A deterministic model for the inventory policy of countries for procurement of vaccines

Abstract

The countries are the units that procure the vaccines during the COVID-19 pandemic. The delivered quantities are huge. The countries must bear the inventory holding cost according to the variation of stock quantities. This cost depends on the speed of the vaccination in the country. This speed is time-dependent. The vaccinated portion of the population can be approximated by the cumulative distribution function of the Cauchy distribution. A model is provided for determining the minimal-cost inventory policy and its optimality conditions are provided. The model is solved for 20 countries for different numbers of procurements. The results reveal the individual behavior of each country.

Introduction

Yellow fever, caused by Flavivirus, a filterable agent, was reported as the first virus infected human beings. It caused pandemic in 1901. Afterward, the 1918–1920 pandemic, Spanish flu (H1N1), is regarded as the greatest medical disease of the time, affecting millions of people worldwide. Furthermore, in 1997, the first human cases of a new and extremely dangerous avian influenza virus—the H5N1 strain—were discovered in Hong Kong. The most recent serious case is the outbreak of the COVID-19 pandemic (Kinlaw et al. 2009; Pradhan et al. 2020).

The outbreak of the 2019 novel coronavirus disease (COVID-19) has brought challenges in many areas of the world (Dwyer et al. 2020; Gabster et al. 2020; Mamun and Ullah 2020; Wollina 2020; Zhou et al. 2020). (World Health Organization, no date), it is a pandemic viral disease that caused more than 620,000,000 people infected and more than 6,540,000 deaths worldwide until 12 October 2022. Vaccination has began at the end of 2020 with the full approval of Pfizer by the Food and Drug Administration (FDA) of the USA. In order to accomplish vaccination; vaccine supplies, people to implement them and people to be vaccinated are required (U.S. Food and Drug Administration, no date; Mills and Salisbury 2021).

Other methods of prevention including wearing a mask, washing hands, and obeying the social distance rule, reduce the risk of transmission in the community as well (Güner, Hasanoğlu and Aktaş 2020). However, it has been discovered that the vaccine must have an efficacy of at least 70% to prevent an epidemic, and this emphasizes the impact of the vaccine on epidemics (Bartsch et al. 2020).

In inventory problems, stockouts and overstock can be handled to meet the demand of dynamic customers (Shekhawat et al. 2016). Both parameters are costly situations for companies and their risks of them are reducible. Improvements in ordering policy cope with stockouts (Antic, Djordjevic Milutinovic and Lisec 2022). From a supply chain management perspective, perishable products such as vaccines are difficult to transport and store and need accurate inventory control models. In other words, inventory optimization strategies need to connect with the right balance to customer demand to avoid inventory holding (O’Neill and Sanni 2018). This is especially true in highly uncertain and dynamic markets, such as those created by the pandemic period. It is of critical importance today due to the large size demand for vaccination required for social needs in the recent Covid-19 epidemic, and because it is subject to public procurement with urgent timing (Patriarca et al. 2020). We assume that the number of vaccines is enough to cover a significant part of the population. Decision-makers are taking a significant role in vaccine allocation in order to reduce problems caused by the disease. The effectiveness of distribution of a limited vaccine inventory in a certain population can be measured by certain outcomes. After a proper distribution model, it is expected to reduce the total number of cases and the total number of losses of life. However, delivering the vaccine to people does not only include the activity of transportation in order to dispatch a certain amount of vaccine and also includes storing in stock points. In this case, some inventory policies should apply which provide a yield of minimizing the cost of inventory holding. It is necessary to provide vaccines uninterruptedly and to create a vaccination policy for this purpose. Properties of perishable products should also take into consideration. On the other hand, decisions depend on the willingness of the population for being vaccinated in the society, because it directly affects the number of requirements of the countries. The paper suggests a new policy for the procurement of vaccines on the level of individual countries. The policy enables the governments to act in a rational and economic way such that still every citizen who volunteers for obtaining the vaccine can get it.

Here, we provide an inventory policy for the pandemic period for the countries. This paper presents a deterministic model for vaccines with a demand rate variable over time for the countries. It is aimed to provide an analytical model to deal with minimization of holding cost and develop inventory policies regarding this aim to be used for a variety of perishable products such as vaccines. In Sect. 2, the saturation process is introduced. In Sect. 3, an approximation of the vaccination curve of the countries has been discussed. In Sect. 4, a deterministic model for inventory policy has been developed. Section 5 discusses the numerical results obtained for 20 countries. The paper is finished with conclusions.

