OPTIMAL STRATEGY FOR A PERIODIC REVIEW INVENTORY SYSTEM WITH DISCOUNTED VARIABLE COST AND FINITE ORDERING CAPACITY

. We study a periodic review inventory system with finite ordering capacity, and assume the variable ordering cost would be discounted when the ordering quantity is the full capacity. Applying the concept of strong 𝐶𝐾 -concavity, we show that the optimal pricing and ordering strategy could be partially characterized by an ( 𝑆, 𝑆 ′ , 𝑝 ) strategy in four regions depending on the starting inventory level per period. Numerical experiments verify the proposed strategy.


Introduction
Limiting purchase quantity and providing discounted prices are two of the most common sales strategies.On the one hand, when the products sell well, which indicates that demand exceeds supply, suppliers are likely to sell goods by limiting purchase quantity.On the other hand, when the products are unsaleable, which indicates that supply exceeds demand, suppliers are likely to stimulate sales by providing discounted prices.In traditional business practices, suppliers will not adopt the two sales strategies simultaneously.
However, in the live streaming market, finite ordering capacity and price discounts have been perfectly unified.On live streaming marketing platforms in China such as Taobao Live and Tik Tok Live, suppliers usually provide limited quantity of products, aiming to promote products but not make profits, while live streaming stores in China, especially popular ones, can often attract hundreds of thousands, millions or even tens of millions of people watching the live stream at the same time.In order to promote their products to more audiences and give more people the opportunity to buy the goods, suppliers often limit purchase quantity for each buyer.At the same time, in order to show that the products sell well and many people are rushing to buy them, suppliers always provide low prices, which are much lower than prices offered by physical stores as well as traditional online stores.Moreover, many suppliers will provide quantity discounts in order to motivate each audience member to buy as many goods as possible so as to enjoy the quantity discounts.For example, one product is sold by the following rule: buy one at the original price, the second at half price, the third at CNY 0.1, and the fourth free.Sometimes, even if the quantity discount is given only when the buyers order up to the capacity limit, products will still be sold out quickly (which is the case we investigate in this paper).Therefore, on these live streaming marketing platforms, finite ordering capacity and price discounts can be unified.
Furthermore, we observe many other cases in the supply chain that finite ordering capacity and price discounts coexist.On the one hand, suppliers usually set a quantity limit on an order due to production, inventory or transportation capacity, or to avoid buyer dependence.On the other hand, the suppliers prefer large-volume orders, and would like to give buyers a discount when they purchase more than a specified number of products.For example, Xiaoxmi, a famous mobile phone manufacturer in China, often adopts "hunger marketing" strategies for its products; it restricts the upper bound of allocated quantity of products to every distributor, and also offers quantity discounts to them.When considering both finite ordering capacity and price discounts, for the buyer, is the classical (, ) policy still optimal?The answer is negative.
Several studies have been conducted to analyze and calculate the optimal procurement (inventory replenishment) policy of an inventory system with price discounts and finite supply capacity.Both Manerba and Mansini [1] and Tan and Alp [2] considered a procurement problem of a company that needs to purchase products from a set of capacitated suppliers.The suppliers offer total quantity discounts, which are independent of the ordering capacity.These models only consider a single period, and do not theoretically construct the optimal procurement strategy.
In this paper, we study a periodic review inventory system in which the ordering quantity is limited.We assume the variable cost would be discounted if the order quantity is full.The objective is to maximize the expected total discounted profit over the whole planning horizon.Supported by the concepts of strong concavity, we point out that the optimal pricing and ordering strategy for the system is partially characterized by an (,  ′ , ) policy in four regions of the starting inventory level per period separately.
Our model has a close relationship with the system in terms of setup cost and finite capacity.Several studies have been conducted to analyze the optimal strategy of an inventory system with setup cost and finite capacity.If there is no setup cost, Federgruen and Zipkin [3,4] pointed out that the optimal strategy for the system is regarded as the modified base-stock policy.However, when the setup cost is fixed, the optimal strategy is very complicated.Chen and Lambrecht [5] and Chen [6] have shown that the optimal ordering policy could be partially characterized in the form of − bands.Then Gallego and Scheller-Wolf [7] refined the policy between the bands into four regions by two thresholds  and  ′ .In Gallego and Toktay [8], the ordering quantity could only be either zero or the maximum capacity.Chao et al. [9] considered a periodic review inventory system with finite ordering capacity and setup cost.
Our research is also related to the studies on quantity-dependent variable costs, especially when the ordering cost functions are concave.Porteus [10] analyzed inventory systems with concave ordering costs.Fox et al. [11] studied a periodic review inventory system with two suppliers, one with a high variable cost but low fixed cost, and the other with a low variable cost but high fixed cost.In Chao and Zipkin [12], if the order quantity is greater than the contracted volume, the fixed cost could be incurred.Chen et al. [13] applied a new preservation property of quasi-K-concavity to study single-product periodic review inventory models with concave ordering cost.Caliskan-Demirag et al. [14] studied a periodic review inventory model in which the order quantity decides the fixed setup cost.
Finally, our work is partly related to the problem of the coordination of pricing and inventory replenishment.Whitin [15] was the first to study the newsvendor problem in which the demand depends on the price.Petruzzi and Dada [16] conducted a further study on the basis of Whitin.Thomas [17] studied a periodic review inventory system with finite horizon and fixed-ordering cost and assumed that the demand is price-dependent.Pekelman [18] considered a joint pricing and production optimization model in which the demand is linear.On the basis of Pekelman [18], Feichtinger and Hartl [19] extended the system into a model in which the demand is non-linear.Federgruen and Heching [20] considered the optimal pricing and inventory strategy for both the infinite and finite horizon problems, which could be considered as a base-stock list-price strategy.Polatoglu and Sahin [21] investigated a model which is similar to the one in Thomas [17], and Polatoglu and Sahin [21] studied a lost-sales system.
The rest of this paper is organized as follows.In Section 2, we present our model.Then, we provide the analytical results in Section 3. The numerical results are provided in Section 4. Finally, we present the conclusion.

