# 5.4: Logistic Growth

In our basic exponential growth scenario, we had a recursive equation of the form

P­n = P­n-1 + r P­n-1

In a confined environment, however, the growth rate may not remain constant. In a lake, for example, there is some maximum sustainable population of fish, also called a carrying capacity.

Carrying Capacity

The carrying capacity, or maximum sustainable population, is the largest population that an environment can support.

For our fish, the carrying capacity is the largest population that the resources in the lake can sustain. If the population in the lake is far below the carrying capacity, then we would expect the population to grow essentially exponentially. However, as the population approaches the carrying capacity, there will be a scarcity of food and space available, and the growth rate will decrease. If the population exceeds the carrying capacity, there won’t be enough resources to sustain all the fish and there will be a negative growth rate, causing the population to decrease back to the carrying capacity.

If the carrying capacity was 5000, the growth rate might vary something like that in the graph shown. Note that this is a linear equation with intercept at 0.1 and slope , so we could write an equation for this adjusted growth rate as:

Substituting this in to our original exponential growth model for r gives

ParseError: EOF expected (click for details)

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Logistic Growth

If a population is growing in a constrained environment with carrying capacity K, and absent constraint would grow exponentially with growth rate r, then the population behavior can be described by the logistic growth model:

ParseError: EOF expected (click for details)

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Unlike linear and exponential growth, logistic growth behaves differently if the populations grow steadily throughout the year or if they have one breeding time per year. The recursive formula provided above models generational growth, where there is one breeding time per year (or, at least a finite number); there is no explicit formula for this type of logistic growth.

### Example 15

A forest is currently home to a population of 200 rabbits. The forest is estimated to be able to sustain a population of 2000 rabbits. Absent any restrictions, the rabbits would grow by 50% per year. Predict the future population using the logistic growth model.

Modeling this with a logistic growth model, r = 0.50, K = 2000, and P­0 = 200. Calculating the next year:

We can use this to calculate the following year:

A calculator was used to compute several more values:

 n 0 1 2 3 4 5 6 7 8 9 10 Pn 200 290 414 578 784 1022 1272 1503 1690 1821 1902

Plotting these values, we can see that the population starts to increase faster and the graph curves upwards during the first few years, like exponential growth, but then the growth slows down as the population approaches the carrying capacity.

### Example 16

On an island that can support a population of 1000 lizards, there is currently a population of 600. These lizards have a lot of offspring and not a lot of natural predators, so have very high growth rate, around 150%. Calculating out the next couple generations:

Interestingly, even though the factor that limits the growth rate slowed the growth a lot, the population still overshot the carrying capacity. We would expect the population to decline the next year.

Calculating out a few more years and plotting the results, we see the population wavers above and below the carrying capacity, but eventually settles down, leaving a steady population near the carrying capacity.

### Try it Now 5

A field currently contains 20 mint plants. Absent constraints, the number of plants would increase by 70% each year, but the field can only support a maximum population of 300 plants. Use the logistic model to predict the population in the next three years.

### Example 17

On a neighboring island to the one from the previous example, there is another population of lizards, but the growth rate is even higher – about 205%.

Calculating out several generations and plotting the results, we get a surprise: the population seems to be oscillating between two values, a pattern called a 2-cycle.

While it would be tempting to treat this only as a strange side effect of mathematics, this has actually been observed in nature. Researchers from the University of California observed a stable 2-cycle in a lizard population in California.[1]

Taking this even further, we get more and more extreme behaviors as the growth rate increases higher. It is possible to get stable 4-cycles, 8-cycles, and higher. Quickly, though, the behavior approaches chaos (remember the movie Jurassic Park?).

### Try it Now Answers

1. Letting n = 0 correspond with 1976, then P­0 = 20,610.

From 1976 to 2010 the number of stay-at-home fathers increased by

53,555 – 20,610 = 32,945

This happened over 34 years, giving a common different d of 32,945 / 34 = 969.

P­n = 20,610 + 969n

Predicting for 2020, we use n = 44

P­44 = 20,610 + 969(44) = 63,246 stay-at-home fathers in 2020.

2. Using n = 0 corresponding with 2008,

P­12 = (1+0.0134)12 (1.14) = about 1.337 billion people in 2020

3. Here we will measure n in months rather than years, with n = 0 corresponding to the February when they went public. This gives P­0 = 45 thousand. October is 8 months later, so P­8 = 60.

P­8 = (1+r)8 P­0

60 = (1+r)8 45

, or 3.66%

The general explicit equation is P­n = (1.0366)n 45

Predicting 24 months (2 years) after they went public:

P­24 = (1.0366)24 45 = 106.63 thousand users.

4. 1.14(1.0134)n = 1.2. n = 3.853, which is during 2011

5.

P2 = 54

P3 = 85

## Global Connected Logistics Market (2021 to 2026) - Growth, Trends, COVID-19 Impact, and Forecasts

Dublin, Feb. 15, 2021 (GLOBE NEWSWIRE) -- The "Connected Logistics Market - Growth, Trends, COVID-19 Impact, and Forecasts (2021 - 2026)" report has been added to ResearchAndMarkets.com's offering.

Connected Logistics help businesses become more customer-centric and efficient by increasing transparency in the business process. The Global Connected Logistics Market is expected to register a CAGR of 17.5% during the forecast period (2021 - 2026).

Increasing adoption of IoT and sensor-based technologies like RFID in various domains, including the Supply Chain Management (SCM) system, will drive the connected logistics market in the forecast period.

Advancement in future technologies like Big Data and Advanced analytics will act as a catalyzer for the adoption of Logistics 4.0. The data collected from smart and connected supply chain and logistics can be converted into actionable insights by using AI systems, which will help businesses to forecast demand accurately and thus improving capacity planning.

Also, the growing demand for cloud-based solutions, RFID, and internet ubiquity across the globe are some more factors driving the market. The introduction of new applications and cloud-based solutions for the transportation and management of products, including tracking of goods and movement of planes, trucks, and ships which carry them. Further, the macroeconomic factors driving the market are the changing lifestyle of consumers, emerging economies, and the rapid rate of smart cities in developing countries such as China, Brazil, and India.

In the United States, due to the COVID-19 outbreak, the government has shut down all the logistic operations in the country, which is heavily impacting the industry. The CEO of Connected Logistics has stated that, due to this outbreak, the American government has shown more flexibility towards telework and online assistance. These factors are expected to drive the market in the forecast period further.

