Inventory decisions on the transportation system and carbon emissions under COVID-19 effects: A sensitivity analysis.

The COVID-19 pandemic has created multiple problems in the existing transportation system. The contribution of this study is to guide logistics managers as they make ordering decisions within a disrupted transportation system. In the overall supply chain system, inventory decisions have been either compromised or challenged. Traditional inventory decisions that consider preplanned transportation facilities (and speeds) are currently becoming obsolete, predominantly in post-COVID times due to delays in the delivery of products and higher delivery costs. Therefore, businesses such as retailers must align ordering and pricing decisions to maintain a sustainable profit. To address this issue, this study investigates optimum inventory decisions under the pandemic's effects while considering the transportation cost as proportional to COVID-19 intensity. This study also considers product deterioration, time-dependent holding costs, price-dependent demands, and carbon emissions from vehicle operation and intends to establish a harmonious relationship among these attributes. The optimization of green technology investment is studied to reduce emissions due to transportation. Some theoretical derivations and numerical examples are given, and they are followed by a sensitivity analysis to extract important managerial insights into the effect of COVID-19. The manager can set an optimal selling price and the cycle length by carefully planning the number of trips in considering the rate of the outbreak and its effect on the increasing transportation cost.

Introduction

Inventory decisions are centered on the number and timing of orders, and they consider various costs and supply chain constraints. Inventory decisions affect company profits and are important regarding stock availability to secure consumer needs. These decisions are influenced by various transportation systems and their characteristics, such as availability, reliability, and cost. Among the many detrimental effects of COVID-19 on human health and the economy, COVID-19 has had an undeniable effect on overall supply chain problems and, more specifically, on transportation systems. Due to unexpected and unplanned lockdowns, the usual logistics and distribution activities have been profoundly disrupted. Twinn et al. (2020) showed some impacts, such as delivery delays, backlogs at container ports, shortages of truck drivers, and uncertain demand, especially for small trucking businesses, while Mogaji (2020) indicated an increase in tariffs in developing countries. Prior studies lack adaptations to these disruptions with respect to inventory and supply chain decisions (Ivanov and Das, 2020). Consequently, overall product availability has been challenged or compromised, which ultimately affects customers, such as in food and grocery markets (Mahajan and Tomar, 2020, Singh et al., 2020). Loske (2020) showed the changing volume and capacity in road transport due to panic buying in the early stage of the pandemic. This study argues that when disruption occurs, the transportation cost is proportional to COVID-19 intensity. Therefore, the inventory decision model must consider the effect of variability in these costs.

The COVID-19 pandemic has been disrupting business systems such that supply, and demand are difficult to match, as the numbers of infections are still fluctuating (Poursoltan et al., 2021, Singh et al., 2020). The effect has continued after one year, and it is not known when it will end. Consequently, making optimal forecasts and decisions is still challenging for logistics managers. Jabbour et al. (2020) suggested managers focus on building smarter and more resilient systems to make a sustainable supply chain. Queiroz et al. (2020) proposed adaptation, digitalization, preparedness, recovery, ripple effects, and sustainability for supply chain operations to sustain the business during the COVID-19 pandemic, while Choi (2021) emphasized the importance of risk analysis in logistics systems to propose innovative solutions and strategies during the pandemic. Ivanov and Das (2020) discussed the existing resilient supply chain practices such as holding more inventory or having subcontracting facilities require further extensions because the COVID-19 pandemic is not limited to a region and time. The length of the pandemic, the likelihood that the transmission rate will re-increase, and the effects on businesses and finances are unknown. Many studies have identified new approaches and models as operational solutions for overcoming logistics disruptions. Alkahtani et al. (2021) investigated a variable production rate as a solution for manufacturers in a supply chain with disrupted demand during the COVID-19 pandemic. Recently, Mashud et al. (2021a) studied the optimum selling price and order decision in a supply chain under the COVID-19 pandemic by suggesting a hybrid payment to deal with any financial crisis that may occur.

Transportation activities, including the transportation of materials and products in the supply chain, are major carbon emitters. Emission levels are influenced by inventory decisions (Tiwari et al., 2018); for example, they may be related to the frequency of transportation and truckloads. Various studies have identified the optimum inventory decisions to minimize these emissions (Tiwari et al., 2019, Taleizadeh et al., 2020, Yu et al., 2020, etc.). Other studies have also involved decisions for green technology investment (Toptal et al., 2014, Mashud et al., 2020b, Mishra et al., 2021; etc.). Sarkis (2020) hoped that the economic recovery after the COVID-19 pandemic will not ignore the regulations and concerns for the environment that have been built so far. Therefore, in line with Sarkis (2020), this study links the effects of pandemics and efforts to reduce carbon emissions in inventory decisions.

Queiroz et al., 2020, Ivanov and Das, 2020, and Choi (2021) suggested innovative solutions to deal with the logistics disruption during the COVID-19 pandemic. Recent studies from Alkahtani et al., 2021, Mashud et al., 2021a developed new inventory models to respond to the disruption in customer demand and financial difficulties of supply chain parties during the pandemic. However, they did not specifically consider the effect of the disrupted transportation system during the COVID-19 pandemic. This study aims to consider the practical influence of the COVID-19 pandemic on the transportation system and proposes an inventory decision model with increasing transportation costs under a pandemic rate function. This study considers a case in which truck drivers are unwilling to drive vehicles during the pandemic or request additional benefits. This situation depends on the rate of outbreaks of COVID-19. If the infectious rate is low, the drivers are willing to go on trips. If the risk is high, then the drivers claim additional payments are owed for their safety, hence, the total transportation cost increases. With this important criterion in mind, a pandemic function is included in the transportation cost, which will regulate the total cost. Further, the pandemic rate is affected by the vaccination program. Hence, how should the logistics manager adjust the ordering and pricing decisions considering the additional cost function? Given the desire to reduce carbon emission levels, how will these decisions affect the required green technology investments? In the end, how do the model parameters affect the total profit? A sensitivity analysis is needed to extract some important managerial insights about the effects of COVID-19 and changes in other parameters on green technology investments and profits. Fig. 1 illustrates the inventory problem of this study. under the transportation disruption during a pandemic.