Saturation processes

The saturation process is best illustrated by the life cycle of a product. The life cycle of a product begins by entering the market. Only a few people know the new product, so consumption is growing slowly. As time goes on, more and more people get to know the product. A nearly constant consumption is formed. Later, other, alternative and substitute products will appear on the market. Therefore, the consumption of the product under investigation slows down and eventually stops. The first finding that is immediately apparent is that it is not possible to an infinite amount of the product. The market is saturating and even the curve of total consumption does not cross a certain threshold. If we look at the curve that shows the volume of all products sold from the product release to a given moment, it will be like an elongated letter S. This particular curve is constantly increasing. If the events occur as described above, the curve will have at least one inflection point. For very famous products, such as the release of a new volume from a well-known book series, or the start of selling a new mobile phone from a popular brand, when interested customers line up for either the book or the phone, the event curve may be different. Namely, consumption is very high in the first period and then decreases later. Thus, the curve of total sales increases steeply first and then decreases. Then the curve has no inflection point. In that case, the curve itself is like a concave function.

The vaccination is very similar to the life cycle of a product. It is not possible to vaccinate more people than the total population. The actual value is even less as some people cannot be vaccinated because of medical reasons, and others resist or are unavailable. Thus, the vaccine is a product with a bounded total demand.

The mathematical definition of a saturation function is as follows:

Definition

A function $f\left(x\right)$ is called saturation function if $f\left(x\right)$ is a real-valued function on the $[-\infty ,\infty ]$ interval such that (i) it is monotone increasing, (ii) $\underset{x\to -\infty }{\lim }f\left(x\right)=0$ , (iii) there is a positive number $L$ such that $\underset{x\to +\infty }{\lim }f\left(x\right)=L$ .

Obviously, it can be seen that the saturation function is very similar to the cumulative distribution function of probabilistic distributions. The only difference is that the limit at $+\infty$ is not necessarily 1 but it can be another positive number. Figure 1 shows a saturation function which is a cumulative distribution function at the same time. It is the cumulative distribution function of the Cauchy distribution.

Fig. 1.

The function $\frac{arctan\left(x\right)}{\pi }+0.5$

In mathematics, a function is called the sigmoid function which is very similar to the saturation function. The sigmoid function must have exactly one inflection point. There is no such a claim in the case of saturation function. On the other hand, the sigmoid function can converge at $-\infty$ to any value. There are many functions which are both sigmoid and saturation. The word saturation expresses better the role of the function in the applications.

Approximation of the vaccination curve of the countries

The total percentage of vaccinated people in a country divided by 100 is a number between 0 and 1 (Our World in Data, no date). This total number is an increasing function of time. Thus, it can be approximated by the cumulative distribution functions of a probability distribution. There are plenty of known functions which are the cumulative distribution function of frequently used probability distributions. The function of Fig. 1 is based on the inverse of the tangent function and its formula is.

$$\frac{arctan\left(x\right)}{\pi }+0.5.$$

It is both a sigmoid and saturation function. The function is not suitable for regression in this pure form. Three parameters must be introduced. One is the position of the inflection point and is denoted by b. The next one determines its steepness at the inflection point. It is a. The last one is the height of the shift vertically such that at the beginning the value of the function is 0. The general formula of the function is

$$\frac{arctan\left(a\left(x-b\right)\right)}{\pi }+c.$$ 1

Many saturation processes have the feature that the saturating property can be uncovered earlier than the process reached the inflection point. It is the case with vaccination as well. It is illustrated in the example of Denmark. The time series of the country consists of 319 days. Denmark did not reach yet the 80 percent vaccination until the end of the period. The inflection point is about the 160th day. Regression is made after 32, 64, 96, and 300 days. Figure 2 shows the final result of the regression after 300 days. Notice that the two curves, i.e. the original time series and the regression function are very close to one another. The first three curves are compared with the last one in Fig. 3. Table 1 summarizes the basic data. The saturation is not recognized yet after 32 days. The regression function in Fig. 3a is almost linear. The reason is that the process is at the beginning. The achieved vaccination is less than 3 percent. The saturation is uncovered after 64 days. The steepness of the curve is somewhat overestimated as Fig. 3 (b) shows. However, the regression function after 96 days is very close and the inflection point is still not achieved, i.e. the process is still in its first half in Fig. 3c.