Key Market Trends

Increasing use of IoT and future technologies in different industries will drive the adoption of Connected Logistics

• According to CISCO, global internet traffic from non-PC devices was 47% of total IP traffic in 2015 and will be 71% of total IP traffic in 2020. Also, Machine-To-Machine (M2M) connections will be half of the total connected devices and connections in the world by 2023 up from 33% in 2018. There will be 14.7 billion M2M connections by 2023.
• Verizon has conducted a survey in 2019 concluding that more than one-third of fleet managers who do not use fleet tracking solutions (fleet tracking solutions are mobile asset tracking solutions primarily fitted to trucks and other vehicles) say they would expect an increase in productivity if they implemented the technology. The survey also indicates that over 53% of companies that make use of asset tracking solutions reported actual productivity increases - with most of the respondents reporting positive growth in productivity (53% respondents) and compliance (52% respondents).
• Walmart is using IoT, machine learning, and Big Data to transform its retail operations and improve its customer experience in more than one ways. The company was one of the early adopters of RFID tags to track their inventories.
• In 2019, King County Department of Transportation, Washington DC, upgraded to Infor Inc's asset tracking solutions - CloudSuite EAM - to efficiently manage its 171 transit base structures/facilities and 1,500 buses. In total, the solution is estimated to track 46,988 pieces of in-service equipment that span across multiple departments and divisions.

Asia-Pacific Region to Exhibit Maximum Growth

• The Asia-Pacific region is expected to show robust growth in terms of revenue in the forecast period, significantly in the developing economies like India and China. Improvements in technology, increasing the use of sensors, and automation is certain factors expected to steer the market growth.
• For instance, China is the world's largest E-Commerce market, with over 50% of global E-Commerce transactions coming from China. According to Dezan Shira & Associates, in 2018, China's online retail sales reached USD 1.33 trillion and are forecasted to reach USD 1.99 trillion by the end of 2019. Also, in 2018, the number of digital buyers in China surpassed 560 million, with the total number projected to reach 634 million in 2020. Further, by 2020, China's E-Commerce market is predicted to be larger than those of the U.S., UK, Japan, Germany, and France combined. The growth of E-commerce industry in recent years in China is attributed to the developments in the internet infrastructure and establishment of global logistics networks.
• Also, in 2019, Singapore began transforming logistics to reinforce its place in APAC's logistics industry by integrating the technologies with the existing processes. The Singapore government has already started transforming logistics as part of its USD 4.5 billion Industry Transformation Programme. The primary objective of the program is to drive excellence in logistics operations and to be leaders in innovation, while also building a strong core of local logistics talent and making Singapore attractive to inward investment. This scheme is named the Logistics Industry Transformation Map (ITM) and follows similar plans for Singapore's food and beverage and precision engineering sectors.

Competitive Landscape

The competitive landscape of the global connected logistics market is moderately fragmented owing to the presence of many existing as well as emerging players in the market. The technological developments in the logistics industry are expected to fuel the rise in investments and product innovations. The competitors are proactively addressing the challenges by crafting strategies that can have the best overall effect on the market's progress. The opportunities for growth in the market have ample scope for development in the forecast period.

• February 2019 - Intel announced the Intel Connected Logistics Platform (Intel CLP), a cost-effective internet of things (IoT) solution enabling users to monitor the condition and location of assets in any environment throughout the supply chain.
• February 2019 - Huawei released the Smart Logistics Solution during MWC 2019, aiming to help enterprises improve efficiency in fields such as transportation, distribution, and warehousing, facilitate agile innovation in the logistics industry, and achieve digital, information-based, and intelligent management.

Key Topics Covered:

1 INTRODUCTION
1.1 Study Assumptions and Market Definition
1.2 Scope of the Study

2 RESEARCH METHODOLOGY

3 EXECUTIVE SUMMARY

4 MARKET DYNAMICS
4.1 Market Overview
4.2 Market Drivers
4.2.1 Increasing Adoption on IoT in various Industries
4.2.2 Adoption of Sensor Based Technologies
4.3 Market Restraints
4.3.1 Lack of Uniform Government Regulations for Smart Technologies
4.3.2 Coronavirus Outbreak Influencing Electronic Industry
4.4 Industry Value Chain Analysis
4.5 Industry Attractiveness - Porter's Five Force Analysis
4.5.1 Bargaining Power of Suppliers
4.5.2 Bargaining Power of Buyers/Consumers
4.5.3 Threat of New Entrants
4.5.4 Threat of Substitute Products
4.5.5 Intensity of Competitive Rivalry
4.6 Assessment of Impact of COVID-19 on the Industry

5 MARKET SEGMENTATION
5.1 By Software
5.1.1 Asset Management
5.1.2 Warehouse IoT
5.1.3 Security
5.1.4 Data Management
5.1.5 Network Management
5.1.6 Streaming Analytics
5.2 By Product Type
5.2.1 Device Management
5.2.2 Application Management
5.2.3 Connectivity Management
5.3 By Transportation Mode
5.3.2 Railways
5.3.3 Airways
5.3.4 Seaways
5.4 By End-user Industry
5.4.1 Automotive
5.4.2 Manufacturing
5.4.3 Oil and Gas
5.4.4 IT & Telecom
5.4.5 Healthcare
5.4.6 IT and Telecommunication
5.4.7 Retail
5.4.8 Food and Beverage
5.4.9 Other End-user Industries
5.5 Geography
5.5.1 North America
5.5.1.1 United States
5.5.2 Europe
5.5.2.1 United Kingdom
5.5.2.2 Germany
5.5.2.3 France
5.5.2.4 Rest of Europe
5.5.3 Asia-Pacific
5.5.3.1 China
5.5.3.2 Japan
5.5.3.3 India
5.5.3.4 Rest of Asia-Pacific
5.5.4 Rest of the World

6 COMPETITIVE LANDSCAPE
6.1 Company Profiles
6.1.1 Bosch Service Solutions GmbH
6.1.2 Cisco Systems, Inc
6.1.3 AT&T Inc.
6.1.4 IBM Corporation
6.1.5 Intel Corporation
6.1.6 SAP SE
6.1.7 Oracle Corporation
6.1.8 Freightgate Inc.
6.1.9 Orbcomm Inc.
6.1.10 HCL Technologies Limited
6.1.11 Honeywell International Inc.
6.1.12 Microsoft Corporation
6.1.13 Siemens AG

## Logistic Growth is Unredeemable

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#### Mewnine

##### Private

I suppose the main reason why I haven't had a lot of trouble with the new s curve mechanic is because clerks have become so crappy, I've turned all their job slots off, and automigration is taking care of the unemployed pops well enough. The empire wide malus means whatever pop growth you can get, you should get.

