
Research Article


Optimizing MultiProduct MultiConstraint Inventory Control Systems with Stochastic Replenishments 

Ata Allah Taleizadeh,
MirBahador Aryanezhad
and
Seyed Taghi Akhavan Niaki



ABSTRACT

Multiperiodic inventory control problems are mainly studied employing two assumptions. The first is the continuous review, where depending on the inventory level orders can happen at any time and the other is the periodic review, where orders can only happen at the beginning of each period. In this study, we relax these assumptions and assume that the periodic replenishments are stochastic in nature. Furthermore, we assume that the periods between two replenishments are independent and identically random variables. For the problem at hand, the decision variables are of integertype and there are two kinds of space and service level constraints for each product. We develop a model of the problem in which a combination of backorder and lostsales are considered for the shortages. Then, we show that the model is of an integernonlinearprogramming type and in order to solve it, a search algorithm can be utilized. We employ a simulated annealing approach and provide a numerical example to demonstrate the applicability of the proposed methodology.





INTRODUCTION
In multiperiodic inventory control models, the continuous review and the periodic review are the major vastly used policies. However, the underlying assumptions of the proposed models restrict their correct usage and utilization in realworld environments. In continuous review policy, the user has the freedom to act at anytime and replenish orders based upon the available inventory level. While in the periodic review policy, the user is allowed to replenish the orders only in specific and predetermined times.
The multiperiodic inventory control problems have been investigated in depth
in different research. Chiang (2003) considered a periodic review model in which
the period is partly long. Where discount has been considered, the costs associated
with his modeling were the purchasing, holding and fixed order costs. The important
aspect of his study was to introduce emergency orders to prevent shortages.
The emergency orders were replenished with a higher cost compared to the normal
ordering costs. He employed a dynamic programming approach to model the problem.
Mohebbi and Posner (2002) investigated an inventory system with periodic review,
multiple replenishment and multilevel delivery. They assumed that the demand
was a Poisson random variable, shortages were allowed and that the lost sale
policy could be employed. Feng and Rao (2007) considered a (R,nT) model in which
in the first level a stochastic demand entered the system and the total unsatisfied
demand were backordered at the second level. Ouyang and Chuang (2000) investigated
a (R,T) model in which the periodlength and leadtime were the decision variables,
demand was a stochastic variable and the service level was a constraint. Chiang
(2006) analyzed a periodic review problem in two cases of backorder and lost
sales and employed the (R, T) policy. Qu et al. (1999) investigated a
transportation model integrated with an inventory model with a periodic review
policy. They designed a multiperiodic model with several providers in which
the demand was stochastic and the planning horizon was finite. Eynan and Kropp
(2007) have propounded the assumption of stochastic demand and variant warehousing
costs on a periodic review system; while assuming nonzero leadtime and safety
stock. Bylka (2005) investigated a model with constraints on the amounts of
orders and backorder shortages in which the leadtime was constant and demand
was stochastic. By analyzing the changes in the leadtime and ordering cost,
they tried to define the optimized ordering time. For a continuous review inventory
model, assuming stochastic leadtime, Mohebbi (2004) considered demand a compound
Poisson random variable.
In summary, while there has been separate emphasis on the stochastic nature of the demands and leadtime, the realworld constraints of the systems have not been completely investigated. Specifically, there is no research in which both the demands and leadtime are considered to be probabilistic. Furthermore, some constraints have been partially studied, the decision variable has been considered integer and constraints such as budget and space have not been investigated.
Three main specifications of the proposed model of this research that have led to its novelty are the stochastic period length, the allowance of both multiproducts and multiconstraints and the fact that the decision variables are integer. By deploying these conditions simultaneously, the created model is different from the other models in the periodic review literature.
PROBLEM DEFINITION
Consider a periodic inventory control model for one provider in which the times
required to order each of several available products are stochastic in nature.
Let the timeperiods between two productreplenishments be identical and independent
random variables. The demands of the products are constant and distinct and
in case of shortage, a fraction is considered backorder and a fraction as lostsale.
The costs associated with the inventory control system are holding, backorder,
lostsales and purchase costs. There are several products and the warehouse
space and service level for each product are considered constraints of the problem.
Furthermore, the leadtime is assumed zero and the decision variables are integer
digits. We need to identify the inventory levels in each cycle such that the
expected profit is maximized. In short, the assumptions involved in the problem
are:
• 
Timeperiods between replenishments are independent and identical
random variables 
• 
There are several products 
• 
There are several constraints 
• 
The decision variables are integer 
• 
There are holding, shortage and purchase costs 
• 
A fraction of a shortage is backordered 
• 
Only one provider exists 
• 
The demands are constant 
• 
All of the purchased product will be sold 
MODELING
For the problem at hand, since the timeperiods between two replenishments are independent random variables, in order to maximize the expected profit of the planning horizon we need to consider only one period. Furthermore, since we assumed that the costs associated with the inventory control system are holding and shortage (backorder and lostsale), we need to calculate the expected inventory level and the expected required storage space in each period. Before doing this, let us define the parameters and the variables of the model.
The parameters and the variables of the model: For, i = 1, 2,....n,
let us define the parameters and the variables of the model as
R_{i} 
: : 
The inventory level of the i^{th} product 
T_{i} 
: 
A random variable denoting the timeperiod between two replenishments
(cycle length) of the i^{th} product 
f_{Ti}(t_{i}) 
: 
The Probability density function of T_{i} 
h_{i} 
: 
The holding cost per unit inventory of the i^{th} product in each
period 
π_{i} 
: 
The backorder cost per unit demand of the i^{th} product 
W_{i} 
: 
The purchasing cost per unit of the i^{th} product 
P_{i} 
: 
The sale price per unit of the i^{th} product 
D_{i} 
: 
The constant demand rate of the i^{th} product 
t_{Di} 
: 
The time at which the inventory level of the i^{th} product reaches
zero 
β_{i} 
: 
The percentage of unsatisfied demands of the i^{th} product that
is backordered 
I_{i} 
: 
The expected amount of the i^{th} product inventory per cycle 
L_{i} 
: 
The expected amount of the i^{th} product lostsale in each cycle 
B_{i} 
: 
The expected amount of the i^{th} product backorder in each cycle 
Q_{i} 
: 
The expected amount of the i^{th} product order in each cycle 
SL_{i} 
: 
The lower limit of the service level for the i^{th} product 
f_{i} 
: 
The required warehouse space per unit of the i^{th} product 
F 
: 
Total available warehouse space 
C_{h} 
: 
The expected holding cost per cycle 
C_{b} 
: 
The expected shortage cost in backorder state 
C_{l} 
: 
The expected shortage cost in lostsale state 
C_{p} 
: 
The expected purchase cost 
r 
: 
The expected revenue obtained from sales 
Z 
: 
The expected profit obtained in each cycle 