Fig. 2.

The final result of the regression after 300 days

Fig. 3.

The results of regression for after a 32 days; b 64 days; c 96 days

Table 1.

Time length of the regression period and vaccination percentage

Day	Achieved percentage of vaccination
32	2.95
64	6.94
96	12.48
300	76.86

The saturation is still not uncovered as can be seen in Fig. 3a. After 64 days. The saturation is uncovered. However, the steepness is overestimated in Fig. 3b. After 96 days, the two regression curves are close to one another in Fig. 3c.

It turned out that function (1) works for every country well. High correlations are achieved. Table 2 summarizes the results for a selected set of countries.

Table 2.

The parameters of the countries

Country	Parameters			Correlation	Available percentage
	a	b	c
Belgium	0.019440264	142.3695511	0.372352365	0.998409235	78
Canada	0.023610645	141.5326286	0.367132913	0.998222367	79
Cyprus	0.045090581	20.01509258	0.224373146	0.994908854	65
Czecia	0.011369405	132.4762014	0.264628608	0.993386019	62
Denmark	0.020214063	159.1129644	0.407710152	0.998316487	81
Finland	0.021794171	94.2677651	0.367909632	0.999739483	77
France	0.016971981	149.3836653	0.381809048	0.998825287	77
Hungary	0.018408277	73.06371651	0.252787089	0.995603327	64
Iceland	0.063514378	54.17233745	0.434359993	0.99793347	84
Ireland	0.018075924	150.5716412	0.389356595	0.998955126	78
Italy	0.018014458	151.3257418	0.377070492	0.999624445	77
Malta	0.022652701	95.41681316	0.39760535	0.999183087	82
Mexico	0.010377526	188.0011619	0.32312548	0.998082526	56
Peru	0.012379217	200	0.349426928	0.995727225	58
Portugal	0.025237491	162.0226288	0.461236454	0.99488844	83
South Korea	0.021931008	161.1951162	0.437882706	0.991209179	79
Spain	0.028466217	107.8294504	0.417967064	0.998390744	82
Turkey	0.012053373	140.355595	0.368700663	0.993605162	68
UK	0.015992445	55.39616283	0.303422349	0.995669144	72
USA	0.013378891	91.25265859	0.249549082	0.99807531	65

A deterministic model for inventory policy

This section elaborates on a model for minimizing the inventory holding cost of vaccines. It is based on the results of the previous section. It was shown that the vaccination process of the countries can be approximated within a short time from the beginning of the vaccination with high accuracy. The model is based on a sequence of assumptions as follows:

1. It is assumed that the vaccination process is deterministic and the total percentage of the vaccinated population has an a priori course in time.

2. This course is described by a saturation function.

3. The number of procuring of the vaccine is given. Each procuring is made by issuing an order.

4. The lead time is 0. i.e. the ordered quantity arrives immediately.

5. The first order is issued at the beginning of the process when still nobody is vaccinated.

6. Any further order is issued when the stock level becomes 0.

7. The process is finished when a certain percentage is achieved. This vaccination level can be different in different countries as the willingness for vaccination is depending on the country.

The inventory holding process is always ahead of the vaccination, i.e. there is no negative stock level. The total vaccinated percentage of the population must be compared to the total arrived vaccine quantity. The latter one is an increasing step function. The first one is a saturation function obtained by regression of the previous section. The two can be seen in Fig. 4.

Fig. 4.

The solution for Denmark is if the number of procurements is 3

The total inventory holding cost is proportional to the area between the two functions as this is the quantity in the stock. It is what should be minimized primarily. However, the area under the saturation function is fixed. See a more detailed mathematical explanation below. Thus, the area between the two functions depends only on the step function. Consequently, it is enough to minimize the area under the step function.

To formalize the mathematical model, some notations are used as follows:

$f(t)$	The saturation function, i.e. the function of the total demand
$n$	The number of orders
$t_i$	The time of order $i(i=1,2,\cdots ,n+1)$
$T$	The time when the target percentage of the vaccination is achieved

The time unit that can be used in real practice is a day. A dummy order is added to the set of orders. It is when the process is finished at the target percentage, i.e. a day $T$. Its ordered quantity is 0.

