Effect of Carbon Emission and Human Errors on a Two-Echelon Supply Chain under Permissible Delay in Payments

Environmental worries in production and inventory models have received large attention in inventory management literature. In this paper, an economic production model is proposed with two-echelon supply chain when trade credit is offered by the supplier. This paper proposes human errors of Type I and Type II due to fatigue and inexperience of the inspector during screening. It considers the use of energy for production along with greenhouse gases (GHG) emission from production and transportation operations. The developed model optimizes the environmental and economic performances of the supply chain. Our aim is to explore the effects of human errors during inspection on the emission cost, transportation cost and delay in payment on the replenishment of order sizes and the expected total profit of the retailer. A mathematical model is developed and numerical examples are provided to illustrate the solution procedure. KeywordsSupply chain, Trade credit, Emission cost, Fuel cost, Human errors.


Introduction
There is an increasingly common agreement that carbon footprints (carbon dioxide emissions and other greenhouse gas emissions), if ignored will lead to major changes in the climatic conditions of our environment system. Administrations are under growing pressure to enact regulation to restrict the emission of the harmful gages. In this direction, firms from all over the world are responding to such regulation and are undertaking initiatives to decrease their carbon footprint. However, these initiatives are more focused on reducing emissions due to the bodily processes involved. For example, replacement of unproductive equipment and facilities, reforming products and wrapping, finding sources of energy that are less polluting, or introducing energy savings programs. The attribute of quality and concern about the ecological surroundings with the perspective of inventory management have received considerable attention from researchers. Today, these topics are analyzed in an in depend manner.
Well known aim of a logistic network is to increase the financial act which was given by Beamon (1999). The expense due to carbon discharges is generally overlooked in finding choices. The significance of precisely estimating, announcing and lessening the carbon impression of a supply chain is expanding. For example, a hostile goal for dropping carbon emission has been set by Mexico, which aimed to reduce 50% of Greenhouse gases emissions by 2050 as compared to Güereca et al. (2013). In another precedent, an ongoing report by Blanco et al. (2016) found that a large number of the expansive firms in the United States revealed 22% of their full extension 3 outflows in 2013.
The order, holding, and setup costs are the main segments of cost in supply chain. The ordering cost involves the cost of fuel as well as transportation among the other costs like salary of driver, a portion of vehicle's purchase cost, and vehicle maintenance cost among other costs which are elaborated in Waters (2003). Aggregation of these costs by the firms is for the easiness of computation and negotiation. In any case, these accumulations of expenses do not demonstrate the impact of a specific cost that could be expanding at a rate quicker than those of alternate expenses in the bundle. Transportation costs should also include fuel cost as its own component.
In general, it is assumed that each engineering system manufactures a product of good quality. Though, in a real situation, a certain quantity of items may be produced which are of imperfect quality. This can be due to the fault of machine or by ignorance of labour. Therefore, inspection turns out to be vital for all items that are produced before selling. However, several researchers overlooked the consequence of human errors (which can be because of fatigue, sickness and ignorance) during inspection process of the quality of the items in the supply chain models. Raouf et al. (1983) was the first one who proposed that mistakes can be made by inspector too. Zhang and Gerchak (1990) proposed an inspection policy for EOQ model where the yield considered is random quantity. Shin et al. (2018) compared the two-echelon supply chain models. The first model was the effect of imperfect quality items while the second was effect of human errors. The effect of transportation and the emission through transportation was calculated. A model is proposed to find the effect of human error and emission through carbon on the supply chain.  Salameh and Jaber (2000) Cárdenas-Barrón (2000) Duffuaa and Khan (2005) Maddah and Jaber ( Two-level supply chain model is proved with credit financing, carbon emission cost and human errors. In this model, the vendor manufactures only one product according to the buyer's demand. It is considered that goods produced or received are of imperfect quality, energy is consumed in the production system and the release of greenhouse gases from production and transportation operations. The model optimizes the economic and environmental performances of the network. The impact of imperfect quality items in the stock along with fuel cost, and emission cost from production and transportation is applied in different scenarios. Contributions of the various authors have been given in Table 1 2

. Assumption and Notation
The following assumptions are considered to formulate the models.

Assumption
(i) A single type of item is delivered by a vendor to a buyer. (ii) Human errors are considered in the inspection process. Two types of errors are considered. (iii) Fixed time period is offered by the vendor to the buyer for settling the accounts. Interest is not charged during this period but will be charged after the completion of this period with certain terms and conditions. (iv) Constant rate of demand and production is considered. Rate of production is greater than the rate of demand. (Wee et al., 2007). (v) Items that are of perfect quality are considered as the imperfect item ( error) and imperfect item as perfect ( error) during inspection. (Shin et al., 2018). (vi) All the trucks are of alike capacity, fuel consumption and emissions. No change in the consumption of fuel and emission over time and change in drivers. (Aljazzar et al., 2018). (vii) Constant rate of emissions from the manufacturing process of product.

