An Optimization of ( Q , r ) Inventory Policy Based on Health Care Apparel Products with Compound Poisson Demands

Addressing the problems of a health care center which produces tailor-made clothes for specific people, the paper proposes a single product continuous review model and establishes an optimal policy for the center based on (Q, r) control policy to minimize expected average cost on an order cycle. A generic mathematical model to compute cost on real-time inventory level is developed to generate optimal order quantity under stochastic stock variation. The customer demands are described as compound Poisson process. Comparisons on cost between optimization method and experience-based decision on Q are made through numerical studies conducted for the inventory system of the center.


Introduction
The research starts from the case study of the health care center, a nonprofit organization which provides tailor-made apparel making services for the disabled and the old.With small-sized inventory capacity, the center cannot have too much cloth material in stock.Due to limited funding, it is necessary to reduce operation cost including inventory cost.Customer orders are stochastic with random arrivals.
From the case study, we summarize the problem as a single item continuous review system with limited inventory capacity and stochastic customer demands.With continuous review, we adopt classic (, ) policy as the inventory control policy: when the inventory position (stock on hand plus stock on order minus backorders) reaches the order point , an order is placed for the fixed amount , the batch size.In this paper, the problem is to determine optimal  in the objective of minimization of average cost including purchase cost and delivery cost which are classic terms defined by Hadley and Whitin [1].
For stochastic inventory system, (, ) policy is the most widely used.There has been no straight-forward solution for computing (, ) policy since Federgruen and Zheng [2] propose a simple and efficient algorithm for determination of an optimal (, ) policy.The algorithm is based on the observation of long-run average cost.Later Forsberg [3] shows how to exactly evaluate holding and shortage costs for a two-level inventory system with one warehouse and  different retailers.To implement such policy, the system needs the inventory space that can store (, ).However, in real practice, the storage space is usually limited due to high cost of land acquisition.Other research on the singleitem (, ) policy includes Hadley and Whitin [1], Lau et al. [4], Zheng [5], and the references cited therein.The issue we consider in this paper is important in the following aspects.Firstly, based on standard method, the model is optimized under the restriction of storage space.Although (, ) inventory problems have been studied and solved, the effort required to find better plans for problems in different cases and backgrounds may vary considerably.Secondly, to the author's knowledge, little attention has been given on the issue of space constraint in literature.Some research have been made for limit production capacity (see [6][7][8][9]).Others have dealt with resource constraint problems for newsvendor models (see [10,11]).Thirdly, also based on author's knowledge, there have been few even no intuitive expressions for a space-limited inventory system.Procedures for determining optimal  for most (, ) models are to solve iteratively (see [12]).The method is complicated in real execution.In Lau et al., 2002, easy-to-use expressions are presented for computing the exact "average on-hand inventory" level for a variety of lead-time-demand distribution forms.However, the expression of  is not applicable in minimizing long-run average system cost.Thus, in our paper, we generate methods not only helpful for academic analysis but reasonable and simple enough for business practitioner's execution.Last, we propose the inventory model with infinitesimal method to compute cost on real-time inventory level and assume customer demands as compound Poisson process.
The rest of the paper is organized as follows.In Section 2, we provide related literature of previous research on (, ) model in stochastic environment.Section 3 mainly describes the notation and preliminary studies.Detailed model formulation and derivations are also provided in Section 3. In Section 4, case based numerical analysis is conducted and comparisons are made.Concluding remarks and potential future studies are given in Section 5.

