1
1
2
1. School of Management, Chongqing Jiaotong University, Chongqing 400074, China;
2. School of Management and Economics, University of Electronic Science and Technology of China, Chengdu 610054,
China control; genetic algorithm
(liumingwu2007@yahoo.cn)
Abstract - This paper considers an (s, Q) Markov queuing-inventory system with two classes of customers, ordinary and priority customers. As and when the on-hand inventory drops to the safety level s, arrival ordinary customers receive service at probability p. Firstly, the inventory level state transitions equation is set up. The steady-state probability distribution and the system’s performance measures which are used for the inventory control are derived. Next, a long-run average inventory cost function is established. An improved genetic algorithm for the optimum control policies is developed. Finally, the optimal inventory control polices and the sensitivities are investigated through the numerical experiments.
Keywords - inventory system; two demand classes; and modeled it as an M/M/S/S queueing system and derived expressions for the average cost. Savin et al.
(2000) provide an analysis of a multi-class environment in the rental business. As pointed out above models that incorporate two or multiple classes of demands considering possible lost sale for rejecting ordinary customers’ demands are rare in the literature. Melchiors et al. (2007) analyzed an (s, Q) inventory system with lost sales and two demand classes for the case of deterministic lead-times and obtained a formula for the total expected cost and presented a numerical procedure for optimization.
They based on this equivalence, and developed a model for cost evaluation and optimization under the assumptions of Poisson demand, deterministic replenishment lead time, and a continuous-review (Q, R) policy with rationing.
This paper presents a selected service discipline in an
I. INTRODUCTION
In practice, inventory systems usually satisfy
(s, Q) priority queuing-inventory system. When ordinary customers arrive, the system makes a decision whether or not to offer service. Our paper introduces a priority parameter p , which is different from the previous papers demands from more than one customer class, each of which may possess respective characteristics, such as affordable price, quality of service, etc. Typically, the inventory demand of customers with different characters on queuing-inventory systems with two classes of customers (Isotupa, 2006; Zhao and Lian, 2011). The parameter p is used for controlling the application of results in various service forms with priority. A classical example of the type of priority inventory demand that we wish to illustrate is booked orders and unscheduled orders. Booked orders, which are from long-term contracts and have much higher shortage lost, must be satisfied preferentially. Meanwhile unscheduled orders from the stochastic demand may bear lower shortage cost, can be lost. The real-life situations and extensive implications stimulate us to consider the queuingpriority. Another difference in this paper is the optimization problem. We setup a mix integer optimization problem and propose an improved genetic algorithm.
II. THE MODEL DESCRIPTION
We consider a single server queueing-inventory inventory system with two demand classes.
Queueing-inventory systems (Schwarz et al., 2006) have captured much attention over the last decades
(Berman and Kim, 1999; Berman and Sapna, 2001). A queuing-inventory system is different from the traditional queuing system because of the way the attached inventory influences the service. As pointed by Zhao and Lian
(2011), “ If there is no inventory on hand, the service will be interrupted. Also, it is different from the traditional inventory management because the inventory is consumed at the serving rate rather than the customers’ arrival rate when there are customers queued up for service”. As analyzed by Dekker et al.
（ 2002 ） a similar system with multiple demand classes, lost sales and Poisson demand system serving two classes of customers and subjecting to selected service. There are two classes of customers: priority customers and ordinary customers. The arrival process for both classes is state independent and each customer needs exactly one item from the inventory.
Priority customers arrive according to Poisson process with intensity

and ordinary customers arrive according
1 to Poisson process intensity

. The service discipline is
2 first-come-first-service (FCFS). And, the service will be finished instantaneously (Compared to the lead-time for the order, the customers’ order processing time can be omitted). The lead-time is exponentially distributed with parameter

. We assume that the replenishment is never

interrupted and there is at most one outstanding order at any time. We also apply the continuous review ( , )
 p
    
1
 p

2
P j
  j

1, 2 , s

1 policy as in Schwarz and Daduna (2006), Zhao and Lian
(2011), but with an additional selected service. As and when the on-hand inventory drops to a fixed level s (safety inventory level), an order for fixed Q (
 s ) units is placed. The condition Q
 s ensures that there is no perpetual shortage. Hence the maximum on-hand inventory is s Q . When the on-hand inventory is not less

P
   
1
 p

2
  
(5)
(6)
The above set of equations together with the normalizing condition written as ,
 j

0

1 determine the steady-state probability distributions uniquely. We solve the equations (1)-(6) by the means of recursive process, and get


1

1

 p

2


 j

1 

1
 p

2
P j

1, 2, , s
(7) than the safety level s , the two classes of customers arrived both can be served. But, when the on-hand inventory drops to the safety level s , arrival ordinary customers receive service at the probability p . These ordinary customers that do not receive service are lost.
Also, when the inventory is empty, both classes of customers are assumed to be lost.
III. THE STEADY-STATE PERFORMANCE
MEASURES
Let ( ),

0 be the on-hand inventory level at time t .