Notations
The following notation are used to develop the model

Modelling
According to Shin et al. (2018) as per Figure 1, every lot delivered contains percentage of imperfect items with a known uniform probability density function, ( ). The lot delivered was inspected at a constant rate, before selling it to the market. The rate of inspection is presumed to be greater than the rate of demand, > for avoiding the shortages. However, during the inspection process, inspector by mistake selects perfect item as imperfect (defective) ( error) and imperfect item as defective ( error). A total expected cost including cost of credit financing, carbon emission cost and human errors in the supply chain is obtained. The lot is inspected with constant rate. 1 items are incorrectly categorized as imperfect from the (1 -) 1 perfect items, and 2 items are incorrectly acknowledged as perfect items from the (1 -2 ) imperfect items. The imperfect items as detected by the inspector are and equal to (1 -) 1 + (1 -2 ). , 1 , 2 and are assumed random variables. Thus, the total expected value of imperfect items is [ ] can be written as (  Shortages are avoided by considering that screened lot is higher than or equal to the demand through inspection 1 = ( / ) i.e., is the fixed period offered by vendor to buyer to settle the account. There will be three different possible cases for the calculation of buyer's cost, ( , ), = 1, 2, 3 Case 1: ≤ 1 ≤ Case 2: 1 ≤ ≤ Case 3: ≥ .
: ≤ ≤ The interest earned by buyer during time period 0 is 2 2 ⁄ and the interest paid for the inventory not sold after the credit period, , is ( -) 2 2 ⁄ + ( 1 - } + The total expected cost of buyer per unit time is From equation (1) By differentiating equation with respect to and equating it to zero we get the optimal order quantity * , [ 1 ( , )] = 0.
Sufficient condition has to be satisfied so that the total expected cost of the supply chain to be convex, 2 [ 1 ( , )] 2 > 0.
To obtain the optimal order quantity in all the three cases DMS Algorithm 2 Shin et al. (2018) is applied.

Numerical Example
Following is example is shown for the effect of human error Shin et al. (2018) and carbon emission on supply chain.

Sensitivity Analysis
Sensitivity analysis is made on different factors to find the effect of and error, effect of rate of change of errors and the carbon emission because of transportation on the total expected cost. International Journal of Mathematical, Engineering andManagement Sciences Vol. 5, No. 2, 225-236, 2020 https://doi.org/10.33889/IJMEMS.2020.5.2.018 233 Figure 3. Effect of rate of change of human error on total cost Here graph clearly proves that growth in the errors ( and error)is directly proportional to the cost for the whole supply chain. The increment of emission from the whole supply chain (including transportation and emission rate) will increase the total cost. The increment in carbon emission will also affect the environment causing more pollution.
From Figure 3 it can be concluded that the variation in the inspection process due to inspector mistakes affect the final cost of the supply chain. More the variation because of tiredness or ill health will lead to higher investment from the buyer. Figure 4 represents the effect of emission of carbon from transportation and production on the supply chain. The carbon emission tax will increase with the emission. Definitely it will increase the total cost of supply chain. The increment in carbon emission affects the supply chain but also the environment dangerously.

Managerial Insights
Some practical applications exist for this paper. The carbon emission from transportation and the production process will not only affect the environment but also upsurge the cost of the model. Human error will lead to addition of another cost in the supply chain. The delay in payment process helps in reduction of cost in supply chain but the addition of emission cost and human error cost increases the total cost.
The effect of different parameters on the total cost has been represented in the figures. The presence of defective items in the lot and emission from the transport increases the cost and the pollution in the environment. We should work on the remedies to reduce pollution like by planting of more trees and maintaining the vehicles.

Conclusion
In this paper, an inventory model is formulated considering the effect of human errors on the supply chain when carbon emission tax is taken into consideration. The effect of presence of human errors and carbon emission can be seen from the graph. As number of imperfect quality items increase the final cost of the vendor and the buyer both increases. The presence of imperfect quality items lead to more demand from the retailer which will indirectly lead to more carbon emission. It will also lead to more demand from the vendor which will ultimately require more production and transportation. Extra transportation means more carbon emission which is harmful to the environment. Therefore, perfect cautions should be taken while handling the lot, during inspection process. For future research rework as well as salvage value can be added for imperfect items.