Fig. 1

Inventory problem under the transportation disruption, price-dependent demand, product deterioration, and green technology investment.

This study contributes guidance to logistics managers of retailers in making ordering decisions given a disrupted transportation system. Simultaneously, managers can specify the optimum price by considering the variable transportation cost, the effect of deterioration, and an opportunity to reduce carbon emissions. This study links the disruption and green issues in the second model. The proposed models can be considered until the end of the COVID-19 pandemic and in similar situations in the future, which cannot be predicted. The major contributions are as follows:

The synergy between pricing strategies and the inventory cycle of deteriorating products has been investigated since the flow of products fluctuates depending on the pandemic outbreak rate and impact of vaccination.

A brief, clear view of the pandemic’s effect on the transportation system is illustrated by constructing a realistic pandemic function linked to the carbon emissions of the system. The pandemic function comprises the relation of infected and vaccinated populations.

A flexible, environmentally sustainable system is presented that can control carbon emissions through green technology investments during the pandemic.

Important managerial implications have been extracted from the experimental results, which will help any future practitioners and researchers.

In this study, Section 2 discusses the existing literature on logistics and supply chain systems during the COVID-19 pandemic, inventory decisions with pricing and deterioration effects, and green logistics that minimize carbon emissions. Section 3 describes the scope of the problem and the assumptions made in solving it. 4, 5 present the proposed mathematical model together with some theoretical derivations. 6, 7 study the system with some numerical examples and sensitivity analysis. Then, Section 8 identifies the managerial insights, and Section 9 provides the research conclusion.

Literature review

The COVID-19 pandemic, which has lasted more than a year and continues to disrupt logistics systems and supply chains in all countries, raises the need for fine-tuning managerial decisions (Kim, 2021, Poursoltan et al., 2021). Through a literature study, Chowdhury et al. (2021) identified four main topics related to logistics systems in response to this pandemic, namely, the impact of the pandemic, strategies for responding to adversity, the role of technology, and the sustainability of the supply chain. Chen et al. (2020) studied the vehicle routing problem for serving some closed gated communities in a certain area. The study focused on food distribution services with contactless service due to communities' isolation during the COVID-19 pandemic, and it optimized the routing to satisfy customers and reduce contact among the couriers. Choi (2020) analyzed mobile service operations to bring services near customers' homes, and he suggested a government subsidy to make the distribution system financially viable. Patriarca et al. (2020) studied order quantity decisions given uncertain demand during a pandemic, and they assumed time-dependent and inventory rate-dependent demand and a variable deterioration rate. The study focused on the effect of demand uncertainty and showed the ability to identify optimum quantity ranges. However, the proposed model did not consider the disruption of transportation. Alkahtani et al. (2021) also investigated the disruption of supply chain demand during the COVID-19 pandemic from the manufacturer's perspective. These authors assumed that the demand depends on the level of emergency or outbreak and proposed a variable production rate as a solution. Mashud et al. (2021a) focused on the financial problems that may occur in the supplier-retailer supply chain. Hence, the study suggested a hybrid payment system as the solution. However, the proposed model also lacks the consideration of different transportation costs from different levels of the outbreak.

Optimum order quantities and replenishment cycles have been considered the main decisions in inventory management. This field considers basic costs such as ordering, purchasing, holding, and transportation costs. According to Swenseth & Godfrey (2002), transportation costs can account for 50% of the total logistic costs. Rahman et al. (2016) incorporated the transportation cost as a function of the shipping weight and distance. The model considered a less-than-truckload transportation mode. Wangsa and Wee, 2018, Wangsa and Wee, 2020 considered freight forwarding services that accumulate products during distribution. Recently, many authors have incorporated carbon emissions from transportation activities to develop green inventory models. The emissions from fuel combustion depend on several factors, such as the delivery distance, shipping quantity, product weight, and fuel efficiency (Tiwari et al., 2018, Daryanto et al., 2019, Wee and Daryanto, 2020, Daryanto and Wee, 2021). Moreover, a cold item supply chain emits higher emissions from temperature-controlled vehicles and warehouses (Bozorgi et al., 2014, Hariga and As’ad, R., Shamayleh, 2017, Babagolzadeh et al., 2020). Similar to these studies, the proposed model incorporates carbon tax regulation to include the emission cost from transportation in the logistics costs.

The economic order quantity (EOQ) model with environmental concern has been studied for several years. Due to climate change, most of the studies incorporate carbon emission reduction. Bonney and Jaber, 2011, Hua et al., 2011 initiated the study by adding emission costs into the traditional EOQ model. The studies considered several emission sources, such as warehousing, transportation, and obsolete items. Hua et al., 2016, Tiwari et al., 2019, Shi et al., 2020, and Mishra et al. (2020) studied EOQ models with carbon emissions for deterioration products. Taleizadeh et al., 2020, Yu et al., 2020 integrated order quantity and price decisions under the effect of carbon emissions. Recently, several researchers have boosted emission reductions by investing in green technology such as energy-efficient equipment and the use of renewable energy (Toptal et al., 2014, Mashud et al., 2020b, Mishra et al., 2020, Mishra et al., 2021, Poursoltan et al., 2021). Lin (2018) focused on emission reduction from transportation activities to create a sustainable EOQ model. This study involves emission reductions through green technology investments by optimizing the investment cost per replenishment cycle.