The stocking process consists of a first-order at time 0, and there are $n-1$ further reorders. A reorder is issued exactly at the moment when the stock becomes 0. The ordered quantity arrives immediately as the lead time is 0. In other words, the ordered quantity is enough exactly until the next reorder point. Thus, the total ordered quantity until the time $t_i$ is $f(t_{i+1})$, as it must be exactly enough until $t_{i+1}$. Hence, the ordered quantity at $t_i$ is $f(t_{i+1}){-f(t}_i)$. The latter quantity is the demand in the time interval $\left[t_i,,,t_{i+1}\right]$.

The inventory holding cost in the time interval $\left[t_i,,,t_{i+1}\right]$ is proportional to the area between the step function and the saturation function. This area is

$$\left(t_{i+1}-t_i\right)f\left(t_{i+1}\right)-\underset{t_i}{\overset{t_{i+1}}{\int }}f\left(t\right)dt.$$

Thus, the total inventory holding cost is proportional to

$$\begin{array}{cc}\hfill \sum _{i=1}^n\left(\left(t_{i+1}-t_i\right)f\left(t_{i+1}\right)-\underset{t_i}{\overset{t_{i+1}}{\int }}f\left(t\right)dt\right)& =\sum _{i=1}^n\left(t_{i+1}-t_i\right)f\left(t_{i+1}\right)-\sum _{i=1}^n\underset{t_i}{\overset{t_{i+1}}{\int }}f\left(t\right)dt\hfill \\ \hfill & =\sum _{i=1}^n\left(t_{i+1}-t_i\right)f\left(t_{i+1}\right)-\underset{0}{\overset{T}{\int }}f\left(t\right)dt.\hfill \\ \hfill \end{array}$$

The last term is independent of the selection of the reorder points. Thus, it is enough to minimize the last but one term. Hence, the optimization problem determining the best inventory policy is as follows:

$$min\sum _{i=1}^n\left(t_{i+1}-t_i\right)f\left(t_{i+1}\right)$$ 2

$$t_1=0$$ 3

$$t_{i+1}\ge t_{i}i=1,\dots ,n$$ 4

$$t_{n+1}=T.$$ 5

The results obtained from the model are in Table 3. Also, population information of the countries is provided in same table (Worldometer, no date).

Table 3.

Direct results obtained from the model

Countries	The number of procurements			Target percentage	Population
	3	4	5
Belgium	118,98	113,38	109,91	70	11,589,623
Canada	119,33	114,21	111,02	70	37,742,154
Cyprus	24,43	22,46	21,31	60	1,207,359
Czechia	119,12	111,61	107,10	60	10,708,981
Denmark	158,97	150,83	145,81	80	5,792,202
Finland	81,15	76,68	73,95	70	5,540,720
France	121,81	115,92	112,28	70	65,273,511
Hungary	70,34	65,59	62,76	60	9,660,351
Iceland	50,05	47,76	46,33	80	341,243
Ireland	120,41	114,86	111,42	70	4,937,786
Italy	125,03	119,15	115,51	70	60,461,826
Malta	105,01	98,13	93,99	80	441,543
Mexico	99,22	94,67	91,86	50	128,932,753
Peru	101,73	97,61	95,04	50	32,971,854
Portugal	137,94	132,63	129,28	80	10,196,709
South Korea	114,42	110,18	107,51	70	51,269,185
Spain	105,09	99,73	96,42	80	46,754,778

Evaluation of the numerical results

The optimality condition of problem (2)–(5) can be obtained by the Karush–Kuhn–Tucker Theorem (May 2020) of optimization theory. However, the same condition can be obtained in a simpler way as well. It is provided in the Appendix. On the other hand, the problem has a very easy structure. Excel solver got the optimal solutions in all cases within 1 min. Thus, the optimality condition was not used directly in the research.

The optimal value given in Table 2 has the unit

$$proportionofthepopulation\times vaccine\times day.$$

The result can be transferred $vaccine\times days$ if it is multiplied by the population. Still, it is necessary to subtract the area under the approximation curve again multiplied by the population.

To obtain the latter quantity, the function given by formula (1) must be integrated between 0 and the day when the target percentage of the vaccination is reached according to the approximation function. Notice, that the stocking policy is determined far before the vaccination process is finished. Thus, the only approximation can be used in the calculations.

