Effects of a quantity-based discount frame in inventory planning under time-dependent demand: A case study of mango businesses in Bangladesh

1.1 Literature review

According to economists and health specialists, consuming seasonal fruits or vegetables is beneficial both from a health perspective for consumers and from an economic perspective for local producers and consumers. Furthermore, Li et al. (2021) revealed that perishable foods account for nearly half of the total stocked items in grocery sales in the United States and, therefore, grocery companies place increasing importance on proper inventory planning for perishable food items. An important and unavoidable characteristic of these perishable foods is deterioration. Because of their storage environment and physical structure, the deterioration of many perishable foods commences late from the storing moment and these items are recognized as non-instantaneous or non-immediately deteriorating items. Wu et al. (2006) described, for the first time, the best inventory planning for non-immediately deteriorating items. Rabbani et al. (2015) discussed a company’s integrated replenishment and marketing strategies for non-immediately deteriorating items, in which an exponential discount is offered on the selling price after deterioration appears. Ai et al. (2017) determined the optimal inventory planning for a company replenishing multiple non-immediately decay items where the consumption rates of all items are constant. To reduce the deterioration cost by slowing down deterioration rates, Li et al. (2019) found the optimal preservation investment cost for non-immediately decay items. Later, Khan et al. (2020) investigated the impacts of disparate deterioration beginning time due to different storage environments in a two-warehouse system on a company’s total. Duary et al. (2022) derived the best selling price decision for a company dealing with inventory problems for non-immediate deteriorating items under a prepayment mechanism imposed by the supplier. Recently, Khan et al. (2022a) analytically characterized when a company’s optimal replenishment policy for non-immediately perishable items involves the effect of deterioration. In the studies described above, no one considers the effects of stocking duration on item consumption rates, but storage duration has a significant effect on consumer demand, especially for all food items.

When a company deals with perishable food items, its inventory planning greatly depends on the reasonable storage duration of the items. Prasad and Mukherjee (2016) studied the resultant consequences of storage duration and currently available stock volume on the consumption rate and then, analyzed the best inventory planning for a decay item. Adopting a decreasing (exponentially) client demand with items’ storage durations along with the impact of selling price reversely on demand, Kumar et al. (2016) further discussed the consequence of time on a company’s overall finances and inventory planning for decay items. Dobson et al. (2017) assumed that consumers evaluate the quality of items by comparing the lifetime and storage time of the items, and, therefore, the demand falls linearly with the age of the items. Integrating a linearly increasing unit holding cost with storage time, Pervin et al. (2018) discussed an inventory planning problem for decay items where the demand also changes linearly with the storage time of the items. To determine the consumption rate of items more accurately, San-José et al. (2019) replaced the linear storage time-dependent demand with a power form and then obtained the best inventory plan analytically. In addition, Cárdenas-Barrón et al. (2021) further investigated the impacts of time on demand in power form after allowing a stock-out scenario. Khan et al. (2022b) found the best inventory policies for both positive stock-ending and stock-out scenarios for a non-immediate decay item when demand is interlinked with the item’s storage duration. Recently, Kumar et al. (2023) further investigated the best replenishment strategy under a time-associated demand pattern by adopting a fractional differential equation approach.

Most seasonal foods have a short shelf life, and therefore, suppliers are very cautious about selling these items in the early stages. For that reason, suppliers of seasonal food items allow several concessions on the unit acquisition cost based on order quantity. Taleizadeh and Pentico (2014) discussed the consequences of an all-unit rebate scheme on a retailer’s inventory decision by adopting a constant demand rate. Again, Taleizadeh et al. (2015) further investigated the results of an incremental rebate scheme on a retailer’s inventory planning. However, both of these studies were conducted at a constant consumption rate. Replacing constant demand with a variable price-sensitive consumption rate, Alfares and Ghaithan (2016) found the best pricing and inventory planning strategies for a retailer under an all-unit rebate frame. Allowing a discount for only imperfectly produced items instead of all items, Cunha et al. (2018) determined the best inventory planning for a production system. Qiu and Lee (2019) studied rail transportation pricing policy under a sole breakpoint-dependent rebate scheme. Shaikh et al. (2019) found the best inventory planning for a company under an all-unit rebate frame for decay items when the demand varies with warehouse stock levels. Taking into consideration the stored items’ maximum lifetime on deterioration, Khan et al. (2019) investigated the consequences of an all-unit rebate frame on both pricing and inventory decisions. Lesmono et al. (2020) described an inventory procedure for a retailer dealing with multiple perishable items under multiple all-unit rebate schemes. Afterward, Khan et al. (2021) proposed a quantity-based integrated advance payment, delay payment, and unit acquisition price scheme and found the best inventory planning for a company. Recently, Khan et al. (2022c) found the best pricing and inventory strategies for a deteriorating item under a purchased quantity-sensitive rebate scheme on the unit acquisition price. Rahman et al. (2022) further scrutinized the significance of an all-unit concession arrangement on the optimal inventory policy in the interval environment. All the above works on the rebate scheme for the unit acquisition price focused on constant, price-dependent, or stock volume-dependent demand. However, the optimal inventory planning for seasonal foods with stocking period-sensitive demand and order-based rebates remains unexplored.