Model
We study a periodic review inventory system, in which the ordering quantity per period cannot exceed .Assume there are  periods, from period 1 to period  .In each period, the variable ordering cost per unit is  when the ordering quantity is less than .Otherwise, the variable ordering cost per unit is  1 , where  1 < .And we assume the ordering leadtime is zero, which implies that, before the demand is realized, an order placed would be fulfilled immediately at the beginning of a period.We further assume that the random demand is price-dependent, which indicates that   (  ) is determined by   .Similar to Petruzzi and Dada [16] and Chen and Simchi-Levi [22], we consider the additive demand.Assume the selling price is   in period , the demand in the same period is given as follows where   is a random variable, and the mean value of   is zero.The selling price   in period  is defined over [, p], and it is a decision variable.(  ) is the average demand, which is decreasing in   .Therefore, when   decreases from p to , (  ) will increase from (p) to ().For simplicity, let d denote (p),  denote (), and  = () denote the inverse function of (), which indicates that () will decrease as  increases.Then the expected revenue is given as We assume () is a concave continuous function with a continuous inverse, therefore, the expected revenue () is also concave.For example, if (  ) is a linear decreasing function of   , then it is clear that () is concave in .
The sequence of events happening in one period is as follows: (1) review the inventory level; (2) release the replenishment order; (3) the replenishment order arrives, (4) set the selling price; (5) the random demand is realized, and (6) all costs are collected.
Suppose the inventory level on hand is   , and the inventory level on hand plus on order is   at the beginning of period , and unsatisfied demands are fully backlogged.Because the ordering quantity is constrained, we have At the end of each period, assume the inventory level is , a cost () is generated, which is a convex function of , and would be the inventory holding cost when  ≥ 0, and be the shortage cost when  < 0. Given that the expected demand for period  is   , the expected holding and shortage cost in period  is given as (  −   ).Then in period , the expected total cost can be written as where 1[] is the indicate function taking value 1 if statement A is true and 0 otherwise.Let   (  ) be the maximum expected total discounted profit from period  to the last period, when the inventory level before the replenishment order is   .Assume  be the discount factor per period,  ∈ [0, 1].The optimality equation is given as follows where the objective is to develop the optimal pricing and ordering strategy that maximizes the expected total discounted profit within the planning horizon.The terminal condition is   +1 () ≡ 0.
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Analytical results
To find the optimal strategy, we need to make some new assumptions and notations firstly.Note that the unit cost will be discounted if the order quantity is full, which means that the cost will be reduced by ( −  1 ) when the ordering quantity is the full capacity.For convenience, we assume ( −  1 ) = .Then, the optimization equation   (  ) will change to: Suppose   = −  − ,   = −  and   (  ) =   (−  − ).Then we will have Then we will introduce the definition of strong -concave which is crucial in finding the optimal strategy in this inventory system (see [9] for detail).
Definition.A function (.) : R → R is strong -concave if, for all  ≥ 0, > 0 and  ∈ [0, ], we have According to Chao et al. [9], there are some properties on strong -concave.E.g., -concavity is a special case of strong -concavity when  = ∞ and  = 0. Based on the definition of strong -concave, Lemmas 1 and 2 in Chao et al. [9] show that if   () is strong -concave, then so is where   () is concave.Furthermore, Theorem 1 in Chao et al. [9] shows that the optimality equation is also strong -concave, and the structure of the optimal inventory and pricing policy for each period is characterized by an (,  ′ , ) policy in four regions of the starting inventory level.Based on Lemmas 1, 2 and Theorem 1 in Chao et al. [9], it is easy to obtain the following lemma.We then characterize the optimal strategy of the system in the following theorem.
Theorem  −  >   , then the optimal ordering strategy is (i') place an order of a quantity of  if   <   ; (ii') no order is placed or place an order of a quantity of  if   ≤   <  ′  − ; (iii') no order is placed or place an order for an amount to bring the inventory level at least up to .This theorem can be proved based on the optimal policy of   (  ).We only characterize the optimal strategy of   (  ) in the scenario of  ′  + ≥   and   >  ′  , and the proof is similar in other scenarios.If  ′  − ≤   and   <  ′  −, the optimal ordering strategy of   (  ) is to place an order of a quantity of , which indicates  *  =  *  + .Note that   = −  −  and   = −  , it implies that when −  −  <  ′  − , the optimal ordering strategy is − *  = − *  , which means that the optimal ordering strategy is no order.Furthermore,   <  ′  −  implies that   > − ′  , therefore, when  ′  −  ≤   and   > − ′  , the optimal ordering strategy of   (  ) is no order.Moreover, the optimal pricing  *  (  ) depends on   .For convenience, we assume  ′  = − ′  and   = −  .Therefore, the optimal ordering strategy of   (  ) is no order when  ′  + ≥   and   >  ′  .