The requested formula of

$$\underset{0}{\overset{T}{\int }}\left(\frac{arctan\left(a\left(t-b\right)\right)}{\pi }+c\right)dt$$

is based on the well-known equation as follows:

$$\int arctan\left(t\right)dt=tarctan\left(t\right)-\frac{ln\left(1+t^2\right)}{2}.$$

Hence

$$\begin{array}{cc}\hfill \underset{0}{\overset{T}{\int }}\left(\frac{arctan\left(a\left(t-b\right)\right)}{\pi }+c\right)dt& ={\left[\frac{\left(t-b\right)arctan\left(a\left(t-b\right)\right)}{\pi }-\frac{\text{ln}(1+\left(a\left(t-b\right){)}^2\right)}{2\pi a}+ct\right]}_0^T\hfill \\ \hfill & =\frac{\left(T-b\right)arctan\left(a\left(T-b\right)\right)}{\pi }-\frac{\text{ln}(1+\left(a\left(T-b\right){)}^2\right)}{2\pi a}\hfill \\ \hfill & +cT+\frac{barctan\left(-ab\right)}{\pi }+\frac{ln\left(1+a^2{b}^2\right)}{2\pi a}.\hfill \\ \hfill \end{array}$$ 6

If the value of formula (6) is subtracted from the optimal value given in Table 2 and the result is multiplied by the population, then the inventory holding cost is obtained in $vaccine\times days$. These results are in Table 4. It is converted to money if this quantity is multiplied by the inventory holding unit cost, i.e. the cost of storing one unit of vaccine for one day.

Table 4.

Numerical results for 20 countries

Counties	a	b	c	Final Day (T)	Formula (F)	3	4	5
Belgium	0.019440264	142.3695511	0.372352365	228	64.29026361	90.19075904	83.57677798	79.64537806
Canada	0.023610645	141.5326286	0.367132913	215	53.24148951	78.03480276	71.64457211	67.86551724
Cyprus	0.045090581	20.01509258	0.224373146	74	27.49832022	33.94322493	32.43122393	31.4920501
Czecia	0.011369405	132.4762014	0.264628608	287	83.11237951	112.3800927	105.1931561	100.8328068
Denmark	0.020214063	159.1129644	0.407710152	300	115.0464986	149.7767118	141.273454	136.0812595
Finland	0.021794171	94.2677651	0.367909632	173	58.29396609	77.20242481	72.47309479	69.63043015
France	0.016971981	149.3836653	0.381809048	241	71.59856407	98.5424194	91.66398699	87.57603406
Hungary	0.018408277	73.06371651	0.252787089	178	55.34248674	72.99695498	68.71574096	66.10102004
Iceland	0.063514378	54.17233745	0.434359993	90	31.90457574	42.43743208	39.78585016	38.19445836
Ireland	0.018075924	150.5716412	0.389356595	233	66.45330793	92.56298544	85.84476148	81.86999212
Italy	0.018014458	151.3257418	0.377070492	241	68.64570778	95.90475771	88.93389157	84.79419076
Malta	0.022652701	95.41681316	0.39760535	235	110.4281795	135.8404753	129.8105321	126.0704568
Mexico	0.010377526	188.0011619	0.32312548	258	49.26915341	66.27082275	60.45730972	57.06268812
Peru	0.012379217	200	0.349426928	242	37.56293105	59.13462631	53.34917208	50.00984351
Portugal	0.025237491	162.0226288	0.461236454	234	72.65974997	99.74732113	92.69207673	88.54613668
South Korea	0.021931008	161.1951162	0.437882706	211	52.2574544	74.99464779	68.90816708	65.39065074
Spain	0.028466217	107.8294504	0.417967064	199	76.64587247	99.72076437	94.03076002	90.57292356
Turkey	0.012053373	140.355595	0.368700663	215	60.35264529	80.66341144	75.43952095	72.35184566
UK	0.015992445	55.39616283	0.303422349	241	117.0078174	137.442241	132.7286793	129.7746195
USA	0.013378891	91.25265859	0.249549082	239	77.73800687	100.7775641	95.22356232	91.8213829