Due to supply disruptions or demand fluctuations, inventory planners sometimes face stock-out situations when customers may go elsewhere for their needs or wait for new items to arrive. If all the customers wait for their needs, then the situation is referred to as ‘complete backordering’ in inventory literature. Prasad and Mukherjee (2016) formulated a deterministic inventory planning problem for time-varying demand when shortages are completely backlogged. San-José et al. (2018) further investigated the complete backordering situation under non-linear stocking duration sensitive demand. Some mentionable recent studies in complete backordering consideration in the literature are: San-José et al. (2020), Lin (2021), and De-la-Cruz-Márquez et al. (2021). In addition, if a fraction of all customers move elsewhere during a stock-out situation and the remaining portion of the customers wait for new items to arrive, then the situation is referred to in the inventory literature as ‘partial backordering’. Taking into consideration a fixed backlogging rate, Taleizadeh and Pentico (2014) analyzed the best inventory policy for a company under an order-based rebate scheme. In this investigation line, some mentionable studies are as follows: Taleizadeh et al. (2015), Lin (2018), Gao et al. (2019), Taleizadeh et al. (2020), and Rahman et al. (2021). On another hand, by adopting a variable backlogging range with clients’ waiting times, Khan et al. (2019) examined the best inventory policy for a company dealing with maximum lifespan-related deteriorating items under an order based rebate scheme. With variable backlogging rate, some mentionable studies are as follows: Panda et al. (2019), Pakhira et al. (2020), Xu et al. (2021), Alshanbari et al. (2021), and Manna et al. (2022).

Though a good number of research works have investigated the effects of order-based rebate facilities on inventory planning, only two studies (Shaikh et al., 2019; Khan et al., 2019) among the research works described above have explored optimal inventory planning for a deteriorating item. It is noteworthy that neither of these (Shaikh et al., 2019 ; Khan et al., 2019) considered the significance of storage duration on item consumption rates, even though consumption of seasonal food items is highly dependent on storage duration. This is a huge research gap in inventory planning for seasonal foods because seasonal food suppliers often allow discounts on unit acquisition costs based on order volume. The main contributions of the study are:

• The first contribution is that this research addresses a research gap in the field of inventory planning for deteriorating items, specifically focusing on seasonal perishable food items.

• The study explores the significance of storage duration on item consumption rates, which is often overlooked in previous research.

• It investigates the impacts of a rebate scheme on unit acquisition price for non-immediately deteriorating items, considering the interplay between stocking period and consumption rate.

• The research provides decision makers with proper inventory planning strategies for seasonal perishable food items under an order-based rebate scheme.

• The introduction of a real example of mango businesses in Bangladesh adds practical relevance and allows for verification of the proposed inventory procedure.

The rest of the study is outlined as follows: Section 2 states the inventory procedure briefly, along with the required notations and assumptions. In Section 3, models are formulated for both with and without stock-out scenarios, whereas Section 4 exposes all the necessary analytical outcomes for ensuring the existence of the optimal solution for both models. Integrating all the analytical outcomes, Section 5 presents two solution algorithms. Section 6 represents a real case study and two additional numerical examples. The sensitivity analysis of the best inventory planning is exposed in Section 7, while several policy directions to manage inventories economically are pointed out in Section 8. Finally, Section 9 highlights the conclusion along with some salient insights for the inventory planner.

2.1 Notation

2.2 Hypothesis

• (i) The consumption rate $D(t)$ of the stored foods increases over time as they ripen and does not increase when deterioration appears. The demand function is given by

When the demand parameter $b$ and $\mu$ are zero, the demand becomes constant over the cycle.

• (ii) The number of deteriorated items depends upon the current stock volume, and the rate of deterioration is $\theta$ where $0<\theta <<1$ .

• (iii) Lead time is too insignificant to consider while the replenishment rate is infinite.

• (iv) The problem is investigated for an infinite planning horizon.

