demand.

Along with the worldwide tragedy of human fatalities, Covid-19 has an impact on other economic sectors and activities, including manufacturing, supply chain logistics, etc. [1]. Small cities in low-income nations like India, Pakistan, Afghanistan, Bangladesh, Nepal, Bhutan, Thailand, Myanmar, and the Maldives are experiencing financial difficulties as a result of the lock-down [2] and closure imposed by their respective governments. The government closes the borders of many cities for security reasons. Therefore, in order to maintain their company, small and medium-sized business owners require the most cost-effective EOQ inventory design for products. A quick increase in demand coupled with a lack of significant raw materials because of disturbances in the global supply chain is the worst-case scenario Covid-19 has brought about in terms of vital stock availability [3]. The creation and distribution of vaccines may potentially be hampered by such a supply chain network breakdown [4].

The rate of perishable product degradation has associated effects. Because a perishable product’s rate of degradation is closely tied to its quality, it affects customer demand in a similar way. For instance, freshness is a crucial aspect of the quality of an Agri-fresh product. The look of an Agri-fresh product is often used as a proxy by customers to assess its freshness [5]. Bhunia and Shaikh [6] looked at a predictive inventory model for things that continually degrade. Dye and Yang [7] looked at how preservation investments affected the optimum course of action for an inventory model for perishable goods. According to [8], heterogeneous product quality with a rate of degradation at various temperatures and stages is taken into account. Tiwari et al. [9] provided a supply chain inventory model with an expiry date to ascertain the ideal pricing and replenishment cycle. For perishable items, Shaikh et al. [10] developed an EOQ method in which degradation starts after a certain length of time has passed since the goods were kept. Yang et al. [11] research focuses on controlling degradation rate by treating degradation as a constant. Garg et al. [12] created a methodology to deal with a two-warehouse inventory model for perishable goods with partial backlog in an effort to increase total profit. Rana et al. [13] built research for two warehouse systems for products that were going bad while taking demand disruption during the Covid epidemic into account. Kumar et al. [14] investigated a model for slowly decaying goods in a two-warehouse system under inflationary circumstances. Das et al. [15] takes into account a two-warehouse system with a demand that varies linearly with time and a two-parameter Weibull distribution that accounts for degradation.

The demand function is crucial in developing strategies for inventory models. It varies according on several factors, including time, quality, special offers, etc. The selling price, supply level, product greenness, timing, and other variables all have an impact on variable demand. Using a time- and price-dependent demand rate, Maihami and Kamalabadi [16] examined a joint pricing inventory model. The multi-period inventory routing problem was dealt with by [17] using stochastic demand. Recent publications in this area include works by [18]–[21] and others. The major aspects of our contribution are compared to published papers in Table 1.

Rana et al. [13] investigated an EOQ model for a two-warehouse system that incorporates linear time-dependent demand while taking Covid-19 lock-down demand disruption into account. The article takes two scenarios, one for a short lock-down time and the other for a lengthy lock-down period, to describe the Covid-19 pandemic issue. Rana et al. [13] does not consider the possibility that a government may be late in declaring lock-down in order to control the Covid-19 transmission rate. This research examines the prospect of an unavoidable scenario similar to Covid-19, in which the government may be late in declaring a lock-down or shut-down, and result in the inventory of Owned Warehouse (OW) being fully depleted before the lock-down start owing to customer over stocking. Additionally, to add realism to the model, Weibull distribution is used to account for the pace of degradation. This model takes the First In, First Out (FIFO) method into account to lower the effect of deterioration.

The remaining portions of the study explains follows; in Section 2, the research classify problem description, mathematical notation and assumption. The required solution procedure and a numerical example taken to validate the research model in Section 3. In Section 4 required sensitivity analysis processed for various parameter. Finally, a justified conclusion draws to the research model in Section 5.

+ ∫ ab tb−1Ir3(t)dt + ∫ ab tb−1Ir4(t)dt].