Fig. 1: 
Presenting the inventory cycle when T_{min}≤T≤t_{D} 

Fig. 2: 
Presenting the inventory cycle when t_{D}≤T≤T_{max} 
Inventory diagram: According to Ertogal and Rahim (2005) and considering
the fact that the timeperiods between replenishments are stochastic variables,
two cases may occur. In the first case the timeperiod between replenishments
is less than the amount of time required for the inventory level to reach zero
(Fig. 1) and in the second case, it is greater (Fig.
2). Figure 3 shows the shortages in both cases.
Single product model–back ordered and lost sales cases
Calculating the costs and the profit: In order to calculate the expected
profit in each cycle, we need to evaluate all of the terms in Eq.
1 (Ertogal and Rahim, 2005).
Based on Fig. 3, L, B, I and Q are evaluated by the following equations:

Fig. 3: 
Presenting shortages in two cases of compact back order and lost sales 
Presenting the constraints: As the total available warehouse space is
F, the space required for each unit of product is f and the upper limit for
inventory is R, the space constraint will be:
Since the shortages only occur when the cycle time is more than t_{D}
and that the lower limit for the service level is SL, then;
In short, the complete mathematical model of the single product inventory is:
s.t:
As introduced later, in the extending phase of the singleproduct model to the multiple product model, in addition to two cases of backorder and lostsales, a combination of backorder and lostsales is also needed.
Multiproduct modelback order and lost sales case: The singleproduct inventory model can be easily extended to a multipleproduct model as follows:
s.t:
In what follows, we consider two probability density functions for T_{i} and hence we develop two models.
T_{i} follows a uniform distribution: In this case the probability
density function of T_{i} is .
Accordingly, (9) will change to:
s.t:
T_{i} follows an exponential distribution: If follows an exponential distribution with parameter λ_{i}, then the probability density function of T_{i} will be f_{Ti} (t_{i}) = λ_{i}eG^{λiTi}. In this case, the model is shown below:
s.t:
Further, we will introduce a heuristic algorithm to solve the problem.
THE SOLUTION ALGORITHM
Since the models in (10) and (11) are integernonlinear in nature, reaching an analytical solution (if any) to the problem is difficult (Gen, 1997). As a result, in this section we will try to solve the problem by the stochastic search algorithm of simulated annealing.
Simulated annealing: Simulated Annealing (SA) is a local search method that finds its inspiration in the physical annealing process studied in statistical mechanics (Aarts and Korst, 1989). It is an effective and efficient method that produces good suboptimal solutions and has diffused rapidly into many combinatorial optimization areas (Kirkpatrick et al., 1994). An SA algorithm repeats an iterative neighbor generation procedure and follows search directions that improve the objective function value. While exploring solution space, the SA method offers the possibility of accepting worse neighbor solutions in a controlled manner in order to escape from local minima. More precisely, in each iteration, for a current solution x characterized by an objective function value f (x), a neighbor x' is selected from the neighborhood of x denoted N (x) and defined as the set of all its immediate neighbors. For each move, the objective difference Δ = f (x')!f (x) is evaluated. For minimization problems, x' replaces x whenever Δ≤0. Otherwise, x' could also be accepted with a probability . The acceptance probability is compared to a number RN 0 Uni (0, 1) generated randomly and x' is accepted whenever P>RN.
The factors that influence the acceptance probability are the degree of objective function value Δ degradation (smaller degradations induce greater acceptance probabilities) and the parameter T called temperature (higher values of T give higher acceptance probability). The temperature can be controlled by a cooling scheme specifying how it should be progressively reduced to make the procedure more selective as the search progresses to neighborhoods of good solutions. There exist theoretical schedules, which require infinite computing time guaranteeing asymptotic convergence toward the optimal solution. In practice, much simpler and finite computing time schedules are preferred even if they do not guarantee an optimal solution.
A typical finite time implementation of SA consists of decreasing the temperature T in S steps, starting from an initial value T_{0} and using an attenuation factor α, (0<α<1). The initial temperature T_{0} is supposed to be high enough to allow acceptance of any new neighbor proposed in the first step. In each step s, the procedure generates a fixed number of neighbor solutions N_{sol} and evaluates them using the current temperature value T_{S} = α^{S}T_{0}. The whole process is commonly called cooling chain or also markov chain. Adaptation of SA to an optimization problem consists in defining its specific components: A neighbor generation procedure, objective function evaluation, a method for assigning the initial temperature, a procedure to change the temperature and a cooling scheme including stopping criteria. These adaptation steps for our new adaptations of SA for the inventory model of (10) and (11) are described below.
