# Optimization solutions

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## Optimization solutions

You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

In this section we are going to look at optimization problems. In optimization problems we are looking for the largest value or the smallest value that a function can take. We saw how to solve one kind of optimization problem in the Absolute Extrema section where we found the largest and smallest value that a function would take on an interval.

In this section we are going to look at another type of optimization problem. Here we will be looking for the largest or smallest value of a function subject to some kind of constraint. The constraint will be some condition (that can usually be described by some equation) that must absolutely, positively be true no matter what our solution is. On occasion, the constraint will not be easily described by an equation, but in these problems it will be easy to deal with as we’ll see.

This section is generally one of the more difficult for students taking a Calculus course. One of the main reasons for this is that a subtle change of wording can completely change the problem. There is also the problem of identifying the quantity that we’ll be optimizing and the quantity that is the constraint and writing down equations for each.

In identifying the constraint remember that the constraint is the quantity that must be true regardless of the solution. In almost every one of the problems we’ll be looking at here one quantity will be clearly indicated as having a fixed value and so must be the constraint. Once you’ve got that identified the quantity to be optimized should be fairly simple to get. It is however easy to confuse the two if you just skim the problem so make sure you carefully read the problem first!

Example 1 We need to enclose a rectangular field with a fence. We have 500 feet of fencing material and a building is on one side of the field and so won’t need any fencing. Determine the dimensions of the field that will enclose the largest area.

In all of these problems we will have two functions. The first is the function that we are actually trying to optimize and the second will be the constraint. Sketching the situation will often help us to arrive at these equations so let’s do that.

In this problem we want to maximize the area of a field and we know that will use 500 ft of fencing material. So, the area will be the function we are trying to optimize and the amount of fencing is the constraint. The two equations for these are,

Okay, we know how to find the largest or smallest value of a function provided it’s only got a single variable. The area function (as well as the constraint) has two variables in it and so what we know about finding absolute extrema won’t work. However, if we solve the constraint for one of the two variables we can substitute this into the area and we will then have a function of a single variable.

Now we want to find the largest value this will have on the interval $$\left[ \right]$$. The limits in this interval corresponds to taking $$y = 0$$ (i.e. no sides to the fence) and $$y = 250$$ (i.e. only two sides and no width, also if there are two sides each must be 250 ft to use the whole 500ft).

Note that the endpoints of the interval won’t make any sense from a physical standpoint if we actually want to enclose some area because they would both give zero area. They do, however, give us a set of limits on $$y$$ and so the Extreme Value Theorem tells us that we will have a maximum value of the area somewhere between the two endpoints. Having these limits will also mean that we can use the process we discussed in the Finding Absolute Extrema section earlier in the chapter to find the maximum value of the area.

So, recall that the maximum value of a continuous function (which we’ve got here) on a closed interval (which we also have here) will occur at critical points and/or end points. As we’ve already pointed out the end points in this case will give zero area and so don’t make any sense. That means our only option will be the critical points.

Setting this equal to zero and solving gives a lone critical point of $$y = 125$$. Plugging this into the area gives an area of $$A\left( \right) = 31250\,<\mbox><<\mbox>^2>$$. So according to the method from Absolute Extrema section this must be the largest possible area, since the area at either endpoint is zero.

Don’t forget to actually read the problem and give the answer that was asked for. These types of problems can take a fair amount of time/effort to solve and it’s not hard to sometimes forget what the problem was actually asking for.

## Two Popular Network Optimization Solutions

In today’s economic atmosphere, IT resources can be stretched thin. Luckily, there are several types of solutions that can help your team with network optimization. This includes network optimization tools that empower you to in-house your optimization efforts, and network optimization managed services that do the heavy lifting for you.

### Network Performance Monitoring Software

Network performance monitoring software is a type of software that helps you track the performance of your network. This type of software enables you to optimize your enterprise network across devices supplied by a host of different vendors.

Enterprise network monitoring software can automate network device discovery and provide intuitive workflows that make it easy to identify when faults arise. Responsive dashboards let you quickly assess your network health or drill down to the device level to quickly and fix network problems.

Entuity Software™ supports network infrastructure monitoring for thousands of devices out of the box across hundreds of vendors. And, Entuity has been consistently recognized as leader in the Network Monitoring & Management category by G2.

### Network Management as a Service

Network management as a service (NMaaS) provides managed engineer assistance for optimizing your network. Network management services can help you with a variety of tasks, including monitoring performance, identifying issues, and implementing solutions.

NMaaS providers can help streamline your network management processes by taking care of hybrid environments, supporting new networking technologies, and ensuring outstanding end-user experiences. Some of the best network infrastructure management providers, like ParkView Managed Services™, are backed by an Enterprise Operations Center (EOC) that ensures a smooth deployment with a tailored onboarding plan that includes custom set-ups for network device management, custom dashboards and reports that cater to your organizational goals.

Authorship:

We want to minimize the cost of the materials subject to the constraint that the volume must be 50ft 3 . Note as well that the cost for each side is just the area of that side times the appropriate cost.

## Optimization solutions

You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

In this section we are going to look at optimization problems. In optimization problems we are looking for the largest value or the smallest value that a function can take. We saw how to solve one kind of optimization problem in the Absolute Extrema section where we found the largest and smallest value that a function would take on an interval.