Before the COVID-19 pandemic, researchers developed inventory models that consider disruptions in terms of supply, demand, and transportation (Wilson, 2007, Hishamuddin et al., 2013, Azad et al., 2014, Snyder, 2014, Paul et al., 2016, Taleizadeh, 2017). For example, Wilson (2007) studied a transportation disruption and investigated the impact on the different echelons of the supply chain using a system dynamic approach. This research focused on the transportation capacity, inventory fluctuation, and unfilled customer demand. Hishamuddin et al., 2013, Azad et al., 2014 assumed a single disruption and considered the recovery time for production and distribution. The studies do not fit the COVID-19-caused disruption, as none of them considers the increasing transportation cost due to lockdowns and health protocols during a pandemic. Those studies are lacked consideration of the variability of transportation costs in different locations with different outbreak levels.

In a situation where the customer demand is price-dependent, the manager also needs to determine the optimal selling price. Naturally, the demand rate decreases with increasing price (Sana, 2011, Teksan and Geunes, 2016). Hence, the demand rate may follow a linear function with a decreasing price coefficient, as in Hasan, Mashud, Daryanto, & Wee, 2021. Another important factor in an inventory system is the deterioration rate. By nature, the quality and quantity of some products, such as fruits and vegetables, decrease over time (Wee, 1999, Widyadana et al., 2011). These losses should not be ignored, and many researchers have sought to reduce them (Hsu et al., 2010, Mashud et al., 2020a). Other researchers have investigated inventory decisions under the effects of both price-dependent demand and product deterioration (Hasan, Mashud, Daryanto, & Wee, 2021, Mashud et al., 2020a, Sana, 2011). This study considers both factors, together with the effects of transportation disruptions and emission reductions.

In summary, this study meets the challenges of inventory decisions under a transportation disruption, such as during the COVID-19 pandemic. COVID-19 has spread all over the world with different outbreak levels in different areas and affected logistics operations. Although many have studied logistics decisions to deal with the COVID-19 pandemic’s impact, these studies investigated inventory decisions and were challenging. Recent studies from Patriarca et al., 2020, Alkahtani et al., 2021, Mashud et al., 2021a proposed inventory decision models to address the effect or disruption caused by this pandemic. However, none of them considers the increase and variability of transportation costs. Queiroz et al. (2020) found that distribution (including transportation) has emerged as one of the main challenges during the pandemic. Considering the variability of transportation costs due to lockdowns and health protocols, this study investigates the optimum replenishment cycle and selling price. Furthermore, we provide a second case in which green investment is optimized to make the inventory model sustainable. Sarkis (2020) argued that the pandemic has shown that sustainability with its three concerns (i.e., the economic profit and cost, social safety and health, and environment) is indivisible.

Problem definition, assumptions and notations

The effect of COVID-19 on the transportation system and inventory decision model is presented in terms of problem descriptions and assumptions. The associated notations are also given.

Problem definition

The supplier-retailer-customer supply chain of deteriorating items is discussed in this study. This study involves a retailer that purchases products from a supplier such that the transportation cost becomes the retailer’s responsibility. During the COVID-19 pandemic, the transportation system has experienced difficulties due to lockdowns and health protocols. Hence, for transporting products to its warehouse, the retailer needs to pay extra money, which depends on the outbreak level and impact of vaccination.

When transporting products, carbon emissions are produced by the vehicle. Hence, green technology investment is considered to reduce carbon emissions. Two cases of systems with and without green investment are investigated to study the effect on the total profit. Furthermore, when certain products are stored in a warehouse, their freshness decreases due to deterioration. Due to the covid pandemic, the transportation cost increases depending on the infection rate. In contrast, the influence of the vaccination process reduces the disruption. The customer demand for the products depends on the selling price such that if the retailer raises the selling price of the product, demand decreases. Therefore, the retailer also tries to increase profit by optimizing the selling price. Considering the total inventory costs and demand characteristics, the order quantity is specified by optimizing the cycle time.

Assumptions

The model was developed based on the following assumptions and notations:

The demand rate is a linear function of the selling price () with and , and this rate is the constant part and price coefficient of the demand function ( and ), which is similar to Hasan, Mashud, Daryanto, & Wee, 2021.

The lead time is negligible or zero.

The transportation costs consist of variable and fixed parts with additional emission costs, and this setup is similar to Daryanto & Wee (2021). Additionally, carbon tax pricing is applied.

A constant deterioration rate is applied, and no replacement or repair of deteriorated items is allowed during the period of the replenishment cycle .

Shortages are not allowed.

The rate of intensity of COVID-19 outbreaks is considered constant during a cycle.

The effectiveness of the vaccine is 100%. Initial vaccinated people must be less than the total people i.e.,. The infection rate lies in [0, 1] and the vaccination rate is positive i.e., . No one will be infected if there are no infected people.

To hold products in a retailing house, the retailer always seeks a fixed arrangement of holding facilities, and variable facilities depend on the quantity of products that must be held. Thus, a realistic holding cost is considered here. This cost has two parts, where one is constant, and the other variable is similar to Mashud et al. (2021c). The complete list of notations is provided in Table 1 .

Table 1

List of Notations.

Notations	Description
	Parameters
c_e	the vehicle fuel consumption when empty
c_f	fixed transportation cost
c_l	fuel consumption per ton of payload
d_s	delivery distance to the retailer warehouse
e_c	carbon emission cost per unit distance of delivery
e_a	carbon emission cost per unit item per unit distance of delivery
E	the efficiency of greener technology for emission reduction, E ≥ 0
f_p	the fuel price
h_u	per unit holding cost
M	Maximum carbon emission reduction from the green technology investment,0<M<1
O_c	ordering cost
P_u	unit purchase cost
r	the rate of intensity of COVID-19 outbreaks
v_r	rate of change of vaccination acceleration
w	product weight
ε	deterioration rate
η	number of trips
	Decision variables
S_p	unit selling price
T	cycle time
G_i	green technology investment per cycle
	Other variables
N	order quantity
I_lt	level of inventory at time t,0⩽t⩽T
ωS_p,T	total profit per unit time for Case 1
ωS_p,T,G_i	total profit per unit time for Case 2

The cycle will be repeated over an infinite planning horizon as similar to the work of (Tiwari et al., 2019, Mishra et al., 2021).

