 )  ✓  Stochastic  Stochastic  (S−1, S)  
Tili et al. (2012 )  ✓  Deterministic  Stochastic  (T, s, S)  
Bahri and Tarokh (2012)  ✓  Stochastic  Deterministic  (R, Q)  
Alimardani et al. (2013)  ✓  Stochastic  Stochastic  (S−1, S)  
Guerrero et al. (2013 )  No shortage  Deterministic  Stochastic  (T, s, S)  
Yu and Dong (2014)  No shortage  Deterministic  Stochastic  (R, Q)  
Baek and Moon (2014)  ✓  Stochastic  Stochastic  (R, Q)  
Deflem and van Nieuwenhuyse (2011)  Combination of both models  Deterministic  Stochastic  (T, S)  ✓  
Ye (2014)  ✓  Deterministic  Stochastic  (R, Q)  ✓  
Ahiska and kurtul (2014)  Combination of both models  Deterministic  Stochastic  (T, S)  ✓  
Salameh et al. (2014)  ✓  Deterministic  Deterministic  (R, Q)  ✓  
Krommyda et al. (2015)  ✓  Deterministic  Deterministic  (R, Q)  ✓  
Our model  ✓  Stochastic  Deterministic  (R, Q)  ✓ 
Model development and analysis
As it has been demonstrated in Fig. 1, states have been put into three sets, set A and set C represent the states which have only one type of inventories and the inventory level will be reduced if a demand for each kind of products inters to the system. Set B represents the states which have inventories for both kinds of products, and no substitution happens.
To prepare a comprehensive model, four lemmas have been proposed to calculate steadystate probabilities in terms of probability of state \( (Q_{1} ,Q_{2} ) \). Henceforth, let \( \pi_{i,\,j} \) denote steadystate probability of state (I, J), \( D_{1} \) denote demand rate for typeone product, and \( D_{2} \) denote demand rate for typetwo product.
Lemma 1
Proof
Arrival flow for this network only goes through node \( (Q_{1} ,Q_{2} ) \). Consequently, each flow goes to state (I,J) comes from state \( (Q_{1} ,Q_{2} \)). Considering flows only move to right or top, each path between node \( \left( {Q_{1} ,Q_{2} } \right) \) and node (I,J), which has I1 lateral movement and J1 vertical movement consists a flow. Also flows only go through these paths, so \( {\text{IF}}_{i,j} \) is sum of flows that go through these paths and could be calculated by Eq. 1.
Lemma 2
Proof
To calculate steadystate probabilities of set A, lemma 3 has been proposed.
Lemma 3
Proof
Considering Eq. 9 and lemma 2, lemma 3 is proved.
Lemma 4
Proof
State of \( \pi_{1,j} \) changes to \( \pi_{0,j} \) by probability of \( \frac{{D_{1} }}{{D_{1} + D_{2} }} \), and then \( \frac{{D_{1} \cdot j}}{{D_{1} + D_{2} }} \) substitution inventories must be used to satisfy demand of product 1. Consequently, Eq. 11 calculates substitution amount for product 1 and similar to product 1, Eq. 12 calculates substitution amount for product 2.
Parameters
 \( h_{1} \)

Holding costs for product 1
 \( h_{2} \)

Holding costs for product 1
 \( A_{1} \)

Ordering cost for product 1 for independently ordering
 \( A_{2} \)

Ordering costs for product 2 for independently ordering
 \( A \)

Ordering costs of simultaneous ordering
 \( D_{1}^{{\prime }} \)

Average demand for inventory of type one in a specific decision period
 \( D_{2}^{{\prime }} \)

Average demand for inventory of type two in a specific decision period
 \( c_{1} \)

Replacement costs for product 1
 \( c_{2} \)

Replacement costs for product 2
Variables
 π _{i,j}

Steadystate probability of state (I, J)
 Q _{ 1 }

Quantity of ordering for product one
 Q _{ 2 }

Quantity of ordering for product two
The objective function is composed of five sections. The first two sections are for calculating holding costs, third section calculates ordering costs. Forth section and fifth sections are to calculate substitution costs.
In this system, when each product is ordered independently, substitution will not used, and optimal ordering quantity for product 1 and 2 can be calculated by \( \sqrt {\frac{{2 \cdot A_{1} \cdot (D_{1}^{{\prime }} )}}{{h_{1} }}} \) and \( \sqrt {\frac{{2 \cdot A_{2} \cdot (D_{2}^{{\prime }} )}}{{h_{2} }}} \).
Solving approach
Numerical results
Example
c _{2}  c _{1}  D _{2}  D _{1}  \( D_{2}^{{\prime }} \)  \( D_{1}^{{\prime }} \)  A  A _{2}  A _{1}  \( h_{2} \)  \( h_{1} \) 