TL TO

tw b−1 −atb−1 βt b+1 α βt CD = Cp [∫ ab t e [W − t (α + ) − at ( + )] dt 0 2 b + 1 b + 2

tw

+ ∫ ab tb−1Xe−atbdt TL b−1 −atb β 2 2 + ∫ ab t e [α(tw − t) + (tw − t ) tw 2 αa b+1 b−1 βa b+2 b+2 + (tw − t ) + (tw − t ) + X] dt b + 1 b + 2

TO

+ ∫ ab tb−1e−atb [X

TL

a b+1 b+1 + (Δd − α) (t − TL + (t − TL )) b + 1 1 2 2 a b+2 b+2 − β ( (t − TL ) + (t − TL ))] dt 2 b + 2 T b−1 −atb a b+1 b+1 + ∫ ab t e [(α + Δd1) ((T − t) + (T − t )) TO b + 1 1 2 2 a b+2 b+2 + β ( (T − t ) + (T − t ))]dt].

2 b + 2

b atb w αabtb w+1 atb w βabtb w+2 atb w CD = Cp[Watw (1 − ) − (1 + ) − (1 + ) 2 b + 1 2b + 1 2(b + 2) b + 1

2 2b+1 1 atb w + αa btw ( + ) 2b + 1 (b + 1)(3b + 1)

2 2b+2 1 atb w b atb w + βa btw ( + ) + Xatw (1 − ) 4(b + 1) (b + 2)(3b + 2) 2

+ ab [α [tw (TL b − tb w) − atw (TL 2b − t2b w ) − 1 (TL b+1 − tb w+1) (16)

b 2b b + 1 a 2b+1 2b+1 + (TL − tw )] 2b + 1 β t2 w b b at2 w 2b 2b 1 b+2 b+2 + [ (TL − tw) − (TL − tw ) − (TL − tw ) 2 b 2b b + 2

a 2b+2 2b+2 + (TL − tw )]] 2b + 2 αa tb w+1 b b atb w+1 2b 2b 1 2b+1 2b+1 + [ (TL − tw) − (TL − tw ) − (TL − tw ) b + 1 b 2b 2b + 1 a 3b+1 3b+1 + (TL − tw )] 3b + 1

βa tb w+2 b b atb w+2 2b 2b 1 2b+2 2b+2 + [ (TL − tw) − (TL − tw ) − (TL − tw ) b + 2 b 2b 2b + 2 a 3b+2 3b+2 1 b b a 2b 2b + (TL − tw )] + X [ (TL − tw) − (TL − tw )] 3b + 2 b 2b

tw b b atw 2b 2b 1 b+1 b+1 + ab [α [ (T0 − TL ) − (T0 − TL ) − (T0 − TL ) b 2b b + 1

a 2b+1 2b+1 + (T0 − TL )] 2b + 1 β t2 w b b at2 w 2b 2b 1 b+2 b+2 + [ (T0 − TL ) − (T0 − TL ) − (T0 − TL ) 2 b 2b b + 2 a 2b+2 2b+2 + (T0 − TL )] 2b + 2

αa tb w+1 b b atb w+1 2b 2b 1 2b+1 2b+1 + [ (T0 − TL ) − (T0 − TL ) − (T0 − TL ) b + 1 b 2b 2b + 1 a 3b+1 3b+1 + (T0 − TL )] 3b + 1

βa tb w+2 b b atb w+2 2b 2b 1 2b+2 2b+2 + [ (T0 − TL ) − (T0 − TL ) − (T0 − TL ) b + 2 b 2b 2b + 2 a 3b+2 3b+2 + (T0 − TL )] 3b + 2 1 b+1 b+1 a 2b+1 2b+1 Tl b b + Δd [ (T0 − TL ) (T0 − TL ) − (T0 − TL ) b + 1 2b + 1 b aTl 2b 2b + (T0 − TL )]

1 2b+1 2b+1 a 3b+1 3b+1 Tlb+1 b b + Δda [ (T0 − TL ) − (T0 − TL ) − (T0 − TL ) 2b + 1 3b + 1 b

aTlb+1 2b 2b 1 b b a 2b 2b + (T0 − TL )] + X [ (T0 − TL ) + (T0 − TL )]] 2b b 2b

T aTb b b 1 2b 2b + ab(Δd1 + α) [ (1 − ) ((T − T0 ) − (T − T0 )) b b + 1 2 a T2b+1 − T0 2b+1 Tb+1 − T0 b+1 a(T3b+1 − T0 3b+1) + (1 + ) ( − + ) b + 1 2b + 1 b + 1 (b + 1)(3b + 1) 2 T2(T2b − T0 2b) 1 aTb T2b+2 − T0 2b+1 + βa b ( ( − ) − 2b 2 b + 2 4(b + 1) a(T3b+2 − T0 3b+2) − )]].