Literature Review
An inventory control system tells us how much to order and when to place the order.To propose a reasonable inventory control system improves operation efficiency.Hence, it brings about advantages like cost-saving, profit earning, and customer satisfaction.Under continuous review, businesses typically track inventory until it reaches a predetermined point of "low" holdings, whereupon an order is made to push holdings back up to a desirable level.(, ) continuous review control policies are mostly used for uncertain demand and lead time.
For stochastic models, the continuous review model can be classified according to whether the demands are discrete or continuous.In the case of discrete demands, Poisson demands in the continuous review models are studied in depth [13].Song and Zipkin [14] consider the case of Markov modulated Poisson demands, while Browne and Zipkin [15] study a model with continuous demand driven by a Markov process.In our research, we discuss, respectively, the optimal (, ) inventory models in continuous demand and discrete demand.
Though different inventory systems are proposed for stochastic situation (see [15][16][17][18][19][20]), (, ) control policies are popular for uncertain demand or random lead time in multiperiod.Song et al. [21] discuss the effect of lead time and uncertain demand in (, ) policy.Berk and Gürler [22] show a continuous review (, ) policy is reasonable good for a perishable inventory system with fixed shelf lives and study the operating characteristics of the system.
Exact analysis and cost evaluation on continuous review installation stock (, ) policy for identical or nonidentical retailers are presented, but with the restriction of items in low demand which is depicted by Poisson or compound Poisson (see e.g., [3,[23][24][25]).A simple heuristic algorithm is proposed to find an approximate near-optimal (, ) policy Inventory under (, ) policy.[26].Heuristic solutions have also been presented to (, ) policy (see [27,28]).
The methods which previous research provide are elaborate and brilliant; however, these algorithms are not easy to execute.This brings out an important aspect as pointed out by Alstrøm [29]; that is, in view of the evidence from a number of studies showing that firms' use of statistical inventory control lags far behind academic interest in the area.The determination of optimal values for the control variables, even in a very simple inventory control system, is a complex task and therefore not suited to practical implementation.Thus, we hope to find a solution easy to carry out for practical inventory control and meaningful to academic research.Reviewing on the research, in the next section we will formulate the problem and produce efficient solutions for (, ) inventory policies.

Overview and Preliminary.
The mathematical model is built in the context of a three-echelon supply chain with customer, the health care center, and supplier (see Figure 1).The health care center provides tailor-made clothes for the disabled or the old.Customer demands are depicted as compound Poisson process.It is a single item inventory system with purchase cost fixed.As profit is not first consideration for the health care center, the model aims to minimize average cost in an order cycle.The inventory is reviewed by (, ) policy continuously (see Figure 2).Lead time is constant.Two optimal policies are proposed for the inventory model.(1)  is determined by service level so that the safety stock is kept at a reasonable level.Optimal  is expressed through derivation.
(2) Genetic algorithm is developed to derive  and .We discuss the system in steady state.Demands that cannot be fulfilled in current state are backordered.
Unfixed customer orders, random order arrivals, stochastic order cycle, and nonlinear stock level increase the complexity of the model.We include ordering cost and inventory cost in the model to find the optimal order quantity so as to achieve minimum average cost on an order cycle.Inventory cost is derived based on real-time inventory level.

Decision Variable
: order quantity (to a supplier), unit per order.

Assumption
(1) the replenishment lead time  is constant (2) in an order cycle, there is no outstanding order at  = 0. Inventory level at  = 0 is  + .

Optimal
Policy with  Determined by Service Level.In this section, reorder point is predefined by service level .Let (⋅) represent the c.d.f. of the standard normal distribution with mean 0 and standard deviation 1. Define  as the solution to () =  and  as the reorder point for a cycle service level of  [30,31].The order arrivals conform to Poisson process while the customer demand  conforms to normal distribution,  ∼ (  ,  2  ).Thus the safety stock and reorder point are still determined by the following equations: problem formulation: min  (AC) , According to Federgruen and Zheng [2], Hadley and Whitin [1], and Sivazlian [32], in the steady state, IL = IP − .IP denotes inventory position and IL inventory level.In this paper, we compute inventory cost based on inventory level.
Thus according to infinitesimal method, during a unit time [,  + ], the expected inventory cost is The average cost on an expected order cycle is To find solutions to this cubic equation, assume Consider the constraints  +  ≤ .
Thus  2 ≥   /ℎ: From ( 16), Q is the extreme point of ( 8).However, it still needs to be compared with the two end points in (18) to obtain the minimum of expected average cost. Thus In this section, we predefine  by service level and obtain the expression of optimal .In the following part, we will relax the policy and derive the optimum of  and  with consideration of the p.d.f of .

Optimal
Policy on  and .If we consider the random variability of , then the average cost (AC) in ( 5) is Through the following generation, V () is obtained The conditional p.d.f of The joint probability function of ,   is Integrate on   to get p.d.f of Thus the proposition is proved.
Thus, (AC) in ( 20) can be written as Problem formulation: min  (AC) Traditional optimization method is not easy to find optimum  for (19) due to its complexity.Thus we will use genetic algorithm to reach the optimums., , ℎ, and  are selected as input parameters. (AC) in ( 25) is regarded as the objective function.The simulation processes can be illustrated as follows (see Figure 3).