Mathematical models

This paper considers the increasing transportation costs as the effect of the COVID-19 outbreak and investigates two different cases. Whereas the first case only involves the impact of the COVID-19 outbreak on transportation, the second case incorporates green technology investment, as the retailer is willing to be environmentally friendly. The research methodology is presented in Fig. 2 . The following sections provide the corresponding mathematical models and discussion for the two cases.

Fig. 2

Research methodology.

Case I (Considering the effect of COVID-19 on transportation)

During the inventory cycle , the level of stock at any time is given by the function , as shown in Fig. 3 . The rate of the inventory level decreases in accordance with the demand rate () and the deterioration rate () of the available inventory level. At the beginning of the cycle (), the inventory level is equal to the order quantity (). At the end of the replenishment cycle (), the inventory level is zero. Therefore, the inventory level at time is governed by the following differential equation:where , and the boundary conditions are and .

Fig. 3

Graphical presentation of the inventory level.

Considering the boundaries, Eq. (1) becomes

The solution of differential Eq. (2) is

As , Eq. (3) becomes

By substituting the value of into Eq. (3), we obtain

Again, using the boundary condition , we derive

Substituting the value of into Eq. (5) and simplify, one can get

The profit function of the system depends on the following costs and revenue component:

Cost for ordering (). The cost of placing an order per cycle is

Purchase cost (). The purchasing cost per cycle is

Holding cost (). The holding cost depends on the inventory level and varies over time. If is known and and are the respective constant part and time-reliant part of the holding cost, then the holding cost per cycle is

Transportation cost (). The transportation cost consists of a fixed transportation cost (), variable transport costs, and a carbon emission cost. The travel distance () by the truck is counted with the incoming and outgoing travels. The variable transport cost is the total fuel consumption () multiplied by the fuel price (). Additional fuel consumption occurs based on the truckload. Hence, a one-way distance () is multiplied by the fuel consumption per ton of payload (), product weight (), order quantity (), and . The carbon emission cost is the travel distance () multiplied by the unit carbon cost and carbon emission cost from the truckload. Therefore, with trips per cycle as like the work of Sultana et al. (2022) the total transportation cost per cycle is

However, due to COVID-19 being around the globe, most places are partially or completely locked down. This circumstance increases transportation and logistics costs. This study considers the situation in which the effect depends on the rate of the COVID-19 outbreaks.

In a town the total population is . The covid spreads among these people with an infection rate . However, the vaccination process is continued; the rate of change of vaccination acceleration is . Then the developed covid function with initial infected people and initial vaccinated people is the Eq. (12). The details are discussed in Appendix A.

For example, for total population ; infected rate ; the rate of change of vaccination acceleration ; initial infected people ; and initial vaccinated people the infected function Eq. (12) is represented in Fig. 4 .

Fig. 4

Infected people in different time.

Normalized covid infected function is as follows

The normalized covid indicator function is . Taking this important criterion in mind, the transportation cost increases times of the original transportation cost that is . This transportation cost becomes highest when the covid indicator function is at peak. Therefore, the complete transportation cost is

Sales revenue (). The retailer's sales revenue refers to the income derived from product sales. Considering the product's selling price and the demand function, the sales revenue is

Finally, the total profit per unit of time can be written as follows:

Case II (Considering the effect of COVID-19 on transportation and green investment)

Although the rush of the transportation system has declined during the pandemic, CO2 is still emitted by transport vehicles, and this gas damages the balance of the environment. To curb CO2 emissions from transportation, green technology is implemented in this case. To implement this technology, capital investment is needed. A fraction of the emission reduction is considered in a similar manner to Mashud et al. (2021b). The value of when there is no green investment () and equals when . The investment cost function is continuously differentiable with and .

The retailer optimizes to reduce the emissions. The Green Technology cost () is

Considering the reduction in transportation carbon emissions, one can rewrite the transportation cost as

Then, the total profit function becomes

Theoretical derivations

This section presents some theoretical derivations for the proposed two cases. Then, the solution algorithms are developed.

For Case I (Considering the effect of COVID-19 on transportation without green investment)

The following propositions confirms the profit maximization of this model when the green investment is not applied though the challenges of COVID-19 on transportation system is considered.

The profit functionin Eq.(16)is concave in terms of selling pricewhen replenishment cycleis considered constant withand the optimalis as follows:

The solution in (a) satisfies the sufficient condition for the retailer's maximum profit.

The retailer profit function from Eq. (16) can be written as:where .

After differentiating Eq. (20) with respect to , we obtain.

By setting the above equation equal to zero, one can find the critical value as

Again, by differentiating Eq. (21) with respect to , we obtain

If , then the second derivative of the profit function becomes negative, which implies that for the critical point , we obtain a maximum value of . The critical point becomes the optimal point . Otherwise, the critical point becomes the minimum value point or a saddle point, which proves part (a) of the proposition.

By substituting for the value of in Eq. (23) with any value of , we obtain

Since , the above second-order derivative is negative, so at the point we have a maximum value of the profit function . This fact proves the second part (b) of the proposition. □.

For any fixed value, the retailer profit functionis concave down. Hence, there exists an optimum point, which is found by the following equation.

This proof is similar to that of Proposition 1. To avoid redundancy, the same solution procedures are omitted in this theoretical derivation. □.

The profit functionis concave with respect toand.

The retailer's profit function from Eq. (16) iswhere .

By inserting the values in Eq. (25), we obtain

Eq. (26) can be written aswhere

Here, are positive.

To maximize the total profit function , we can calculate first-order partial derivatives of from Eq. (27) with respect to and . Then set these partial derivatives equal to zero. Hence, we obtain

By solving and simultaneously from Eqs. (29), (30), we obtain the critical points of and , which are denoted by and , respectively.