• (v) When the company faces the stock-out condition, the backlogging rate is interrelated with clients’ waiting times and is expressed as $B(t)=e^{-\sigma (T-t)}$ , where $\sigma ⩾0$ and $t_1⩽t⩽T$ . If $\sigma =0$ , then a complete backordering case raises.

• (vi) Using $n-$ number of quantity breaks ( $q_i$ where $i=1,2,3,......,n$ and $q_1<q_2<......<q_n<\infty$ ), a rebate scheme on the unit acquisition price based on the purchased quantity is offered by the supplier where the unit acquisition price ( $c_j$ ) decreases stepwise based upon the purchased quantity $(Q)$ such that $c_1>c_2>\cdots \cdots >c_n$ . This graphic illustration of this rebate scheme with four quantity breaks is provided in Fig. 2.

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Fig. 2

Graphic illustration of an all-unit discount scheme with four quantity breaks.

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3.1 Model for the zero-end inventory situation

Suppose a company’s stocking procedure begins with $Q$ units of seasonal food, and the complete length of the cycle is $T$ , where Fig. 3 reveals the graphic illustration of the inventory cycle. In Fig. 3, the point $\mu$ specifies the time duration for deterioration-free inventory volume while $T$ indicates the termination of the cycle. Consequently, the inventory volume drops due to the consumption of the items only over $[0,\mu ]$ , and then it decreases due to both deterioration and consumption of the items over $[\mu ,T]$ . Let $I_1(t)$ and $I_2(t)$ represent the inventory volume in the storage at any time $t$ during $[0,\mu ]$ and $[\mu ,T]$ , respectively.

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Fig. 3

Graphical illustration of the inventory framework without shortage.

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Now, the differential equation for the inventory volume $I_1(t)$ over $[0,\mu ]$ is

Again, the differential equation for the inventory volume $I_2(t)$ over $[\mu ,T]$ is

Taking into account $I_1(0)=Q$ , the solution of Eq. (1) is

Applying another boundary condition $I_1(\mu )=S_1$ in (3), one has

On solving Eq. (2) with the auxiliary condition $I_2(T)=0$ , one has

Again, adopting $I_2(\mu )=S_1$ in (6), one finds

Combining Eqs. (5) and (7), one has

and

Cost Components:

The total cost of this inventory system per replenishment cycle consists of the following cost components:

1. Ordering cost (OC): $c_o$

2. Purchasing cost (PC): $c_{j}Q$

3. Holding cost (HC): The cumulative holding cost for the periods $[0,\mu ]$ and $[\mu ,T]$ is

Hence, the company’s total cost per unit of time is: $$\mathit{TVC}(T)=\frac{1}{T}\left[\mathit{OC}+PC+HC\right]$$

3.2 Model for the inventory procedure with a stock-out situation

Suppose a company places a requisition for $Q=S+R$ units of a decaying item early in a business cycle consisting of three different phases, such as $\left[0,\mu \right]$ , $\left[\mu ,t_1\right]$ , and $\left[t_1,T\right]$ . When the company receives the lot, $R$ units are consumed to satisfy the accumulated backlogging demands; thus, the remaining available inventory level falls to $S$ units. At this stage, the stock level falls just for satisfying clients’ desires over the interval $\left[0,\mu \right]$ . At time $t=\mu$ , the inventory level becomes $S_1$ . Because of the consolidated impacts of deterioration and client desire, the stock declines throughout the next phase $\left[\mu ,t_1\right]$ of the cycle and eventually, falls to a zero level at $t=t_1$ . After this phase, a stock-out situation is faced by the company when shortages are amassed depending on the clients’ waiting time during the period $\left[t_1,T\right]$ . At this stage, another order is submitted, and the entire stocking procedure is repeated. The complete inventory procedure is illustrated in Fig. 4.

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Fig. 4

Graphical illustration of the inventory framework with shortages.

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The differential equations to illustrate the behavior of the present stock at any moment of the business cycle are:

with $I_1(0)=S$ , $I_1(\mu )=S_1$ , $I_2(\mu )=S_1$ , $I_2(t_1)=0$ , $I_3(t_1)=0$ and $I_3(T)=-R$ , where $k_0=a+b\mu$ .