A neighbor generation procedure: The neighbor generation is an important component of SA. It has to be designed to allow easy neighbor generation and fast calculation of Δ and guarantee accessibility for the entire solution space.
In this study, the initial solutions are generated in two groups. In the first group, the procedure randomly chooses the initial solution from the possible solution space. For the other groups of initial solutions the procedure employs some solutions of a Genetic Algorithm.
Evaluating the objective function: After creating each solution, the objective function should be evaluated to compare the j^{th} and the i^{th} solutions. In a maximization problem, if the objective function of the new solution (j) is bigger than the previous solution (i), then (j) will be accepted. Otherwise, by generating a random number the better solution will be selected.
Assigning the initial temperature: Temperature is one of the important parameters of the SA algorithm that affects the acceptance or nonacceptance of the objective function changes. The initial temperature should be chosen such that at the first level a big number of inappropriate solutions are accepted. In this way, more changes are possible and hence more solutions are explored. Furthermore, the time required for the process repetition along with the annealing process depends on the initial temperature. In this research, the large values of 1000, 2000 and 5000 are chosen for initial temperatures.
Changing the temperature: One of the primary aspects of the annealing
process in a SA algorithm is the range of temperature change. The temperature
plays an important role in the possibility of selecting a bad solution. On one
hand, when the temperature assumes a high value, a big number of bad solutions
are accepted, leading to selecting a local optimum point. On the other hand,
when the temperature is low, the probability of the solution to be a local optimum
is high. In this paper, we change the temperature of the SA algorithm based
on a geometric function shown in Eq. 12 with α = 0.9,
0.95 and 0.99.
Cooling scheme and stopping criterion: In a specific temperature of a SA algorithm, it is necessary to analyze the equilibrium state after a couple of renitence to see if the annealing process will be continued in that temperature or it will be stopped and moved to the next temperature. An epoch is the number changes that should occur in a specific temperature.
Different types of stopping criteria have been presented for different SA algorithms
in the literature. Some of them are:
• 
A lower bound on the final temperature 
• 
An upper bound on the total number of states 
• 
Reaching a solidification point based on the value of a function evaluated
on the value of the objective function and the temperature 
• 
An upper bound on the total number of exchanges accepted along the annealing
process 
• 
An upper bound on the total number of refused changes 
In this research, we select the first criterion and stop the algorithm after it reaches the final temperature T_{i}. Furthermore, we use 100, 200 and 300 for different values of N (t).
In short, the steps involved in the SA algorithm used in this research are:
(1) 
Choosing an initial solution i from the group of the feasible solutions
S. 
(2) 
Choosing the initial temperature T_{0}>0. 
(3) 
Selecting the number of iterations N (t) at each temperature. 
(4) 
Selecting the final temperature T_{F}. 
(5) 
Determining the process of the temperature reduction until it reaches
T_{F}. 
(6) 
Setting the temperature exchange counter n to zero for each temperature.
(Balancing process). 
(7) 
Creating the j solution at the neighborhood of the i solution. 
(8) 
Evaluating the objective function f = Z at any temperature. 
(9) 
Calculating Δ = f (j)!f (i). 
(10) 
Accepting the solution j, if Δ<0. Otherwise, generating a random
number RNUniform. If then selecting the j solution. 
(11) 
Setting n = n+1. If n is equal to N (t) then go to 12. Otherwise, go to
7. 
(12) 
Reducing the temperature. If it reaches T_{F} then stop. Otherwise,
go to 6. 
In order to demonstrate the proposed SA algorithm and evaluate its performances, in the next section we bring a numerical example used in Ertogal and Rahim (2005). In this example, two cases of the uniform and the exponential distributions for the timeperiod between two replenishments are investigated.
NUMERICAL EXAMPLES
Consider a multiproduct inventory control problem with eight products and
general data shown in Table 1. Table 2 and 3 show the parameters of the uniform
and the exponential distributions used for the timeperiod between two replenishments,
respectively. The total available warehouse space is 18000.
Table 2 : 
Data for uniform distribution 