After finding the second derivative of the profit function from Eq. (27), we obtain where

The Hessian matrix of the profit function is

The first principal minor iswhere is the coefficient of the price in the demand rate, which is always greater than 0. Hence, . Additionally, the second principal minor is

The Hessian matrix for will be negative definite if the second principal minor is . For these the condition arises as follows

Considering the condition in Eq. (34), the Hessian matrix for is negative definite, one can easily conclude that the profit function is a strictly concave function with respect to and . This implies that, there are only one critical point which is optimal point . Hence, our objective function reaches the global maximum value at point . □.

For Case II (Considering the effect of COVID-19 on transportation with green investment)

The challenges of COVID-19 on transportation system are considered. The green investment is applied Simultaneously.

The profit functionin Eq.(19)is concave in terms of selling price, replenishment cycle, and green investment. There exists a unique optimal pointwhich is find by solving the following system of equation:where,

And are describe in Eq. (28).

Using Eq. (19), we see that the objective function for Case II is highly nonlinear. Indeed, this function is very similar to Eq. (16). Thus, to avoid the redundancy of the same solution procedures (i.e., theories and their corresponding proofs), we have omitted the theoretical derivations. □.

Algorithms

For numerically optimize of profit function () one can use the following algorithm.

Algorithm for Case I with a single decision variable

Step 1. Input all the parameters value (,,,,,,,,,,,,,,,,,,,,,,, ).

Step 2. Evaluate the value of from Eq. (22).

Step 3. Evaluate profit function from Eq. (20) using all the parameters and the value of .

Step 4. Output the optimal value of and .

Step 5. End.

Step 1. Input all the parameters value (,,,,,,,,,,,,,,,,,,,,,,, ).

Step 2. Declare left hand side of the Eq. (24).

Step 3. Take where () and iterative variable .

Step 4. Evaluate using all the parameters and the value of .

Step 5. IF , Go to Step 3.

Step 6. Set .

Step 7. IF Go to Step 8 (where is a small value).

Step 8. Update and . Go to Step 5.

Step 9. Calculate from Eq. (16) using all the parameters.

Step 10. Output the optimal value of and optimal profit .

Step 11. End

Algorithm for Case I with two decision variables

Step 1. Input all the parameters value (,,,,,,,,,,,,,,,,,,,,,,).

Step 2. Evaluate additional sub equations from Eq. (28) using the parameters.

Step 3. Declare and from Eqs. (29), (30).

Step 4. Take where (, ) and iterative variable .

Step 5. Evaluate .

Step 6. IF and , Go to Step 3. ELSE IF and , Go to Step 11.

Step 7. Evaluate and.

Step 8. Set and .

Step 9. IF and , Go to Step 11 (where is small value).

Step 10. Update, and . Go to Step 5.

Step 11. Check condition from Eq. (34), IF it’s False Go to Step 1.

Step 12. Calculate profit function from Eq. (27).

Step 13. Output the optimal value of , and optimal profit .

Step 14. End

Case study

This section illustrates the problem with a case study that contains numerical examples.

Link to a real situation

The anticipated model is based on a transportation company called Volvo Truck Agency, Bangladesh. We talked with the company manager and asked him questions about the transportation system during the pandemic. The manager said that the situation is worse than ever. In as much as most truck drivers and helpers are unwilling to drive trucks, the daily basic transportation system faces some unavoidable scenarios. The drivers demand high wages according to the condition of the place of delivery (a red, yellow, or green zone). If the delivery place is within a red zone, drivers are sometimes unwilling to go or ask a very high price to transport the products. Due to the rapid vaccination rate, the situations rapidly change, resulting in slightly lower transportation wages than the previous fully pandemic situation. Fig. 5 illustrates the situation at a truck station during the pandemic. Based on the importance of this problem, this study was formed. To validate our model, some information is requested from the manager in our query.

Fig. 5

A situation during the pandemic at a truck station.

Numerical examples

Based on Case I of the study, we assume the following parameters from Mashud et al. (2020a) with some additional parameters when the selling price is considered the decision variable.

The initial expense for placing an order , the constant part of the demand rate , the coefficient of the demand rate , the number of trips per cycle , the cost for purchasing a product , the deterioration rate , the holding cost , and the constant part of the holding cost. In addition, the time-reliant part of the holding cost , the distance to travel , the fixed transportation cost , the fuel price , the vehicle fuel consumption when empty , the extra vehicle fuel consumption , the product weight , the carbon emission cost , the additional carbon emission cost . Consider in a city, total population , a variant of covid affected people meanwhile already vaccinated people of the same type of variant of covid, the rate of the intensity of covid variant outbreaks , rate of change of vaccination acceleration and peak cost of transportation due to covid .

When replenishment cycle is constant . We obtain the optimal solutions: the selling price , order quantity and total profit per unit time . Fig. 6 (a) shows the concavity of the retailer's profit function () regarding the individual decision variables of .

Fig. 6

Profit function () with regard to: (a) selling price (); (b) cycle time ().

When selling price is constant . We obtain the optimal solutions: the replenishment cycle , order quantity and total profit per unit time . Fig. 6 (b) shows the concavity of the retailer's profit function () regarding the individual decision variables of .

We assume the following parameters when the selling price () and replenishment cycle () are considered the decision variables.

In this example, all of the values are the same as in Example 1, and we will solve for two decision variables: the selling price () and replenishment cycle ().

We obtain the optimal solutions: the selling price , cycle time, order quantity and total profit per unit time . Fig. 7 illustrates the concavity of profit () with respect to selling price () and cycle time ().

Fig. 7

Profit function () with regard to selling price () and cycle time ().

Based on Case II of the study, we assume the following parameters when only one variable is decision variable other as constant. All of the values of the parameters are the same as in Example 1 with the maximum reduced amount of carbon emissions and the efficiency of green technology .

Then we obtain the optimal selling price , optimal cycle time , optimal green technology investment . Fig. 8 shows the concavity of the retailer's profit function ().

Fig. 8

Profit function () with regard to: (a) selling price (); (b) cycle time (); (c) green technology investment.