Counties	3-F	4-F	5-F	Population (P)	3-F*P	4-F*P	5-F*P
Belgium	25.90049544	19.28651438	15.35511445	11,589,623	300,176,977.6	223,523,430.6	177,959,987.6
Canada	24.79331325	18.4030826	14.62402773	37,742,154	935,753,046.7	694,571,977.5	551,942,306.8
Cyprus	6.444904705	4.932903707	3.993729877	1,207,359	7,781,313.699	5,955,785.687	4,821,865.711
Czecia	29.26771315	22.08077662	17.72042726	10,708,981	313,427,384	236,462,617.3	189,767,718.8
Denmark	34.73021323	26.22695542	21.03476097	5,792,202	201,164,410.6	151,911,823.6	121,837,584.6
Finland	18.90845872	14.1791287	11.33646406	5,540,720	104,766,475.4	78,562,581.95	62,812,173.15
France	26.94385533	20.06542292	15.97746999	65,273,511	1,758,720,037	1,309,740,604	1,042,905,563
Hungary	17.65446823	13.37325421	10.7585333	9,660,351	170,548,359.9	129,190,329.7	103,931,207.9
Iceland	10.53285634	7.881274419	6.289882625	341,243	3,594,263.497	2,689,429.727	2,146,378.417
Ireland	26.10967751	19.39145355	15.41668419	4,937,786	128,924,000.1	95,750,847.85	76,124,287.36
Italy	27.25904993	20.28818379	16.14848298	60,461,826	1,648,131,934	1,226,660,638	976,366,768.1
Malta	25.41229585	19.38235261	15.64227735	441,543	11,220,621.35	8,558,142.119	6,906,738.067
Mexico	17.00166933	11.1881563	7.793534703	128,932,753	2,192,072,033	1,442,519,793	1,004,841,885
Peru	21.57169526	15.78624103	12.44691246	32,971,854	711,258,786.8	520,501,634.3	410,397,780.4
Portugal	27.08757117	20.03232676	15.88638671	10,196,709	276,204,080.7	204,263,806.6	161,988,862.3
South Korea	22.73719339	16.65071268	13.13319634	51,269,185	1,165,717,374	853,668,468.9	673,328,272.9
Spain	23.0748919	17.38488756	13.92705109	46,754,778	1,078,861,448	812,826,558.4	651,156,182.1
Turkey	20.31076614	15.08687566	11.99920037	84,339,067	1,712,991,067	1,272,413,017	1,012,001,364
UK	20.43442357	15.72086188	12.76680214	67,886,011	1,387,211,503	1,067,226,602	866,687,270.5
USA	23.03955727	17.48555545	14.08337603	331,002,651	7,626,154,535	5,787,765,209	4,661,634,801

The final inventory cost can be obtained if the order costs are added. In the case of the order cost, only the transportation cost can be a non-negotiable amount of money. The decision-maker can solve the model easily for different numbers of procurements. The best solution can be selected after taking into account the order cost.

Conclusions

The countries bear the costs of vaccination in case of a pandemic. The paper intends to reduce public expenses. The main contributions of the paper are as follows:

• The increase in the proportion of the vaccinated population is analyzed in the COVID-19 pandemic.

• It is shown on selected 20 countries that the countries have individual behavior.

• However, the cumulative distribution function of the Cauchy distribution is a suitable tool to model the increase of the aforementioned proportion in the case of each country.

• A simple model is elaborated to minimize the inventory holding cost if the number of procurements is fixed. The model can be solved for a country within one minute by an excel solver. The numerical results are presented for the 20 countries. Three results are provided for each country such that the number of procurements is 3, 4, and 5, respectively.

Appendix: The optimality condition of (2)–(5)

The Karush–Kuhn–Tucker optimality conditions are first order conditions. The easy way to obtain the same conditions in this particular case is as follows:

Assume that the times of orders $i-1$ and $i+1$ are fixed. What is the optimal time of order $i$? The area under the step functio in the interval $[t_{i-1},t_{i+1}]$ is

$$g\left(t_i\right)=\left(t_i-t_{i-1}\right)f\left(t_i\right)+\left(t_{i+1}-t_i\right)f\left(t_{i+1}\right).$$

Notice that the only unknown quantity in the formula is $t_i$. The derivative of $g\left(t_i\right)$ must be 0 at the optimal position of $t_i$. Hence,

$$g^{\prime }\left(t_i\right)=f\left(t_i\right)+\left(t_i-t_{i-1}\right)f\left(t_i\right)-f\left(t_{i+1}\right)=0.$$ *

Thus, the optimality condition is that (*) holds for $i=1,2,\cdots ,n$.

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Conflict of interest

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