Numerical experiments
In this section, we conduct several numerical experiments for this periodic review inventory system, where the variable unit ordering cost is discounted when the ordering quantity is the full capacity.We further investigate the effects of various parameters on the control strategy and the profit.
To validate our analytical results, we consider a 4-period inventory problem, and set the discount factor  to 0.9.In period , the demand is given as   (  ) = (  )+  , where (  ) = 10−  , and   is a random variable, with the probability mass function  {  = 1} =  {  = −1} = 0.5,  = 1, . . ., 4. Therefore, () = (10 − ).Other parameters are given as follows: the unit holding cost ℎ = 2, the unit shortage cost  = 4, the ordering capacity  = 10, the variable unit ordering cost  = 3 for the case when the ordering capacity is less than , and the variable unit ordering cost  1 = 2.5 for the case when the ordering capacity is equal to .

Effects of the discounted variable ordering cost
In this subsection, we illustrate the optimal pricing and ordering strategy for different values of the discounted variable unit ordering cost  1 .The results are shown in Figures 1-3, where the -axis is the initial inventory level on hand .Here, the value of  increases from −10 to 15, with the increment of 1.In Figure 1, the -axis displays the maximum expected total discounted profit.In Figure 2, the -axis displays the optimal post-order inventory level .In Figure 3, the -axis displays the optimal average selling quantity ().
In Figure 1, when the initial inventory level on hand  is small, the maximum expected total discounted profit is increasing in .However, after the maximum expected total discounted profit reaches the maximum value, it will decrease in .Obviously, the profit is not concave in the initial inventory level .In addition, when  is large enough, the maximum total expected discounted profit will not change with respect to the discounted variable ordering cost  1 .That is because when  is large enough, the optimal ordering quantity will be always less than the full capacity.Therefore,  1 will not affect the profit.
In Figure 2, the optimal post-order inventory level  is increasing in the initial inventory level on hand  when  is small.This is because when  is small, the optimal ordering strategy is ordering the full capacity.Then  will decrease when  increases to   −  + 1, and the reason is that the optimal ordering strategy changes to at most up  ′  .Then  is still non-decreasing in  when  >   − , because the optimal ordering strategy is ordering at most up to  ′  when   −  <  ≤  ′  , and the optimal strategy is ordering nothing when  >  ′  .In Figure 3, when the initial inventory level on hand  ≤   − , the optimal average selling quantity () is increasing in .However, () will decline when  =   −  + 1.Then () keeps non-decreasing in  again when  >   − .Furthermore, compared with Figure 2, we find that when the optimal post-order inventory level  keeps constant, () also keeps constant, the reason is that the optimal selling price  *  () depends on both  and ().
In addition, from Figures 1 to 3, we could see that the maximum total expected discounted profit, the optimal post-order inventory level , and the optimal average selling quantity () are all non-increasing in  1 .First, it is obvious that the profit should be non-decreasing in the cost  1 .Second, when ordering the full capacity, there will be more cost saving if  1 is smaller, therefore, if the initial inventory level on hand is given, it is better to order the full capacity when  1 is smaller.Third, the average selling quantity () is non-decreasing in , therefore, () is also non-increasing in  1 .
Finally, we find that  ′ 1 is always equal to  1 in all these cases, and both  1 and  ′ 1 are non-increasing in  1 , which is consistent with our intuition, and the more the discount is, the more we prefer to order.These findings are shown in Figure 4.