Exploiting $I_1(0)=S$ , the solution of Eq. (11) is given by

Applying $I_1(\mu )=S_1$ in Eq. (14), one has

$I_1(\mu )=S_1=S-a\mu -\frac{1}{2}b{\mu }^2$ , i.e.,

On solving Eq. (12) with $I_2(t_1)=0$ , one finds

Again, applying $I_2(\mu )=S_1$ in Eq. (15), one has $$I_2(\mu )=S_1=\frac{k_0}{\theta }\left[e^{\theta \left(t_1-\mu \right)}-1\right]$$

From Eqs. (15) and (17), one has

Solving Eq. (13) with $I_3(t_1)=0$ , one has

Applying $I_3(T)=-R$ in Eq. (19), one finds

Consequently, the total number of requisition amounts is

From Eq. (21), the business cycle duration is expressed as

The total cost of the entire inventory framework per renewal cycle consists of the following components:

1. Ordering cost (OC): $c_0$

2. Purchasing cost (PC): $c_{j}Q=c_j(S+R)$

3. Holding cost (HC): The total holding cost for the period $\left[0,\mu \right]$ and $\left[\mu ,t_1\right]$ is

• (4)

  Shortage cost (SC): $-c_s{\int }_{t_1}^T{I}_3(t)dt=c_s\frac{k_0}{\sigma }\left[\frac{1}{\sigma }-\frac{e^{-\sigma (T-t_1)}}{\sigma }-\left(T-t_1\right)e^{-\sigma (T-t_1)}\right]$

• (5)

  Lost Sale cost (LC): ${\mathrm{c}}_{\mathrm{l}}{\int }_{{\mathrm{t}}_1}^{\mathrm{T}}\left[1-{\mathrm{e}}^{-\sigma (\mathrm{T}-\mathrm{t})}\right]{\mathrm{k}}_0\mathrm{dt}={\mathrm{c}}_{\mathrm{l}}{\mathrm{k}}_0\left[(\mathrm{T}-{\mathrm{t}}_1)-\frac{1}{\sigma }+\frac{{\mathrm{e}}^{-\sigma (\mathrm{T}-{\mathrm{t}}_1)}}{\sigma }\right]$

Consequently, the company’s total cost per unit of time is:

Now the goal is to bring out the optimal period for stock available inventory $t_1^{*}$ , and the optimal length of the fill-up cycle $T^{*}$ to minimize the company’s total cost per unit of time. For that, the following section deals with the theoretical findings for achieving the company’s best inventory planning.

4.1 Model for the zero-end inventory situation

Utilizing the important result mentioned above, the following theorem is proposed to examine the curvature of the cost function in Eq. (10).

Proof: See Appendix A in the Supplementary material file.

Now, to calculate the point $T^{*}$ , it is necessary to compute the first-order derivative of $\mathit{TVC}(T)$ and set the result of the derivative as zero. After some simplifications, the condition for achieving $T^{*}$ is:

On solving Eq. (25), the value of $T^{*}$ is found, which guarantees the global minimum cost $\mathit{TVC}(T^{\ast })$ for the company. Next, our aim is to show the joint pseudo-convex of the company’s total cost function (23) for the inventory system in a stock-out situation.

4.2 Model for the inventory procedure with stock-out situation

For achieving the best inventory planning for minimizing the company’s total cost $\mathit{TVC}(t_1,T)$ when a stock-out situation appears, compute the first-order partial derivatives of $\mathit{TVC}(t_1,T)$ with respect to $t_1$ and $T$ , then set them equal to zero. $$\frac{\partial }{\partial t_1}\left[\mathit{TVC}(t_1,T)\right]=0$$ i.e.,

and $\frac{\partial }{\partial T}\left[\mathit{TVC}(t_1,T)\right]=0$

i.e.,

On solving Eqs. (26) and (27) simultaneously, the optimal period for stock available inventory $t_1^{*}$ and the optimal length of the fill-up cycle $T^{*}$ to minimize the company’s cumulative cost per unit of time are achieved. Now, the existence of the point $(t_1^{*},T^{*})$ and the global optimality of the company’s total cost $\mathit{TVC}(t_1,T)$ at the point $(t_1^{*},T^{*})$ are both examined in the following theorem.

Proof: See Appendix B in the Supplementary material file.

5.1 If the purchased quantity of the company $Q^{\ast }$ stands at any quantity break under a stock-out situation

This subsection determines the necessary condition for obtaining the best inventory plan for the company when the best $Q^{\ast }$ stands at any quantity break under a stock-out situation. After substituting the expression of $T$ with the assistance of Eq. (22), the company’s total cost $\mathit{TVC}$ in Eq. (23) involves only one decision variable, $t_1$ . Thus, it is necessary to determine the condition for computing $t_1^{\ast }$ under which the company’s total cost $\mathit{TVC}(t_1)$ is optimized.