(b + 2)(3b + 2)

Total cost per cycle for the given model,

TC = OC + CH + CpQ + CD, (17) where OC, CpQ, CH and CD are consider for total ordering costs, purchasing costs, holding costs and degradation costs for the given model respectively.

The entire cost of the inventory system is calculated using the following steps, and the graph shows how different factors affect the system overall cost.

Step 1. Configure each input parameter.

Step 2. Analyze the pace at which demand is changing D(t).

Step 3. Determine how many items are stored in RW (X).

Step 4. Analyze the stock quantity in OW and RW at every given time t, IO1, and Ir1, Ir2, Ir3, Ir4 respectively.

Step 5. Calculate tw.

Step 6. Determine the Total Costs (TC) per cycle.

Step 7. Display the total cost curves based on several factors.

Here we use two demand parameters like α (the starting demand rate), β (the rate at which demand rises over time) and two Weibull distribution degradation parameters like a (scale parameter), b (shape parameter). And we use another two must important factors i.e., Δd (reduced demand rate due to, lock-down imposed), Δd1 (rise in demand rate due relax the lock-down). We consider G as holding costs per unit stocks in OW and E as holding costs per unit stocks in RW. The lock-down period Z = (TO − TL) plays an important role in this model. Here 20% and 10% changes in parameter consider and also percentage changes in total cost are calculated in Table 3 and graphical sensitivity analysis shown in Fig. 2.

I. We can see that any modification to these factors has a substantial impact on the result. When the lock-down period Z increases, the degradation cost as well as the holding cost increases So the total cost increases.

II. The parameter and β are related to demand function, so when α or β increases the demand is increasing.

Therefore, effect of degradation decreases and the holding time of stocks at both warehouse decreases which implies a decrease in total degradation cost and holding cost. So total cost decreases.

III. When the holding costs per unit item of OW (G) and RW (E) rises, the total holding costs also rises while other costs remain constant, which results in an increase in the total costs.

IV. Δd is the amount of demand decrease due to lock-down imposed by government. So, when Δd rises the demand rate fall so stocks are kept at warehouse for a longer time and effect of degradation increases and due to this, the holding cost and degradation cost was increases. Therefore, total cost of the model increases. V. Lock-down was relaxed by government so Δd1 amount of demand increases because panic buying behaviour of the customer. So, when Δd1 increases, the holding period of stocks at warehouse decreases and effect of degradation decreases. So, the total holding costs and degradation cost decreases. Therefore, total costs of the model decreases.

a. Total cost (TC) varies with respect to .

b. Total cost (TC) varies with respect to .

c. Total cost (TC) varies with respect to and time .

In the majority of countries, the Covid-19 outbreak has dramatically changed the economic landscape. During Covid-19 epidemic, many victims are owners of medium- and small-sized businesses. A country’s economy will expand in the business sector, which is the pillar of the economy. This article investigates the effect Covid-19 epidemic and suggests an inventory approach. This mathematical research was investigating, (a) the impact of demand interruption, i.e., the decline and increasing demands due to Covid-19 lock-down, (b) the effect of time frame of lock-down on the total costs, and (c) importance of two warehouse inventory system. We solve a mathematical scenario to show how the lock-down duration influences total expenditures. This research examines the importance of FIFO policy to measure the effect of degradation. Due to discounts provided for big orders as well as other factors, the merchant makes more orders than the warehouse can store. The merchant is therefore needed to keep the excess supplies in any RW in this model. This research demonstrates that increasing OW capacity may lower total holding costs since RW has higher holding costs than OW, hence lowering the system’s overall cost. It has a number of limitations, for instance, it ignores inflation and partial backlogs and employs a fixed rise and drop in convenience demand because genuine demand disruption is difficult to assess.

Problem finding, article writing and sensitivity analysis: D. Das; Article reviewing, designing and editing: G. C. Samanta

The author D Das is thankful to University Grant Commission (UGC), Govt. of India for providing JRF (Ref No. 201610142889) to carry out his research.

No external data use for this research work.

The authors have no conflicts of interest to declare that are relevant to the content of this article. All authors contributed equally to this research work.