Edit: Neither system works, but due to the horrible design of the empire malus, the s curve gets a pass for now, because it would be even worse without it.

#### The Founder

##### Field Marshal

Yes, it really is reaching the point where they should make a venting thread for this. Like they did with FTL. And Sectors.
Because nothing much is comming from these discussion other then "me no like, remove."

However unlike FTL and Sectors, this is something that can be removed via modding. If enough people agree, then you should have no issues finding multiplayer games with the same mod. So you should still be able to play the game you want, without these rules.

25 jobs on any given planet (exact number varies slightly based on capacity), allowing players willing to micro to double organic growth more or less forever. Most of the other issues with logistic growth follow from this one.

2) It increases population counts dramatically. Logistic growth effectively doubles organic growth. Right now, the empire-wide malus "hides" this to an extent, but if that is removed or reworked (and given the extreme backlash against it, there's a good change it will be. If not, mods removing it already have tens of thousands of subscribers), it will make lategame 2.8 performance issues come back worse then ever.

5) The AI has no idea how to handle it. While players can get x2 organic growth forever, the AI has absolutely no idea how to do this, and just plays as if it didn't exist. The game is already full of powerful mechanics that the AI has no idea how to use. Adding another one, and one that is incredibly powerful, makes the single player experience much worse. The AI already struggles (to put it generously) to deal with the other major economic changes in 3.0 (industrial districts and the empire-wide malus in particular) adding another one is a bad idea.

6) It incentivizes bizarre micro/gameplay incredibly strongly. Getting such a massive boost for leaving all of your planets half-developed feels silly and requires quite a bit of micro (having to do a new math problem for each planet). It also makes prebuilding non-city districts a terrible idea.

Two simple fixes:
1. Capacity is the lesser of Jobs or current capacity
2. As long as there are places to migrate too, this growth reduction is fed into the migiraiton system. However it must be substracted even if there is no place to migrate too - that is why trying to solve it via migration always failed.

"Intelligent, but nor artificial."
The Forumname is based on a Character from Universe at War. The forum would not allow me to just create a new account, I HAD to resuse this one.

"I don't like Vikings.
They are coarse, rough and irritating.
And they get everyhwere."
- Anakin Caelus Pedes

#### Mial42

##### Colonel

Minor counter-point: Machines are way too good otherwise. Higher per-pop efficiency of machine jobs and lack of habitability penalties means that machines should have slower growth for the game to be somewhat balanced. I think in most previous patches whenever I checked out multiplayer games people often banned machines for being way ahead of the curve, a mechanic helping organic empires is needed (but perhaps not this particular mechanic).

I agree with your other points though.

In my experience, machines were quite a bit weaker then normal organics in 2.8 due to lacking ascensions, ecumenepoli, and slavery, meaning they couldn't eco-boom as well as organics mid-lategame. They were stronger early on thanks to better metallurgists and habitability (although migration treaties make this a near-nonfactor very quickly), but since they couldn't conquer efficiently*, they couldn't turn this into a lasting advantage. Compared to something like techno-slavers => synth ascension or militarist slaver conquest snowballing, they got left in the dust very quickly and couldn't catch up.

Regardless, IMO the ideal nerf to machines (if one is necessary) would be removing their replicators at the lvl 4 capitals, which has the added benefits of reducing potential micro and lategame pop growth.

*Except assimilators, which were probably the strongest non-glitched empire in 2.8 and are definitely the strongest in 3.0.

## R-bloggers

Posted on July 24, 2015 by Nicole Radziwill in R bloggers | 0 Comments

If you’ve ever wondered how logistic population growth (the Verhulst model), S curves, the logistic map, bifurcation diagrams, sensitive dependence on initial conditions, “orbits”, deterministic chaos, and Lyapunov exponents are related to one another… this post attempts to provide a simplified explanation(!) in just 10 steps, each with some code in R so you can explore it all yourself. I’ve included some code written by other people who have explored this problem (cited below) as portions of my own code.

It all starts with a hypothesized population… and a process where the size of the population changes over time. We want to understand how (and under what conditions) those changes occur, so we choose a model that characterizes population changes: the logistic growth model. It’s been used in biology, ecology, econometrics, marketing, and other areas.

1. The logistic growth model describes how the size of a population (P) changes over time (t), based on some maximum population growth rate (r). There is a limiting factor called the carrying capacity (K) which represents the total population that the environment could support, based on the amount of available resources. dP/dt is the rate of change of the population over time.

2. You can simplify the logistic growth model by defining a new variable x to represent the portion of the population that’s alive, compared to the total population that the environment could support (and keep alive). So with x = N/K, you get a new differential equation in terms of x. Now we are looking at the rate of change of the population fraction over time. Once x = N/K = 1, the environment can’t support any more members in the population:

3. You can solve this equation by integration! Then, you’ll have an expression that you can use to calculate x (which is still the population fraction) for any time t. This is called the sigmoid or (more commonly), the S Curve. To compute x at any time t, all we need to know is how big the population was when we started looking at it (x0) and the maximum growth rate r:

4. The equation for the S Curve is deterministic and continuous. If we want to solve it numerically, we have to discretize it by chopping up that continuous axis that contains time into little tiny pieces of time. That’s what produces the difference equation that we recognize as the logistic map. It’s a map because it “maps” each value of the sequence onto the next value in the sequence. As long as you know one of those values for x (indicated by the subscript n), you’ll be able to figure out the next value of x (indicated by the subscript n+1). The value x[n] is the population fraction of the current generation, and the value x[n+1] is the population fraction for the next generation. This makes the logistic map a Markov chain. If you plot x[n] on the x axis and x[n+1] on the y axis, this expression will produce the familiar upside down parabola:

5. The logistic map behaves differently depending upon the maximum growth rate (r) that describes your population. This parameter is also called fecundity and represents how rabbit-like your population is reproducing. The higher the r, the more productive, like rabbits (although I’m not sure precisely which r you’d choose if you were studying rabbits). Here is an R function that you can use to generate the last M iterations from a sequence of N total, developed and described at Mage’s Blog:

6. The logistic map has many interesting properties, but here are two in particular (the first in Step 6 and the second in Step 7). First, for several values you can choose for r, the chain converges to a single value (or fixed point) when n gets really big. For other values of r, the value of x will eventually bounce between two values instead of converging (a limit cycle of 2). For other values of r, the value of x will eventually bounce between four values instead of converging. Sometimes, x will bounce around a near limitless collection of values (a condition called deterministic chaos). The eventual values (or collection of eventual values, if they bounce between values) is called an orbit. For example, when the growth rate r is 2.6, the logistic map rapidly converges to an orbit of about 0.615:

7. Sometimes, it can be nice to take a look at how the values bounce around, and where they eventually converge (or not). To do this, we use cobweb diagrams (which are also sometimes called web diagrams). I used a function that I found at http://bayesianbiologist.com to plot the behavior of the orbits for r=2.6, r=3.2, and r=3.9:

8. (Remember to dev.off() before you continue.) Second, for some values of r, the logistic map shows sensitive dependence on initial conditions. For example, let’s see what happens for two different growth rates (r=3 and r=3.9) when we start one iteration with an x[n] of 0.5 COLORED BLACK, and another one with an x[n] of 0.5001 COLORED RED. It’s a small, small difference that can lead to big, BIG variations in the orbits. In the r=3 case, the chain produced by the logistic map with x[n] of 0.5 (in black) is IDENTICAL to the chain produced by the logistic map with x[n] of 0.5001 (in red). That’s why you can’t see the black… the values are the same! But for the r=3.9 case, the chain produced by the logistic map with x[n] of 0.5 (in black) RAPIDLY DIVERGES from the chain produced by the logistic map with x[n] of 0.5001 (in red). They are very different, despite a very tiny difference in initial conditions! The logistic map for r=3.9 shows a very sensitive dependence on initial conditions.

9. For any chain, we can determine just how sensitive the logistic map is to initial conditions by looking at the Lyapunov exponent. Very simplistically, if the Lyapunov exponent is negative, the chain will converge to one or more fixed points for that value of r. If the Lyapunov exponent is positive, the chain will demonstrate deterministic chaos for that value of r. If the Lyapunov exponent is zero, there is a bifurcation: a 1-cycle is doubling to a 2-cycle, a 2-cycle is doubling to a 4-cycle, or so forth. The top chart shows an approximation of the Lyapunov exponent based on the first 500 iterations (ideally, you’d use an infinite number, but that would eat up too much computing time), and the bottom chart shows a bifurcation diagram.You’ll notice that the Lyapunov exponents are zero where a bifurcation occurs. To interpret the bifurcation diagram, just remember that each vertical slice through it represents the results of ONE COMPLETELY CONVERGED CHAIN from the logistic map. So it shows the results from many, many, many completely converged chains – and provides an excellent way for us to look at the behavior of MANY different types of populations in just one chart:

10. Notice that in the bifurcation diagram, we can easily see that when r is between 0 and 1, the population converges to extinction. This makes sense, because the growth rate is smaller than the size of the population – it can’t sustain itself. You might like to zoom in, though, and see what the orbits look like for some smaller portions of the diagram. Here’s how you can do it (but be sure to refresh your graphics window with dev.off() before you try it). Try changing the plot character (pch) too, or maybe the size of the characters with cex=0.2 or cex=0.5 in the last line:

Find out more information on these other web pages, which are listed in order of difficulty level:

## Maximizing the total population with logistic growth in a patchy environment

This paper is concerned with a nonlinear optimization problem that naturally arises in population biology. We consider the population of a single species with logistic growth residing in a patchy environment and study the effects of dispersal and spatial heterogeneity of patches on the total population at equilibrium. Our objective is to maximize the total population by redistributing the resources among the patches under the constraint that the total amount of resources is limited. It is shown that the global maximizer can be characterized for any number of patches when the diffusion rate is either sufficiently small or large. To show this, we compute the first variation of the total population with respect to resources in the two patches case. In the case of three or more patches, we compute the asymptotic expansion of all patches by using the Taylor expansion with respect to the diffusion rate. To characterize the shape of the global maximizer, we use a recurrence relation to determine all coefficients of all patches.

This is a preview of subscription content, access via your institution.

## Logistic Growth Formula

Differential equation of logistic growth:

G : Growth maximum value k : Logistic growth rate

With the growth function for the inital values t0 = 0 and y0 = y(0)

y = G 1 + e - k G t G y 0 - 1

With the growth function for the general inital values t0 and y0 = y(t0)

y = G 1 + e - k G t - t 0 G y 0 - 1

Turning point of the logistic growth function:

At the turning point of the logistic growth function value equal to half the saturation limit.

t W = t 0 + ln G y 0 - 1 k G

The maximum growth rate is achieved at the turning point.

Growth of populations with limited resources

## 5.4: Logistic Growth

Exponential growth cannot continue forever. Exponential models, while they may be useful in the short term, tend to fall apart the longer they continue. Consider an aspiring writer who writes a single line on day one and plans to double the number of lines she writes each day for a month. By the end of the month, she must write over 17 billion lines, or one-half-billion pages. It is impractical, if not impossible, for anyone to write that much in such a short period of time. Eventually, an exponential model must begin to approach some limiting value, and then the growth is forced to slow. For this reason, it is often better to use a model with an upper bound instead of an exponential growth model, though the exponential growth model is still useful over a short term, before approaching the limiting value.

The logistic growth model is approximately exponential at first, but it has a reduced rate of growth as the output approaches the model’s upper bound, called the carrying capacity. For constants a, b, and c, the logistic growth of a population over time x is represented by the model

Figure 6 shows how the growth rate changes over time. The graph increases from left to right, but the growth rate only increases until it reaches its point of maximum growth rate, at which point the rate of increase decreases.