Extension of the Model.
The previous model can be extended for discrete customer demands (see Figure 4).Assume probability distribution function of demands: (() = ). According We will also adopt genetic algorithm to obtain the optimal  and  for discrete demands.The procedures are the same as those in Section 3.2.Numerical analysis will be conducted in Section 4.

Numerical Analysis and Insights
To verify the solution, numerical analysis is given with real data obtained from the health care center from January to March, 2013.Table 1 shows the parameters of the mathematical models.Table 2 shows optimal  when  is determined by service level.Table 3 reflects the optimal results of genetic algorithm while Table 4 shows the optimization for discrete customer demands.
Table 5 is the policy determined by human being in real practice.The inventory manager of the center usually counts on his previous experience to propose the order quantity, while our optimization system is developed on account of   3 and  5, and difference can be found between real practice and optimization system.The expected average cost is generally reduced at around 17∼20% with the models introduced in the paper.The inventory manager is usually conservative and reorder point is higher than optimal .When customer demands are considered as discrete in Table 4, the cost is also less than that of real practice.The optimization models provide referenced  and  for the inventory manager to achieve cost minimization.

Insights Analysis.
Sensitivity analysis is conducted with variation of the input parameters in genetic algorithm to analyze their effect on the optimal results.Take the product of wheelchair raincoat for example.The increase of  increases the ordering cost and thus correspondingly has an increasing effect on the average cost (see Table 6).We find that  derived by genetic algorithm also increases.It can be explained that the larger  is, the less the ordering cost on each unit increases.From Table 7, the increase in  leads to increase in  and .The larger  means customer demands per unit time increases.To avoid the occurrence of stock-out,  and  both increase.Thus the average cost increases.
From Table 8, the increase in  brings about increases in  and average cost.When stock-out cost per unit time becomes larger, the shortage cost adds on.To avoid stock-out situation, the level of reorder point has to be raised.
From Table 9, the increase of ℎ increase inventory cost thus correspondingly has an increasing effect on the average cost. derived by genetic algorithm also increases.It can be explained that larger  offsets the increase in the inventory cost on each unit.

Conclusion and Future Work
An inventory control system tells us how much to order and when to place the order.Large amounts of research have been made on inventory policy and lots of optimal proposals have been raised for inventory models; however, investigation on the situation of limited inventory capacity is not much, especially under a stochastic environment.In this paper, how to find an optimal order quantity under certain constraints to minimize average cost including inventory and ordering cost is the key issue.We propose an inventory system with restriction on capacity for the case of health care center based on (, ) policy.Firstly, we derive the expression of optimal order quantity with polynomial time computational complexity, provided reorder point determined by service level.Secondly, without service level definition, genetic algorithm is proposed to generate optimal  and  for the capacitated inventory system.Thirdly, we derive a model extension on discrete demands.With comparisons to the results of experiencebased decision on  and , the optimal policies we provided lead to an average cost reduction by 17∼20%.
In this paper, we discuss the capacitated single item inventory system.However, the resource can be commonly shared across different items, rather than exclusively used in part by a specified item.Then, an optimization problem for multi-item (, ) policies with a constraint of such a resource becomes more complex and will be a direction for future research.
Parameter AC: average cost on an order cycle : lead time of material procurement ℎ: holding cost per unit per time unit : stock-out cost per time unit : ordering cost per procurement SS: safety stock  (): stock level (continuous random variable) at time   (, ): probability density function of stock level  at time ,  (() = )  (, ): cumulative distribution function of  (, )   : customer order quantity per order,   ∼ (  ,  2  ),  = 1, 2, . . . (): accumulated customer demand at time   (, ): probability density function of ,  (): customer order arrivals during [0, ] : order cycle between two adjacent customer order arrivals V (): probability density function of   (): the probability of material arrival at time  ( < ) : Service level.

Figure 3 :
Figure 3: The procedures of genetic algorithm in finding min ((AC)).

Figure 4 :
Figure 4: Inventory variation on discrete customer demands.

Table 1 :
Parameters of the model.

Table 2 :
Optimal policy of Q in Section 3.1.

Table 3 :
Optimal policy of  and  in Section 3.2 (genetic algorithm).

Table 4 :
Optimal policy for discrete customer demands.

Table 6 :
The Effect of variation of  on  and  in Section 3.2.

Table 7 :
The effect of variation of  on  and  in Section 3.2.

Table 8 :
The effect of variation of  on  and  in Section 3.2.

Table 9 :
The effect of variation of ℎ on Q and r in Section 3.2.