Effects of the ordering capacity
In this subsection, we investigate the effects of ordering capacity  on the optimal pricing and ordering strategy.The results are shown in Figure 5-7, where the -axis is the initial inventory level on hand .Here, the value of  increases from −10 to 15, with the increment of 1.In Figure 5, the -axis displays the maximum expected total discounted profit.In Figure 6, the -axis displays the optimal post-order inventory level .In Figure 7, the -axis displays the optimal average selling quantity ().
In Figure 5, the maximum total expected discounted profit is also increasing in the initial inventory level  when  is small.Then it is decreasing in  after the maximum expected total discounted profit reaches its largest value.This observation is similar to what is shown in Figure 1.However, different values of  could make different effects on the maximum expected total discounted profit when  is small.
In Figure 6, when the initial inventory level on hand  ≤ −  − , the optimal post-order inventory level  is increasing in , because  =  +  always holds in this case.When  = −  −  + 1, the optimal post-order    inventory level  suddenly drops.After that  starts to be non-decreasing in  when  > −  − .And there also exist some quantities that buyers will never order.These observations are also similar with what is shown in Figure 2.However, different values of  could also make different effects on the optimal post-order inventory level  when  is small.In   it continues to be non-decreasing in .Furthermore, compared with Figure 6, () keeps constant when the optimal post-order inventory level  keeps unchanged.
In this subsection, we also find that  ′ 1 is always equal to  1 in all these cases. 1 and  1 are nondecreasing in .Moreover,  ′ 1 − , the upper limit for the full capacity order, is decreasing in , which indicates that the more we have to order to enjoy the discount, the less we prefer to order the full capacity.These findings are shown in Figure 8.

Conclusion
In this paper, we study a periodic review inventory system with finite ordering capacity.In addition, if the system orders the full capacity, the variable ordering cost will be discounted, and both the price and the ordering quantity are decision variables.The optimal pricing and ordering strategy is partially characterized by an (,  ′ , ) strategy in four regions, which depends on the initial inventory level per period.We find that  and  ′ turn out to be the same in the numerical results.Moreover, both  and  ′ are increasing in  1 and decreasing in .Future research directions include designing effective algorithms to calculate values of  and  ′ .

Figure 1 .
Figure 1.Maximum total expected discounted profit for different  1 .

Figure 2 .
Figure 2. Optimal post-order inventory level  for different  1 .
C. ZHU ET AL.

Figure 5 .
Figure 5. Maximum total expected discounted profit for different .

Figure 6 .
Figure 6.Optimal post-order inventory level  for different .

Figure 7 ,
when  = 10 and  = 12, the observations are similar as what is shown Figure 3.When the initial inventory level on hand  ≤ −  − , the optimal average selling quantity () is increasing in .At the point when  = −  −  + 1, () suddenly drops.Then () starts to be non-decreasing in  again when  > −  − .When  = 8, the observation has a little difference that the curve drops twice before
1. Given that   (  ) is -concave.Suppose   is the inventory level on hand, and   is the inventory level on hand plus on order at the beginning of period .The optimal pricing and ordering strategy of   (  ) is characterized by the optimal pricing  *  (  ) and two threshholds   and  ′  ,  *  (  ) depends on   and   ≥  ′  ≥ 0. If  ′  +  ≥   , then the optimal ordering strategy is given as (i) order nothing if   >  ′  ; (ii) order at most up to  ′  if   −  <   ≤  ′  ; (iii) order either capacity  or at most up to  ′  if  ′  −  <   ≤   − ; and (iv) order capacity  if   ≤  ′  − .Else if  ′  +  <   , then the optimal ordering strategy is given as (i') order nothing if   >   − ; (ii') order either capacity  or nothing if  ′  <   ≤   − ; (iii') order either capacity  or at most up to  ′  if  ′  −  <   ≤  ′  ; and (iv') order capacity  if   ≤  ′  − . (  ) depends on the post-order inventory level   and   ≤  ′  .If  ′  −  ≤   , the optimal ordering strategy is (i) place an order of a quantity of  if   <  ′  − ; (ii) place an order for an amount to bring the inventory level at least up to  ′  if  ′  −  ≤   <   ; (iii) no order is placed or place an order for an amount to bring the inventory level at least up to  ′  if   ≤   <  ′  ; and (iv) no order is placed if   ≥  ′  .Else if  ′ [9]of.Because   (  ) is  − , according to Theorem 1 in Chao et al.[9], the optimal strategy of   (  ) is characterized by the optimal pricing  *  (  ) and two threshholds   and  ′  , such that  *