Table 3 : 
Data for exponential distribution 

Table 4 : 
The parameters of the SA algorithm 

Table 4 shows different values of the parameters of the SA
method. In this research all the possible combinations of the parameters in
SA (N (t), T_{0} and α) method are employed and using the max(max)
criterion the best combination of the parameters has been selected. Table
5 and 6 shows the best result and best combination, respectively.
Furthermore, the convergence paths of the objective function values of the SA
algorithm in uniform and exponential distributions are shown in Fig.
4 and 5.
Table 5: 
The best result for R_{i} 

Table 6: 
The best combination of the SA parameters 


Fig. 4: 
The convergence paths of the best result in Uniform example
of the SA 

Fig. 5: 
The convergence paths of the best result in Exponential example
of SA 
CONCLUSION
In this study, a stochastic replenishment multiproduct inventory model was investigated. Two mathematical modeling for two cases of uniform and exponential distribution of the time between two replenishments have been developed and shown to be integernonlinear programming problems. Then, a metaheuristic solution algorithm of SA has been proposed to solve the integer nonlinear problems. Finally, two numerical examples were given to demonstrate the applicability of the proposed methodology.
Some recommendation on future works follow:
• 
As we may consider the model parameters fuzzy, the analysis may be performed
using fuzzy systems. 
• 
In addition to the SA algorithm, some other metaheuristic algorithms
like Genetic Algorithm, TabuSearch and AntColony optimization may be employed
to solve the integer nonlinear problems. 
• 
Some other probability density functions may be considered for the time
between replenishments. 
• 
The discount factor may be augmented to the problem. 

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