When the selling price (), replenishment cycle () and green investment () are considered the decision variables. We obtain the optimal solutions: the selling price , cycle time , investment in green technology , order quantity , and total profit per unit time. Fig. 9 illustrates the concavity of the retailer's profit function ().

Fig. 9

Profit function () with regard to: (a) selling price () & cycle time (); (b) selling price () & green technology investment; (c) cycle time () & green technology investment.

Key findings and discussions on profits and costs

In this section, let us consider that the COVID-19 outbreak rate varies for both Case I and Case II; hence, the profit also varies.

A graphical representation of the effects of the COVID-19 outbreak rate on the total profit function is illustrated in Fig. 10 . When the effects of the COVID-19 outbreak rate increase, the total profit decreases exponentially. The effect of COVID-19 on the total profit is significant when no green investment is implemented, while a green investment slows the fluctuations in profit. A slightly higher profit has been achieved without any carbon emissions, while a higher COVID rate brings a decline in profit in all cases.

Fig. 10

COVID-19 effect on profit function Case I and Case II.

The effect of the rate of change of vaccination acceleration on the total profits of the chain is presented in Fig. 11 . The total profit increases with increasing rate of change of vaccination acceleration. The rapid vaccination process provides a higher profit in all cases. The total profit is significant when no green investment is implemented, while a green investment slows the fluctuations in profit. However, without any carbon emission helps to achieve a slightly higher profit.

Fig. 11

COVID-19 vaccination effect on profit function Case I and Case II.

From Fig. 12 , we see that the effects of COVID-19 on the total profit and cost significantly increase with an increased rate of COVID. The increased rate of COVID-19 on the purchase cost produces a significant impact that has been identified from the figure, while a less sensitive movement of the COVID effect has been acknowledged for other cases. After the purchase cost, the impacts of the ordering cost are also notable, and proper management of the ordering cost brings greater profit.

Fig. 12

COVID-19 effect on profit and associated cost functions (Case I).

Sensitivity analysis

We present a sensitivity analysis to study the impacts of different parameters on the optimal values of the total profit (), cycle length (), selling price (), order quantity (), and green investment () for the case with green technology. For this purpose, the parameter value is changed by −20% to + 20%, and the remaining values are fixed.

Based on the obtained results of the sensitivity analysis outlined in Table 2 , the following observations can be made:

Table 2

Sensitivity analysis without and with green technology investment.