## Logistic Growth Formula

In A12 I have the value 8855
In J24 I have the price $346,400 In K24 I have the value 39.1 In N24 I have the value 2 In O24 I have the value 50 In P24 I have the formula, =(O24-K24)*$A$12*IF((N24>=1)*(N24<9),(N24/8),1), which gives the price$24,088

The key area I’d like to focus on in the formula is N24/8, in this instance 2/8. The denominator will always be 8, but the numerator may change. We can see if I change the value of N24 to 1, I get $12,044, and if I change the value to 3, I get$36,131 … That is, the price rises by 1/8, or $12,044 each time. The value in N24 is number of weeks, and what I am measuring is profit increase over time. However, I would like the value in P24 to rise in logistic growth model fashion - that is, profit should rise by a greater amount early, and lesser amount late. For example, the profit increase between week 7 and week 8 should be <$12,044, while the profit increase between week 1 and 2 should be > $12,044, instead of the current formula, which is the same amount ($12,044) each week.

My question, how can I rewrite the formula in P24 according to logistic growth model?
Many thanks.

### Excel Facts

#### KRice

##### Well-known Member

Ideally there would be some theory or other basis for describing the profit growth model, and that information could be used to establish the formula. If that information is lacking, and you are looking for a monotonically increasing function (meaning every time step leads to a greater value), but whose derivative is monotonically decreasing (meaning every subsequent time step leads to smaller and smaller increases), then there are several approaches. You could simply define a lookup table of expected profit percentages and use an interpolating polynomial, if necessary, to return the percentage of maximum profit gained for any input value of elapsed time. Alternatively, you could select any of numerous mathematical functions with the desired characteristics to determine if they yield results that are considered reasonable representations of reality. On this second point, I played around with a 2-parameter Weibull distribution, specifically its cumulative distribution function (cdf). The form of this equation is F(x) = 1 - exp(-((x/lambda)^kappa)), where lambda is a scale parameter and kappa is a shape parameter. These two parameters cause the function to take on substantially different shapes as the cdf values increase from 0 to 1. although 1 is approached asymptotically (very gradually) at large values of x (time in this case). This asymptotic behavior is inherent in many of the functions mentioned (logistic, logarithmic, and the current exponential), and that potentially limits their usefulness for your application. To overcome this issue, I propose making an adjustment by evaluating the cdf at the time when maximum profit is known to be achieved (x=T=8 weeks in this case), and then using that value to normalize the entire cdf. This forces the modified function to produce values between (0,1) over the time domain (0,8), and this can be used to determine the fraction of maximum profit achieved.

I would suggest first trying out the modified Weibull cdf to produce a curve that makes sense. This is done by changing the values lambda and kappa in cells T8 and T9, respectively. I've added some scroll bars that are linked to cells under them, and those are used to change T8:T9 in increments of 0.1. This makes it convenient to see how the curve changes as the parameters are varied. Note that the plotted curve is normalized, and cell T10 represents the time in weeks when 100% profit is to be achieved. For this example, lambda = 2.9 and kappa = 0.8 is one feasible curve, although you might experiment with other parameters to find a more sensible curve. You can review the difference between adjacent function values beginning in cell T13 and down to confirm that the increases are diminishing. A snapshot of the curve is shown below.

## Global Food Delivery Logistic Market: Industry Size, Growth, Analysis and Forecast of 2026

Food Delivery Logistic Industry

Wiseguyreports.Com Adds “Food Delivery Logistic -Market Demand, Growth, Opportunities and Analysis Of Top Key Player Forecast To 2023” To Its Research Database

Food delivery logistics market deals with the business of delivering food products at a desired location. Advancements in technology has led to the rapid growth of third-party ordering & delivering services. Global presence of food delivery services enables the supply of food products, which are scarce or unavailable within the specified time.

This report focuses on the global Food Delivery Logistic status, future forecast, growth opportunity, key market and key players. The study objectives are to present the Food Delivery Logistic development in United States, Europe and China.

The key players covered in this study

Allen Lund Company, LLC (U.S.)
Alliance Shippers, Inc. (U.S.)
C.H Robinson Worldwide, Inc. (U.S.)
Deutsche Bahn AG (Germany)
Schneider National, Inc.(U.S.)
Bender Group (U.S.)
CaseStack, Inc. (U.S.)
Echo Global Logistics, Inc. (U.S.)
H&M Bay, Inc. (U.S.)
Hellmann Worldwide Logistics GmbH & Co. (Germany)
Henningsen Cold Storage Co. (U.S.)

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Market segment by Type, the product can be split into
Seaways
Airways
Freight/Railways

Market segment by Application, split into
Sea Food & Meat Products
Fruits & Vegetables
Cereals & Dairy Products
Oils & Beverages

Market segment by Regions/Countries, this report covers
United States
Europe
China
Japan
Southeast Asia
India
Central & South America

The study objectives of this report are:
To analyze global Food Delivery Logistic status, future forecast, growth opportunity, key market and key players.
To present the Food Delivery Logistic development in United States, Europe and China.
To strategically profile the key players and comprehensively analyze their development plan and strategies.
To define, describe and forecast the market by product type, market and key regions.

In this study, the years considered to estimate the market size of Food Delivery Logistic are as follows:
History Year: 2013-2017
Base Year: 2017
Estimated Year: 2018
Forecast Year 2018 to 2025

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1 Report Overview
1.1 Study Scope
1.2 Key Market Segments
1.3 Players Covered
1.4 Market Analysis by Type
1.4.1 Global Food Delivery Logistic Market Size Growth Rate by Type (2013-2025)
1.4.2 Seaways
1.4.3 Airways
1.4.4 Freight/Railways
1.5 Market by Application
1.5.1 Global Food Delivery Logistic Market Share by Application (2013-2025)
1.5.2 Sea Food & Meat Products
1.5.3 Fruits & Vegetables
1.5.4 Cereals & Dairy Products
1.5.5 Oils & Beverages
1.6 Study Objectives
1.7 Years Considered

2 Global Growth Trends
2.1 Food Delivery Logistic Market Size
2.2 Food Delivery Logistic Growth Trends by Regions
2.2.1 Food Delivery Logistic Market Size by Regions (2013-2025)
2.2.2 Food Delivery Logistic Market Share by Regions (2013-2018)
2.3 Industry Trends
2.3.1 Market Top Trends
2.3.2 Market Drivers
2.3.3 Market Opportunities