20  20  10  5  200  100  30  30  30  8  10 
Comparison between using substitution and independently ordering
Using substitution  Independently ordering  

Product 1  Product 2  Product 1  Product 2  
Ordering quantity  10  37  24  38 
Costs of inventory system  405  554  
Substitution costs  33  0 
Decision for this system, can be influenced by different parameters. For that matter, this section has analyzed the model due to different parameters. For the previous example, Fig. 2 illustrates sensitivity of the model due to A. Decision of using substitution or not depends on value of A. For this example, if A be less than 57 it would be affordable to use substitution, otherwise it can cause more costs than independently ordering. Consequently, it can be inferred that decision of independently ordering or simultaneous ordering (using substitution) are dependent on parameter of A.
Real example
The proposed model can be appropriately used for Dairy supply chains. KALE Company produces Dairy products, and this company has two factories located in Amol and Karaj cities, and also 25 cities consisting majority of KALE customers.
In this supply chain, 25 retailers exist, where 18 retailers are for KALE and the other 7 retailers are acting in a decentralized supply chain. For most of these retailers, transportation is more than 3 h and there are only five retailers near KALE factories, which have less transportation time than 3 h. For this reason, we only considered one retailer which has less transportation time than the other retailers. Ahamd Abad retailer is located at Tehran city and it acts in a centralized supply chain with KALE.
KALE has some characteristics that make it suitable for our model. First its production rate is high, and it has a small transportation time for Ahmad Abad retailer, so we could consider lead time as an ignorable parameter. On the other hand, demand of Ahmad Abad is high. For this reason, we considered poison distribution for demand of each retailer.
Comparison between using substitution and independently ordering for cream cheese and Amol cheese
Using substitution  Independently ordering  Current policy in company  

Product 1  Product 2  Product 1  Product 2  Product 1  Product 2  
Ordering quantity  1,520  3,640  3,330  3,560  2,100  21,000 
Costs of inventory system  1,513  2,633  2,124 
Comparison between using substitution and independently ordering for lowfat Tetra milk and lowfat Manshori milk
Using substitution  Independently ordering  Current policy in company  

Product 1  Product 2  Product 1  Product 2  Product 1  Product 2  
Ordering quantity  1,330  2,640  2,630  2,780  1,850  1,850 
Costs of inventory system  1,851  3,210  2,520 
Comparison between using substitution or independently ordering IML yogurt and Ps yogurt
Using substitution  Independently ordering  Current policy in company  

Product 1  Product 2  Product 1  Product 2  Product 1  Product 2  
Ordering quantity  2,160  1,210  2,300  2,120  1,450  1,450 
Costs of inventory system  1,100  1,950  1,760 
As demonstrated in these tables, current policy is to order in same quantity for each kind of substitute products. Using this policy, their costs are lesser than when they use independent ordering. But using substitution flexibility they can improve inventory costs significantly, where major reason is decrease in ordering costs. Using substitution, overall ordering quantity has been increased, where 1794 items have been increased for cheese, 270 items increased for milk, and 470 items for yogurt; however, 611,000,000 Rials for cheese, 669,000,000 Rials for milk and 660,000,000 Rials for yogurt would diminish.
Conclusion
In this research, an inventory system with two substitute products with ignorable lead time and stochastic demand has been considered, and by means of queuing theory a mathematical model has been proposed. For this system, steadystate equations have been solved and all of the steadystate probabilities have been calculated in terms of \( \pi_{{Q_{1} ,Q_{2} }} \).
In this paper, the model has been analyzed due to different parameters and their behaviors have been discovered. Using substitution requires simultaneous ordering and it can be compared with the situation when different items use independent ordering. It is clear that, sometimes decision of using substitution and simultaneous ordering is not affordable and it depends on values of problem parameters, so we analyzed this too.
We also prepared a real example, which is for Kale Company. Inventory costs of this Company for three kinds of substitution products have been analyzed, and effectiveness of the model has been revealed.
For future studies, this research can be extended by considering substitution stock control system for more than two products with different substitution relations, this may increase complexity of the problem but the model would become more realistic. Another extension of this research is possible by considering rate of corruption for perishable inventories.
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