1
  s 

1
 p

2 1

2
P j
 
1, s

2, , Q
(8)







1


1

 p

2

 s
 
 
1 j
 
1, Q

2, , Q
 s


1
 p

2







2
P
From the model assumptions, the state space of ( ) is E

{0,1, , , , ,1

Q , ,

} . For the
Poisson input process and the exponential distribution lead-time, the inventory level state next period depends only on the current state and not on any past states. The inventory level process ( ) constitutes a Markov process on state space E . And define the steady-state probability distributions for ( ) as P j .
(9)
P


1
 p

2

 Q

 
1

2
 p
 
2
 


1



1
 p

2
 s


(10)
Inserting (10) in (7)-(9) respectively, we have the analytical steady-state probability distributions of the inventory level.
In the long run equilibrium, the steady-state probability distributions of the inventory level
( ) satisfy the following equations (1)-(6). The balance Let I denote the average inventory level. Using equations can be obtained by the fact that transition out of a state is equal to transition into a state for a Markov
I
 j

, we have

1 process. For example, let us consider a type inventory level state j that line in the range Q j Q s 1 . The equation is represented in Eq. (2). When j is in this range, there is no order pending, and then transition out this state can be only duce to either an ordinary demand arrival or a priority demand arrival. This fact represented on the left-


1





1
 p

2


1
 p

2
 s
 s

 s

1
 p
 
1

2
 
2

Q

1
 p

2

2



Q 2

2

2
  
1

2


1
 p



2



1
 p

2

2
P (0)



P (0) hand side of equation Eq. (2). Either an ordinary demand or a priority demand in state j

1 will cut down the 



Q


1
 p



2

 

1 p
 
1



2
2



P (0) inventory level by one unit, thus bring it to state j .
State j can also be reached from state j Q when a
(11)

 replacement arrives. The only two possible ways of reaching state j are reflected on the right side of Eq. (2).


 
1

2
2
 
    j

,

1, ,

 
1
Q s

1
2
 
 
. (1)
   
,
(2) p

2
Let us denote
R
 the mean reorder rate
  the mean shortage rates for the priority
1 customers
  the mean shortage rates for the ordinary
2 customers
  
2
 
  
2
  j s 1, , Q

1 (3)
The mean reorder rate, the mean shortage rates for the priority customers and the mean shortage rates for the
  
1

(4) ordinary customers, respectively, are written by

R
   
1

2
 
1



1
(12)

1
 
1
P
(13)

2
  p )

2
 s j

1

  p

) 1


  


1
 p

2 p

 s


1
2


P
 p

2
 p

 
2
P s

P
(14)
IV. INVENTORY CONTROL
A. Inventory cost formulation
This section, we will define an inventory cost function and discuss the analytical property. We establish a steady-state average inventory cost function per unit time, in which s and Q , as well as p are the decision variables. Let us denote the following cost parameters: h : the inventory holding cost per unit per unit time k : the fixed ordering cost per order c
1
: the purchase price per unit g
1
: the shortage cost per unit for the priority customers lost g
2
: the shortage cost per unit for the ordinary customers lost
The system incurs inventory holding costs, fixed ordering cost, purchase costs and costs of lost sales for possibly rejecting two classes of customer orders. Using the definitions of each cost component numbered up above, the total long-run average cost function per unit time is given by

, ,
  hI ( k )
 g
1

1
 g
2

2
 h



1



1
 p

2
 s



 s

1
 p

 
1

2

2  
Q
2 
2

2
 h



1


 h Q




1
 p

2

1
 p


 s
 

Q
 

1
 p

2

2

2

 

1 p
 
1



2
2





1
P (0)
 p

 
2
  
1

2


 P (0)

1
 p

2

2



P (0)

 g
2



1


1

 p


(1
 p

) 1



2


 s

P


1
 p

2



 s
 g

P p

 
2
P
B. Optimization problem
A mixed integer optimization model is constructed here to obtain the optimal inventory control policy which consists of priority rule and order policy. The mixed integer optimization model is presented below:
M in Cost
 