12 International Players Profiles
12.1 Allen Lund Company, LLC (U.S.)
12.1.1 Allen Lund Company, LLC (U.S.) Company Details
12.1.2 Company Description and Business Overview
12.1.3 Food Delivery Logistic Introduction
12.1.4 Allen Lund Company, LLC (U.S.) Revenue in Food Delivery Logistic Business (2013-2018)
12.1.5 Allen Lund Company, LLC (U.S.) Recent Development
12.2 Alliance Shippers, Inc. (U.S.)
12.2.1 Alliance Shippers, Inc. (U.S.) Company Details
12.2.2 Company Description and Business Overview
12.2.3 Food Delivery Logistic Introduction
12.2.4 Alliance Shippers, Inc. (U.S.) Revenue in Food Delivery Logistic Business (2013-2018)
12.2.5 Alliance Shippers, Inc. (U.S.) Recent Development
12.3 C.H Robinson Worldwide, Inc. (U.S.)
12.3.1 C.H Robinson Worldwide, Inc. (U.S.) Company Details
12.3.2 Company Description and Business Overview
12.3.3 Food Delivery Logistic Introduction
12.3.4 C.H Robinson Worldwide, Inc. (U.S.) Revenue in Food Delivery Logistic Business (2013-2018)
12.3.5 C.H Robinson Worldwide, Inc. (U.S.) Recent Development
12.4 Deutsche Bahn AG (Germany)
12.4.1 Deutsche Bahn AG (Germany) Company Details
12.4.2 Company Description and Business Overview
12.4.3 Food Delivery Logistic Introduction
12.4.4 Deutsche Bahn AG (Germany) Revenue in Food Delivery Logistic Business (2013-2018)
12.4.5 Deutsche Bahn AG (Germany) Recent Development
12.5 Schneider National, Inc.(U.S.)
12.5.1 Schneider National, Inc.(U.S.) Company Details
12.5.2 Company Description and Business Overview
12.5.3 Food Delivery Logistic Introduction
12.5.4 Schneider National, Inc.(U.S.) Revenue in Food Delivery Logistic Business (2013-2018)
12.5.5 Schneider National, Inc.(U.S.) Recent Development
12.6 Bender Group (U.S.)
12.6.1 Bender Group (U.S.) Company Details
12.6.2 Company Description and Business Overview
12.6.3 Food Delivery Logistic Introduction
12.6.4 Bender Group (U.S.) Revenue in Food Delivery Logistic Business (2013-2018)
12.6.5 Bender Group (U.S.) Recent Development
12.7 CaseStack, Inc. (U.S.)
12.7.1 CaseStack, Inc. (U.S.) Company Details
12.7.2 Company Description and Business Overview
12.7.3 Food Delivery Logistic Introduction
12.7.4 CaseStack, Inc. (U.S.) Revenue in Food Delivery Logistic Business (2013-2018)
12.7.5 CaseStack, Inc. (U.S.) Recent Development
12.8 Echo Global Logistics, Inc. (U.S.)
12.8.1 Echo Global Logistics, Inc. (U.S.) Company Details
12.8.2 Company Description and Business Overview
12.8.3 Food Delivery Logistic Introduction
12.8.4 Echo Global Logistics, Inc. (U.S.) Revenue in Food Delivery Logistic Business (2013-2018)
12.8.5 Echo Global Logistics, Inc. (U.S.) Recent Development
12.9 H&M Bay, Inc. (U.S.)
12.9.1 H&M Bay, Inc. (U.S.) Company Details
12.9.2 Company Description and Business Overview
12.9.3 Food Delivery Logistic Introduction
12.9.4 H&M Bay, Inc. (U.S.) Revenue in Food Delivery Logistic Business (2013-2018)
12.9.5 H&M Bay, Inc. (U.S.) Recent Development
12.10 Hellmann Worldwide Logistics GmbH & Co. (Germany)
12.10.1 Hellmann Worldwide Logistics GmbH & Co. (Germany) Company Details
12.10.2 Company Description and Business Overview
12.10.3 Food Delivery Logistic Introduction
12.10.4 Hellmann Worldwide Logistics GmbH & Co. (Germany) Revenue in Food Delivery Logistic Business (2013-2018)
12.10.5 Hellmann Worldwide Logistics GmbH & Co. (Germany) Recent Development
12.11 Henningsen Cold Storage Co. (U.S.)

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## Global Logistics Market Information Report: By Transportation Type (Airways, Waterways, Railways, Roadways), Logistic Type (First Party, Second Party, Third Party) End User (Industrial and Manufacturing, Retail, Healthcare, Oil & Gas) - Forecast 2027

Impact of covid-19 on the Global Logistics Market

In 2019, the global logistics market recorded 1.2% in export trade volume growth due to the sluggish global economy, geopolitical uncertainties, trade disputes, and environmental regulations. In Q1 2020, the COVID-19 outbreak reversed the growth change in every section of the economy, including the global supply chain. Additionally, as economies across the globe are actively focusing on containing the outbreak, the global supply chain and trade is taking a severe hit owing to a rapid drop in global investment flow.

The global maritime industry has been affected by the COVID-19 outbreak, both directly and indirectly. Slumping manufacturing confidence and declining demand for raw materials and commodities are increasing the ambiguity for the ocean freight & transportation market. Stringent containment actions imposed by government bodies across the world and protective measures to lessen the outbreak impact have led to declined cargo volumes and trade across ports in North Europe and the West US.

China accounts for two-thirds of the top 10 busiest ports across the world and is responsible for more than 40% of the maritime trade in the world. The lockdown in the country has resulted in supply deficiencies as the pandemic is taking a toll on several global maritime segments from container lines to oil tankers. The drop in cargo volume has led to increased carrier service delays and cancellations. Considering the uncertain market situation, this trend is anticipated to continue to cause demand/supply imbalance. If the situation persists, carriers are likely to opt for substantial capacity reduction actions, which is expected to impact the fleet-order.

The airfreight segment of the logistics market has been a crucial partner in ensuring that the global supply chain remains effective for important and time-bound shipments. As the COVID-19 outbreak has spread across the world, many air carrier companies have grounded their fleets owing to travel restrictions and reducing demand. Air cargo demand fell by 3% year-on-year in the first two months of 2020 due to the COVID-19 crisis. This deterioration has been partially offset by augmented demand for air shipments of crucial relief supplies and other transitional goods. The air carrier companies based in the US, South Korea, and countries in Europe have begun offering their idled aircraft and passenger jets to transport food supplies and medical/pharmaceutical essentials across important international freight centers. Government bodies are likely to eliminate economic barriers and terminal slot limitations for air cargo actions, ensuring the flow of sensitive and essential goods.