Parameter (Base)	Change in %	Changed Value	Case I				Case II
			Sp∗	T^∗	N^∗	ω^∗	Sp∗	T^∗	N^∗	Gi∗	ω^∗
O_c (400)	20%	480	644.551	6.121	105.146	2693.674	623.631	4.192	78.035	1.801	3362.172
	10%	440	644.151	6.045	103.996	2700.251	623.429	4.161	77.507	1.802	3371.750
	−10%	360	643.375	5.900	101.788	2713.647	623.025	4.099	76.452	1.803	3391.122
	−20%	320	642.998	5.831	100.731	2720.466	622.823	4.068	75.925	1.804	3400.918
α (65)	20%	78.0	719.358	4.147	101.831	6256.052	704.760	3.361	85.953	1.855	7146.543
	10%	71.5	679.789	4.724	99.233	4324.019	663.568	3.677	81.402	1.832	5114.938
	−10%	58.5	612.952	8.712	114.039	1444.487	584.739	4.899	73.538	1.762	1954.066
	−20%	52.0	576.115	10.261	95.001	520.976	553.186	7.072	77.031	1.681	856.883
β (0.075)	20%	0.090	584.284	8.765	113.730	1069.544	555.188	4.791	73.776	1.767	1585.944
	10%	0.083	612.044	7.559	113.909	1770.481	585.633	4.410	75.220	1.787	2374.010
	−10%	0.068	687.600	5.211	99.430	3947.044	669.931	3.910	78.861	1.816	4677.886
	−20%	0.060	745.086	4.771	99.166	5588.827	728.912	3.729	80.786	1.828	6369.301
P_u (200)	20%	240	670.233	7.128	108.851	2056.468	645.232	4.372	74.220	1.787	2668.974
	10%	220	656.711	6.477	105.365	2371.874	634.178	4.243	75.582	1.795	3016.867
	−10%	180	631.481	5.610	101.784	3060.673	612.361	4.028	78.401	1.809	3762.434
	−20%	160	619.654	5.337	101.567	3432.436	601.566	3.937	79.841	1.816	4159.857
h_u (3)	20%	3.6	640.999	4.969	86.218	2592.497	623.333	3.833	71.317	1.810	3302.388
	10%	3.3	641.770	5.320	92.169	2646.024	623.243	3.967	73.883	1.807	3340.825
	−10%	2.7	650.566	8.119	137.084	2789.838	623.340	4.338	80.909	1.798	3424.690
	−20%	2.4	651.484	9.666	163.793	2914.479	623.714	4.632	86.390	1.791	3471.710
ε (0.01)	20%	0.012	642.982	5.623	97.583	2667.348	623.416	4.043	75.580	1.805	3353.545
	10%	0.011	643.313	5.782	99.995	2686.855	623.320	4.085	76.265	1.804	3367.406
	−10%	0.009	644.368	6.207	106.428	2727.625	623.138	4.176	77.724	1.801	3395.531
	−20%	0.008	645.193	6.506	110.930	2749.171	623.052	4.224	78.502	1.800	3409.806
g (2)	20%	2.4	642.958	5.523	95.269	2648.205	623.631	4.005	74.488	1.805	3336.283
	10%	2.2	643.258	5.720	98.639	2677.002	623.426	4.065	75.695	1.804	3358.657
	−10%	1.8	644.591	6.315	108.567	2738.254	623.035	4.199	78.352	1.801	3404.535
	−20%	1.6	645.911	6.808	116.645	2771.598	622.853	4.273	79.829	1.799	3428.092
h (1.2)	20%	1.44	640.823	5.169	89.854	2644.840	622.868	3.932	73.328	1.808	3345.491
	10%	1.32	641.850	5.465	94.718	2673.796	623.024	4.024	75.023	1.805	3363.010
	−10%	1.08	649.489	7.508	127.001	2750.655	623.496	4.256	79.299	1.799	3400.834
	−20%	0.96	652.868	9.125	153.207	2820.789	623.868	4.413	82.164	1.796	3421.576
η (15)		11	633.412	4.255	76.043	3013.654	618.472	3.417	64.702	1.786	3661.693
		13	637.748	4.930	86.770	2848.137	620.820	3.765	70.738	1.796	3515.573
		17	652.105	7.712	129.021	2597.165	625.767	4.527	83.665	1.807	3258.063
		19	656.142	8.677	143.117	2510.483	628.548	4.978	91.156	1.809	3145.027
c_f (3)	20%	3.6	644.047	6.025	103.697	2702.283	623.354	4.149	77.311	1.802	3374.762
	10%	3.3	643.902	5.998	103.284	2704.590	623.291	4.140	77.145	1.802	3378.077
	−10%	2.7	643.618	5.945	102.475	2709.237	623.163	4.120	76.813	1.803	3384.729
	−20%	2.4	643.478	5.920	102.080	2711.577	623.100	4.110	76.648	1.803	3388.066
c_e (2)	20%	2.4	650.240	7.290	122.751	2623.403	625.859	4.541	83.907	1.792	3254.011
	10%	2.2	646.938	6.589	112.248	2662.741	624.519	4.330	80.363	1.797	3316.306
	−10%	1.8	641.214	5.513	95.834	2755.434	621.971	3.938	73.712	1.808	3449.382
	−20%	1.6	639.110	5.154	90.270	2807.683	620.739	3.752	70.528	1.813	3520.386
f_p (0.15)	20%	0.180	671.406	9.208	141.257	2205.658	640.587	4.630	80.350	1.783	2758.072
	10%	0.165	659.622	7.855	126.899	2437.648	631.696	4.350	78.353	1.794	3064.157
	−10%	0.135	632.999	5.328	95.911	3001.623	615.034	3.944	75.921	1.810	3709.066
	−20%	0.120	623.689	4.960	92.670	3310.925	607.043	3.781	75.034	1.818	4046.683
c_l (1)	20%	1.2	669.053	8.541	132.146	2262.613	637.399	4.151	72.887	1.796	2884.104
	10%	1.1	655.156	6.823	112.016	2474.117	630.290	4.137	74.887	1.799	3129.059
	−10%	0.9	634.878	5.657	101.178	2950.915	616.201	4.127	79.157	1.806	3641.116
	−20%	0.8	626.665	5.470	101.197	3204.021	609.205	4.130	81.417	1.808	3908.207
w (3)	20%	3.6	669.053	8.541	132.146	2262.613	637.399	4.151	72.887	1.796	2884.104
	10%	3.3	655.156	6.823	112.016	2474.117	630.290	4.137	74.887	1.799	3129.059
	−10%	2.7	634.878	5.657	101.178	2950.915	616.201	4.127	79.157	1.806	3641.116
	−20%	2.4	626.665	5.470	101.197	3204.021	609.205	4.130	81.417	1.808	3908.207
d_s (100)	20%	120	676.588	9.877	148.001	2071.800	643.393	4.871	83.595	1.817	2668.746
	10%	110	663.989	8.587	136.295	2359.526	632.924	4.442	79.631	1.812	3016.972
	−10%	90	629.596	5.021	91.559	3111.433	613.959	3.873	74.844	1.789	3760.502
	−20%	80	617.806	4.511	86.117	3544.662	604.979	3.648	72.915	1.772	4153.493
e_c (0.3)	20%	0.36	650.240	7.290	122.751	2623.403	624.257	4.289	79.675	1.821	3329.076
	10%	0.33	646.938	6.589	112.248	2662.741	623.739	4.209	78.316	1.812	3355.010
	−10%	0.27	641.214	5.513	95.834	2755.434	622.721	4.052	75.660	1.792	3408.252
	−20%	0.24	639.110	5.154	90.270	2807.683	622.220	3.976	74.357	1.781	3435.574
e_a (0.1)	20%	0.12	648.212	6.214	105.041	2601.770	624.480	4.131	76.600	1.822	3335.980
	10%	0.11	645.920	6.079	103.769	2654.055	623.853	4.130	76.789	1.812	3358.660
	−10%	0.09	641.689	5.883	102.239	2760.287	622.601	4.129	77.169	1.792	3404.197
	−20%	0.08	639.685	5.807	101.784	2814.164	621.975	4.129	77.361	1.782	3427.054
M (0.6)	20%	0.72	–	–	–	–	619.875	3.902	73.652	1.850	3531.826
	10%	0.66	–	–	–	–	621.539	4.014	75.296	1.827	3456.018
	−10%	0.54	–	–	–	–	624.945	4.250	78.714	1.776	3307.965
	−20%	0.48	–	–	–	–	626.697	4.377	80.517	1.746	3235.718
E (4.5)	20%	5.40	–	–	–	–	623.226	4.130	76.978	1.536	3381.703
	10%	4.95	–	–	–	–	623.227	4.130	76.978	1.658	3381.564
	−10%	4.05	–	–	–	–	623.228	4.130	76.979	1.977	3381.200
	−20%	3.60	–	–	–	–	623.228	4.130	76.980	2.191	3380.955
r (0.3)	20%	0.36	652.950	7.745	129.077	2662.732	626.292	4.338	79.927	1.801	3296.911
	10%	0.33	649.339	6.975	117.741	2678.119	624.701	4.217	78.160	1.802	3336.778
	−10%	0.27	639.874	5.451	95.300	2749.640	621.841	4.070	76.277	1.802	3430.627
	−20%	0.24	637.010	5.169	91.374	2802.824	620.520	4.033	75.983	1.800	3484.369
v_r (0.5)	20%	0.60	651.513	8.028	134.883	2764.191	623.828	4.259	79.245	1.799	3392.261
	10%	0.55	648.104	6.976	118.430	2730.162	623.509	4.190	78.040	1.801	3386.699
	−10%	0.45	641.720	5.541	96.128	2689.595	622.975	4.076	76.029	1.804	3376.326
	−20%	0.40	640.455	5.282	92.020	2674.987	622.748	4.027	75.170	1.806	3371.451
P (1000)	20%	1200	630.764	3.869	69.793	2623.740	615.751	3.217	61.517	1.827	3409.653
	10%	1100	636.151	4.492	79.432	2617.419	619.427	3.558	67.172	1.819	3366.569
	−10%	900	645.436	7.496	129.163	2931.769	625.958	5.053	93.564	1.773	3497.943
	−20%	800	637.865	7.520	134.023	3256.131	623.826	5.860	109.920	1.737	3751.897
b (300)	20%	360	654.877	8.118	134.327	2643.040	632.104	5.109	92.211	1.785	3228.826
	10%	330	651.653	7.439	124.534	2665.086	627.853	4.587	84.069	1.795	3293.874
	−10%	270	635.339	4.870	86.577	2789.774	618.324	3.743	71.040	1.807	3494.975
	−20%	240	628.033	4.214	77.026	2916.876	613.204	3.422	66.178	1.809	3637.550
V (500)	20%	600	577.898	4.350	96.295	5709.483	573.762	4.085	91.602	1.593	5988.893
	10%	550	578.206	4.393	97.162	5699.965	574.069	4.129	92.511	1.593	5978.835
	−10%	450	578.960	4.497	99.247	5676.286	574.818	4.235	94.682	1.592	5953.962
	−20%	400	579.432	4.561	100.525	5661.214	575.283	4.299	96.007	1.591	5938.230
M_p (5)	20%	6	669.961	9.629	149.117	2274.724	636.162	4.527	80.063	1.816	2889.236
	10%	5.5	660.005	8.293	134.024	2467.919	629.531	4.306	78.262	1.810	3132.108
	−10%	4.5	633.748	5.245	94.070	2978.716	617.143	3.980	75.988	1.793	3636.617
	−20%	4	625.521	4.853	89.937	3264.758	611.224	3.849	75.174	1.783	3897.435