{ hI k )
 g

1 1
 g

2 2
}
(16)
Subject to: s Q 1
(17)
0 1
(18) s

0
(19)
, are integer variables
(20)
The objective function minimizes the inventory cost that is represented by (15). Constraint (17) is the assumption that there is no perpetual shortage. Constraint
(18) is the priority identification rule. E.q. (19) is the nonnegative constraint. Constraint (20) characterizes the order policy variables.
C. Algorithm designing
Suggested algorithms by Isotupta (2006) or Zhao and
Lian (2011) fail to solve our problem when certain variables are integer-valued. We adopt a real coded genetic algorithm genetic named MI-LXPM which is introduced by Deep et.al. (2009) for solving integer and mixed integer constrained optimization problems. MI-
LXPM algorithm is an extension of LXPM algorithm, which is efficient to solve integer and mixed integer constrained optimization problems. In MI-LXPM,
Laplace crossover and Power mutation are modified and extended for integer decision variables. Readers who want to read the MI-LXPM algorithm in detail may refer to
Deep et.al. (2009).
MI-LXPM algorithm can be put in practice on
MATLAB R2012a. In the following numerical examples, the initial population contains 30 individuals, crossover probability Pc=0.8, mutation probability Pm=0.01. The algorithm stops when the number of generations reaches the value of Generations (2000) or the weighted average change in the fitness function value is less than Function tolerance (1e-6).
V. NUMERICAL EXAMPLE
In this section, we present an ( , ) order policy with service discipline p to minimize the long-run average
(15)
Where, P
 
is given by (10). inventory cost function. In the following, we examine the effect of the selected service and investigate the

sensitivities of system parameters. The numerical results are summed up as follows.
(1) Case 1: The shortage lost cost of priority customers is twice over the shortage lost cost of ordinary customers, but shortage lost costs of the two classes of customers are small, g
1

20, g
2

10 . Let us consider the parameter

1


in a situation where
2
10,
 
5 . h

10, k

100, c
1

10,
From the table 1, the long-run inventory cost is increasing in accordance to

. The optimal reorder
2 point s

and reorder quantity Q
 are increasing in accordance to

. When
2
 
2
1,10 , the reorder point s
 
0 , the optimal p
 is an arbitrary value in[0, 1].
The system is indistinctive between priority service and no-priority service rule. The shortage rate of the two classes of customer is equal. But, the arrival rate of the ordinary customers increase to 100 or 1000, the optimal p

0 . That is to say the priority service rule will better the inventory control.
THE OPTIMAL CONTROL POLICY WITH THE CHANGE OF

2
T ABLE I
TWO SMALL SHORTAGE COSTS
( ,

, p

) Cost


2
I
WITH
1
10
(0,14,*)
(0,18,*)
256.2963
386.8182
6.4815
7.7727
100 (2,32,0) 1.4350e+03 10.8571
1000 (3,47,0) 1.0594e+04 6.1447
R
0.6790
0.9091
2.0798
4.0793
The optimal p


1
1.3580
1.8182
1.8487
2.4174

2

* means p is an arbitrary value in [0, 1] .
0.1358
1.8182
41.5966
815.8564
(2) Case 2. The shortage lost cost of priority customers is twice over the shortage lost cost of ordinary customers, but shortage lost costs of the two classes of customers are big, g
1

2000, g
2

1000 . Table 3 summarizes the numerical results as in Table 2. The optimal service rule is also no-priority service.
From Case 1 and Case 2, we can conclude that when the two classes of shortage cost are big, the optimal service rule is no-priority for the inventory control, even though the shortage cost of ordinary customers is obviously less than the shortage cost of priority customers.
THE OPTIMAL CONTROL POLICY WITH THE CHANGE OF

2
T ABLE II
TWO BIGGER SHORTAGE COSTS
( ,

, p

) Cost


2
WITH
I
1 (12,18,1) 392.4398 19.2982
10 (21,25,1) 623.7207 29.9926
100
1000
(103,104,1)
(911,912,1)
2.7976e+03
2.4341e+04
133.4361
1.1649e+03
VI. CONCLUSION
In this paper, we consider a Markov queueinginventory system with two classes of demands, (s, Q) replenishment policy, and subjected to an additional selected service discipline. The main contribution of this paper is the introduction an additional selected service discipline into the queueing-inventory system that endues the inventory management polices much more flexibilities. The current work can be generalized to the situation where demands arrival is not a Poisson process.
ACKNOWLEDGMENTS
This work is supported by Chongqing Science and
Technology Commission, Chongqing, China under grant cstc2011jjA30014.
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