Apart from the factors mentioned above, consumer purchasing behavior sentiments are also expected to impact the overall logistics market as people are not willing to invest in any business deals owing to limited per capita income and fear of market fluctuations. Thus, considering the current scenario, logistics services are not expected to remain affordable for the stakeholders.

consumer price index (cpi%) of prominent countries

1 Executive Summary

2 Scope of the Report

2.4 Market size Estimation

3 Market Landscape

3.1 Porter’s Five Forces Analysis

3.1.1 Threat of New Entrants

3.1.2 Bargaining power of buyers

3.1.3 Bargaining power of Suppliers

3.1.4 Threat of substitutes

3.2 Value Chain/Supply Chain Analysis

4 Market Dynamics

5 Global Logistics Market By Transportation Type

5.2.1 Market Estimates & Forecast, 2020-2027

5.2.2 Market Estimates & Forecast by Region, 2020-2027

5.3.1 Market Estimates & Forecast, 2020-2027

5.3.2 Market Estimates & Forecast by Region, 2020-2027

5.4.1 Market Estimates & Forecast, 2020-2027

5.4.2 Market Estimates & Forecast by Region, 2020-2027

5.5.1 Market Estimates & Forecast, 2020-2027

5.5.2 Market Estimates & Forecast by Region, 2020-2027

6 Global Logistics Market, By Logistic Type

6.2.1 Market Estimates & Forecast, 2020-2027

6.2.2 Market Estimates & Forecast by Region, 2020-2027

6.3.1 Market Estimates & Forecast, 2020-2027

6.3.2 Market Estimates & Forecast by Region, 2020-2027

6.4.1 Market Estimates & Forecast, 2020-2027

6.4.2 Market Estimates & Forecast by Region, 2020-2027

7 Global Logistics Market, By End-User

7.2 Industrial and Manufacturing

7.2.1 Market Estimates & Forecast, 2020-2027

7.2.2 Market Estimates & Forecast by Region, 2020-2027

7.3.1 Market Estimates & Forecast, 2020-2027

7.3.2 Market Estimates & Forecast by Region, 2020-2027

7.4.1 Market Estimates & Forecast, 2020-2027

7.4.2 Market Estimates & Forecast by Region, 2020-2027

7.5.1 Market Estimates & Forecast, 2020-2027

7.5.2 Market Estimates & Forecast by Region, 2020-2027

7.6.1 Market Estimates & Forecast, 2020-2027

7.6.2 Market Estimates & Forecast by Region, 2020-2027

8 Global Logistics Market, By Region

8.2.1 Market Estimates & Forecast, 2020-2027

8.2.2 Market Estimates & Forecast by Transportation Type , 2020-2027

8.2.3 Market Estimates & Forecast by Logistic Type , 2020-2027

8.2.4 Market Estimates & Forecast by End-User , 2020-2027

8.2.5.1 Market Estimates & Forecast, 2020-2027

8.2.5.2 Market Estimates & Forecast by Transportation Type , 2020-2027

8.2.5.3 Market Estimates & Forecast by Logistic Type , 2020-2027

8.2.5.4 Market Estimates & Forecast by End-User , 2020-2027

8.2.6.1 Market Estimates & Forecast, 2020-2027

8.2.6.2 Market Estimates & Forecast by Transportation Type , 2020-2027

8.2.6.3 Market Estimates & Forecast by Logistic Type , 2020-2027

8.2.6.4 Market Estimates & Forecast by End-User , 2020-2027

8.3.1 Market Estimates & Forecast, 2020-2027

8.3.2 Market Estimates & Forecast by Transportation Type , 2020-2027

8.3.3 Market Estimates & Forecast by Logistic Type , 2020-2027

8.3.4 Market Estimates & Forecast by End-User , 2020-2027

8.3.5.1 Market Estimates & Forecast, 2020-2027

8.3.5.2 Market Estimates & Forecast by Transportation Type , 2020-2027

8.3.5.3 Market Estimates & Forecast by Logistic Type , 2020-2027

8.3.5.4 Market Estimates & Forecast by End-User , 2020-2027

8.3.6.1 Market Estimates & Forecast, 2020-2027

8.3.6.2 Market Estimates & Forecast by Transportation Type , 2020-2027

8.3.6.3 Market Estimates & Forecast by Logistic Type , 2020-2027

8.3.6.4 Market Estimates & Forecast by End-User , 2020-2027

8.3.7.1 Market Estimates & Forecast, 2020-2027

8.3.7.2 Market Estimates & Forecast by Transportation Type , 2020-2027

8.3.7.3 Market Estimates & Forecast by Logistic Type , 2020-2027

8.3.7.4 Market Estimates & Forecast by End-User , 2020-2027

8.3.8.1 Market Estimates & Forecast, 2020-2027

8.3.8.2 Market Estimates & Forecast by Transportation Type , 2020-2027

8.3.8.3 Market Estimates & Forecast by Logistic Type , 2020-2027

8.3.8.4 Market Estimates & Forecast by End-User , 2020-2027

8.3.9.1 Market Estimates & Forecast, 2020-2027

8.3.9.2 Market Estimates & Forecast by Transportation Type , 2020-2027

8.3.9.3 Market Estimates & Forecast by Logistic Type , 2020-2027

8.3.9.4 Market Estimates & Forecast by End-User , 2020-2027

8.4.1 Market Estimates & Forecast, 2020-2027

8.4.2 Market Estimates & Forecast by Transportation Type , 2020-2027

8.4.3 Market Estimates & Forecast by Logistic Type , 2020-2027

8.4.4 Market Estimates & Forecast by End-User , 2020-2027

8.4.5.1 Market Estimates & Forecast, 2020-2027

8.4.5.2 Market Estimates & Forecast by Transportation Type , 2020-2027

8.4.5.3 Market Estimates & Forecast by Logistic Type , 2020-2027

8.4.5.4 Market Estimates & Forecast by End-User , 2020-2027

8.4.6.1 Market Estimates & Forecast, 2020-2027

8.4.6.2 Market Estimates & Forecast by Transportation Type , 2020-2027

8.4.6.3 Market Estimates & Forecast by Logistic Type , 2020-2027

8.4.6.4 Market Estimates & Forecast by End-User , 2020-2027