N.B. (–) means infeasible solution or invalid.

When the ordering cost increases, the retailer needs to increase the selling price. As the retailer needs to invest more in ordering a product, it always tries to sell the product at an increased price. Consequently, the cycle time and the number of ordering quantities also increase, but the total profit decreases. When considering green technology, the total profit and green investment (G) decrease with increasing ordering cost.

When the number of trips increases, the selling price, ordering quantity and cycle time also increase. However, the total profit decreases because an increasing number of trips means increases to the transportation cost.

When the purchasing cost per unit increases, the order quantity and total profit decrease because the retailer always has limited capital. As a result, the retailer will have fine-tuned the selling price to avoid losses, which causes the selling price to increase.

With increases of the outbreak COVID-19 rate (r), deterioration, and holding cost per unit time, the total profit, cycle time and ordering quantity decrease.

When the fixed transportation cost, product weight, fuel price, quantity of fuel consumed when the transport is empty, additional fuel consumption per ton and traveled distance increase, the transportation cost increases. For this reason, the retailer raises the selling price to prevent any losses. However, the total profit decreases with higher transportation expenses.

When the emission cost per unit distance and the additional emission cost increase, the total carbon cost also increases, and for this reason, the total profit decreases. To raise profits, green investment is applied to reduce emissions.

When the maximum reduced carbon emissions are increased, the profit from green technology increases. Additionally, green investment increases. In a similar manner, when the efficiency of green technology increases, the total profit also increases.

Managerial insights

This study provides some insights for managers to respond to disruptions in transportation systems that can be implemented in the supply chain during a pandemic. A retailer’s manager can consider some critical decisions as follows:

By implementing this model, a retailer can discover the optimal cycle length and order quantity by responding to the increasing transportation cost after considering the rate of the outbreak. The manager must carefully plan the number of trips, as this number affects the transportation costs, hence, it decreases the total profit.

Proper management of carbon emissions can be achieved through the use of green technology, and managers need to know how much to invest to curb emissions.

A manager can make decisions in which the optimal price will result in maximum profit by considering the whole cost increase in the transportation system and technology investment.

Concluding remarks

The effect of COVID-19 on transportation systems has been illustrated with considerations of the environmental impacts. The proposed model introduces a cost factor in the transportation cost of the inventory model. This model considers a realistic transportation cost that hinges on the rate of the COVID-19 outbreak under price-sensitive demand. This research shows that the intensification of the COVID-19 rate decreases the retailer's profit. Further study reveals some realistic results, such as:

A retailer can decide what extra expense he needs to provide to the driver to transport products to higher-risk areas.

This model allows decisions on how green technology efficiently works within a limited budget under some regulations.

Some pricing strategies with simultaneous investments in emissions reduction that directly impact profit-making decisions have been suggested for deteriorating items.

This study has certain limitations; hence, further research to extend the proposed model can be performed in the future. This study failed to show how COVID-19 affects customers’ purchases instead of the effects on the transportation system. Therefore, a variable time-varying demand may be considered because the pandemic may influence customer demand. This type of work could be a great extension of this study. Furthermore, the limitations of employing a carbon emission reduction process, such as cap-and-trade and carbon offsets, were missing from this study.

CRediT authorship contribution statement

Abu Hashan Md Mashud: Conceptualization, Writing-original draft, Writing-review & editing. Sujan Miah: Conceptualization, Writing-original draft, Writing-review & editing. Yosef Daryanto: Conceptualization, Writing-original draft, Writing-review & editing. Ripon K. Chakrabortty: Conceptualization, Writing-original draft, Writing-review & editing. S.M. Mahmudul Hasan: Conceptualization, Writing-original draft, Writing-review & editing. Ming-Lang Tseng: Conceptualization, Writing-original draft, Writing-review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.