From Eq. (21), one has

In this circumstance, the condition for the best $t_1^{*}$ is found by setting $\frac{d}{d{t}_1}\left\{\mathit{TVC}(t_1)\right\}=0$ with the assistance of Eq. (28). After performing some simplifications, the condition is given by

Now we present two algorithms: one for achieving the best inventory plan under zero-end inventory, and another for achieving the best inventory plan under a stock-out situation, considering the rebate scheme on unit acquisition price.

5.2 Algorithm for the company’s best inventory plan under the zero-end inventory situation

5.3 Algorithm for the company’s best inventory plan under the stock-out situation

6.1 Case study

Mahim Natural Mango (MNM) Company, a wholesale business of fresh mangoes in Bangladesh, operates with its head office in Rajshahi and 20 wholesale centers nationwide. During the May to August harvesting period, the centers procure mangoes directly from farms and sell them to local retailers. We conducted a case study on Manihari Mango Shop, a prominent retailer in Dhaka, gathering information from historical records and interviews with the shop’s manager.

Mahim Natural Mango (MNM) Company, a wholesaler of mature mangoes with a short shelf life, offers a concession on unit acquisition cost based on order quantity for retailers. The cost per mango box (containing 10 kg of mangoes) is $6 for quantities [0, 1350), $5.5 for [1350, 1500), and $5 for 1500 boxes or more. The goal of the Monihari Mango Shop manager is cost-effective inventory planning. The mangoes are non-instantaneous deteriorating items, with deterioration starting 2 days after storage. Demand increases with ripening, starting at 350 boxes/day and increasing by 3.5 boxes/day. Ordering cost is $300/order, deteriorating rate is 0.02 box/day, and holding cost is $0.01/box/day. In a stock-out situation, backlogging follows a negative exponential function with a waiting time sensitivity coefficient of 0.03. Shortage cost is $3/box/day, and opportunity cost is $7/box/day. All aforementioned information is expressed with notation as follows: $a=350$ , $b=3.5$ , $c_{°}=300$ , $k_{°}=357$ , $\theta =0.02$ , $\mu =2$ , $h_c=0.01$ , $\sigma =0.03$ , $c_s=2$ , $c_l=7$ and the discount scheme as

Again, the best inventory planning for the Monihari Mango Shop under a stock-out situation is: $t_1^{\ast }=4.063$ days, $T^{\ast }=4.186$ days, $Q^{\ast }=1503$ boxes and $\mathit{TV}{C}_{\min }(t_1^{\ast },T^{\ast })=\mathrm{\$}1875.2$ . The detailed calculations of the optimal inventory planning for both situations are supplier in Appendices C and D in the Supplementary material file.

Since $\mathit{TV}{C}_{\min }(t_1^{\ast },T^{\ast })=\mathrm{\$}1875.2<TV{C}_{\min }(T)=\mathrm{\$}1875.5$ , adopting our model with partial backlogged shortages is better for the Monihari Mango Shop to run the mango business with a total minimum cost.

6.2 Numerical illustration

To examine the working performance of the algorithms 5.2 and 5.3 derived in the previous section, two numerical examples are illustrated under the zero-end inventory and stock-out situations.

Suppose the following rebate scheme on the unit acquisition price based upon the purchased amount is presented by the supplier.

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Fig. 5

Behavior of the company’s total cost in Example 1.

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This example adopts the same rebate scheme on unit acquisition price and other data in Example 1 along with $\sigma =0.04,c_s=3,c_l=5$ to adopt the model under a stock-out situation. The company’s minimum cost per unit time is $\mathit{TV}{C}_{\min }(t_1^{\ast },T^{\ast })=474.32$ with the optimal $t_1^{\ast }=4.212$ , $T^{\ast }=4.413$ and $Q^{\ast }=465$ (the solution process is provided in Appendix F in the Supplementary material file). Fig. 6 exhibits the behavior of the company’s cost $\mathit{TVC}(t_1,T)$ under stock-out situation against variables $t_1$ and $T$ . As seen from Fig. 6, the company’s cost $\mathit{TVC}(t_1,T)$ is jointly convex in $t_1$ and $T$ , and therefore, $\mathit{TVC}(t_1,T)$ holds the global minimum value at the unique point $(t_1^{*},T^{*})$ . To examine the behavior of the company’s total cost $\mathit{TVC}(t_1,T)$ more explicitly, two additional graphs (Figs. 7 and 8) are provided against each variable separately.

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Fig. 6

Behavior of the company’s total cost in Example 2.

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Fig. 7

Behavior of the company’s total cost with respect to $t_1$ in Example 2.

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Fig. 8

Behavior of the company’s total cost with respect to $T$ in Example 2.

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