8.4.7.1 Market Estimates & Forecast, 2020-2027

8.4.7.2 Market Estimates & Forecast by Transportation Type , 2020-2027

8.4.7.3 Market Estimates & Forecast by Logistic Type , 2020-2027

8.4.7.4 Market Estimates & Forecast by End-User , 2020-2027

8.4.8.1 Market Estimates & Forecast, 2020-2027

8.4.8.2 Market Estimates & Forecast by Transportation Type , 2020-2027

8.4.8.3 Market Estimates & Forecast by Logistic Type , 2020-2027

8.4.8.4 Market Estimates & Forecast by End-User , 2020-2027

8.5.1 Market Estimates & Forecast, 2020-2027

8.5.2 Market Estimates & Forecast by Transportation Type , 2020-2027

8.5.3 Market Estimates & Forecast by Logistic Type , 2020-2027

8.5.4 Market Estimates & Forecast by End-User , 2020-2027

9 Competitive Landscape

10 Company Profile

10.1 C.H. Robinson Worldwide, Inc. (U.S.)

10.1.2 Products/Services Offering

10.2.2 Products/Services Offering

10.3.2 Products/Services Offering

10.4.2 Products/Services Offering

10.5 Expeditors International of Washington, Inc. (U.S.)

10.5.2 Products/Services Offering

10.6 DHL International GmbH (Germany)

10.6.2 Products/Services Offering

10.7 DSV Global Transports and Logistics (Denmark)

10.7.2 Products/Services Offering

10.8 A.P. Moller – Maersk (Denmark)

10.8.2 Products/Services Offering

10.9.2 Products/Services Offering

10.10.2 Products/Services Offering

10.11 DTDC Express Limited (India)

10.11.2 Products/Services Offering

Table 1 Global Logistics Market: By Region, 2020-2027

Table 2 North America Logistics Market: By Country, 2020-2027

Table 3 Europe Logistics Market: By Country, 2020-2027

Table 4 Asia-Pacific Logistics Market: By Country, 2020-2027

Table 5 RoW Logistics Market: By Country, 2020-2027

Table 6 Global Logistics Market by Transportation Type , By Regions, 2020-2027

Table 7 North America Logistics Market by Transportation Type , By Country, 2020-2027

Table 8 Europe Logistics Market, by Transportation Type , By Country, 2020-2027

Table 9 Asia-Pacific Logistics Market by Transportation Type , By Country, 2020-2027

Table 10 RoW Logistics Market by Transportation Type , By Country, 2020-2027

Table 11 Global Logistics by Logistic Type Market: By Regions, 2020-2027

Table 12 North America Logistics Market by Logistic Type : By Country, 2020-2027

Table 13 Europe Logistics Market by Logistic Type : By Country, 2020-2027

Table 14 Asia Pacific Logistics Market by Logistic Type : By Country, 2020-2027

Table 15 RoW Logistics Market by Logistic Type : By Country, 2020-2027

Table 16 Global Logistics Market by End-User , By Regions, 2020-2027

Table 17 North America Logistics Market by End-User , By Country, 2020-2027

Table 18 Europe Logistics Market by End-User , By Country, 2020-2027

Table 19 Asia Pacific Logistics Market by End-User , By Country, 2020-2027

Table 20 RoW Logistics Market by End-User , By Country, 2020-2027

Table 21 Global Transportation Type Market: By Region, 2020-2027

Table 22 Global Logistic Type Market: By Region, 2020-2027

Table 23 Global End-User Market: By Region, 2020-2027

Table 24 North America Logistics Market By Country

Table 25 North America Logistics Market By Transportation Type

Table 26 North America Logistics Market By Logistic Type

Table 27 North America Logistics Market By End-User

Table 28 Europe: Logistics Market By Country

Table 29 Europe: Logistics Market By Transportation Type

Table 30 Europe: Logistics Market By Logistic Type

Table 31 Europe: Logistics Market By End-User

Table 32 Asia-Pacific: Logistics Market By Country

Table 33 Asia-Pacific: Logistics Market By Transportation Type

Table 34 Asia-Pacific: Logistics Market By Logistic Type

Table 35 Asia-Pacific: Logistics Market By End-User

Table 36 RoW: Logistics Market By Region

Table 37 RoW Logistics Market By Transportation Type

Table 38 RoW Logistics Market By Logistic Type

Table 39 RoW Logistics Market By End-User

FIGURE 1 RESEARCH PROCESS OF MRFR

FIGURE 2 TOP DOWN & BOTTOM UP APPROACH

FIGURE 4 IMPACT ANALYSIS: MARKET DRIVERS

FIGURE 5 IMPACT ANALYSIS: MARKET RESTRAINTS

FIGURE 6 PORTER’S FIVE FORCES ANALYSIS

FIGURE 7 VALUE CHAIN ANALYSIS

FIGURE 8 GLOBAL LOGISTICS MARKET SHARE, BY RAW MATERIAL, 2020 (%)

FIGURE 9 GLOBAL LOGISTICS MARKET BY TRANSPORTATION TYPE , 2020-2027 (USD MILLION)

FIGURE 10 GLOBAL LOGISTICS MARKET SHARE, BY END-USER, 2020 (%)

FIGURE 11 GLOBAL LOGISTICS MARKET BY END-USER , 2020-2027 (USD MILLION)

FIGURE 12 GLOBAL LOGISTICS MARKET SHARE (%), BY REGION, 2020

FIGURE 13 GLOBAL LOGISTICS MARKET BY REGION, 2020-2027 (USD MILLION)

FIGURE 14 NORTH AMERICA LOGISTICS MARKET SHARE (%), 2020

FIGURE 15 NORTH AMERICA LOGISTICS MARKET BY COUNTRY, 2020-2027 (USD MILLION)

FIGURE 16 EUROPE LOGISTICS MARKET SHARE (%), 2020

FIGURE 17 EUROPE LOGISTICS MARKET BY COUNTRY, 2020-2027 (USD MILLION)

FIGURE 18 ASIA-PACIFIC LOGISTICS MARKET SHARE (%), 2020

FIGURE 19 ASIA-PACIFIC LOGISTICS MARKET BY COUNTRY, 2020-2027 (USD MILLION)

FIGURE 20 REST OF THE WORLD LOGISTICS MARKET SHARE (%), 2020

FIGURE 21 REST OF THE WORLD LOGISTICS MARKET BY COUNTRY, 2020-2027 (USD MILLION)