In this section we are going to look at another type of optimization problem. Here we will be looking for the largest or smallest value of a function subject to some kind of constraint. The constraint will be some condition (that can usually be described by some equation) that must absolutely, positively be true no matter what our solution is. On occasion, the constraint will not be easily described by an equation, but in these problems it will be easy to deal with as we’ll see.

This section is generally one of the more difficult for students taking a Calculus course. One of the main reasons for this is that a subtle change of wording can completely change the problem. There is also the problem of identifying the quantity that we’ll be optimizing and the quantity that is the constraint and writing down equations for each.

In identifying the constraint remember that the constraint is the quantity that must be true regardless of the solution. In almost every one of the problems we’ll be looking at here one quantity will be clearly indicated as having a fixed value and so must be the constraint. Once you’ve got that identified the quantity to be optimized should be fairly simple to get. It is however easy to confuse the two if you just skim the problem so make sure you carefully read the problem first!

Example 1 We need to enclose a rectangular field with a fence. We have 500 feet of fencing material and a building is on one side of the field and so won’t need any fencing. Determine the dimensions of the field that will enclose the largest area.

In all of these problems we will have two functions. The first is the function that we are actually trying to optimize and the second will be the constraint. Sketching the situation will often help us to arrive at these equations so let’s do that.

In this problem we want to maximize the area of a field and we know that will use 500 ft of fencing material. So, the area will be the function we are trying to optimize and the amount of fencing is the constraint. The two equations for these are,

Okay, we know how to find the largest or smallest value of a function provided it’s only got a single variable. The area function (as well as the constraint) has two variables in it and so what we know about finding absolute extrema won’t work. However, if we solve the constraint for one of the two variables we can substitute this into the area and we will then have a function of a single variable.

Now we want to find the largest value this will have on the interval $$\left[ \right]$$. The limits in this interval corresponds to taking $$y = 0$$ (i.e. no sides to the fence) and $$y = 250$$ (i.e. only two sides and no width, also if there are two sides each must be 250 ft to use the whole 500ft).

Note that the endpoints of the interval won’t make any sense from a physical standpoint if we actually want to enclose some area because they would both give zero area. They do, however, give us a set of limits on $$y$$ and so the Extreme Value Theorem tells us that we will have a maximum value of the area somewhere between the two endpoints. Having these limits will also mean that we can use the process we discussed in the Finding Absolute Extrema section earlier in the chapter to find the maximum value of the area.

So, recall that the maximum value of a continuous function (which we’ve got here) on a closed interval (which we also have here) will occur at critical points and/or end points. As we’ve already pointed out the end points in this case will give zero area and so don’t make any sense. That means our only option will be the critical points.

Setting this equal to zero and solving gives a lone critical point of $$y = 125$$. Plugging this into the area gives an area of $$A\left( \right) = 31250\,<\mbox><<\mbox>^2>$$. So according to the method from Absolute Extrema section this must be the largest possible area, since the area at either endpoint is zero.

Don’t forget to actually read the problem and give the answer that was asked for. These types of problems can take a fair amount of time/effort to solve and it’s not hard to sometimes forget what the problem was actually asking for.

#### First Derivative Test for Absolute Extrema

Let $$I$$ be the interval of all possible values of $$x$$ in $$f\left( x \right)$$, the function we want to optimize, and further suppose that $$f\left( x \right)$$ is continuous on $$I$$ , except possibly at the endpoints. Finally suppose that $$x = c$$ is a critical point of $$f\left( x \right)$$ and that $$c$$ is in the interval $$I$$. If we restrict $$x$$ to values from $$I$$ (i.e. we only consider possible optimal values of the function) then,

The first way to use the second derivative doesn’t actually help us to identify the optimal value. What it does do is allow us to potentially exclude values and knowing this can simplify our work somewhat and so is not a bad thing to do.

Suppose that we are looking for the absolute maximum of a function and after finding the critical points we find that we have multiple critical points. Let’s also suppose that we run all of them through the second derivative test and determine that some of them are in fact relative minimums of the function. Since we are after the absolute maximum we know that a maximum (of any kind) can’t occur at relative minimums and so we immediately know that we can exclude these points from further consideration. We could do a similar check if we were looking for the absolute minimum. Doing this may not seem like all that great of a thing to do, but it can, on occasion, lead to a nice reduction in the amount of work that we need to do in later steps.

The second way of using the second derivative to identify the optimal value of a function is in fact very similar to the second method above. In fact, we will have the same requirements for this method as we did in that method. We need an interval of possible values of the independent variable in function we are optimizing, call it $$I$$ as before, and the endpoint(s) may or may not be finite. We’ll also need to require that the function, $$f\left( x \right)$$ be continuous everywhere in $$I$$ except possibly at the endpoints as above.

Now, suppose that $$x = c$$ is a critical point and that $$f”\left( c \right) > 0$$. The second derivative test tells us that $$x = c$$ must be a relative minimum of the function. Suppose however that we also knew that $$f”\left( x \right) > 0$$ for all $$x$$ in $$I$$. In this case we would know that the function was concave up in all of $$I$$ and that would in turn mean that the absolute minimum of $$f\left( x \right)$$ in $$I$$ would in fact have to be at $$x = c$$.

## Our clients

Our solutions work for both B2C and B2B environments. We use methods that can process large amounts of clickstream data in B2C settings, and we use advanced techniques to handle sparse data in B2B setups. The complex data come in, but we manage it well to ensure your company has the most effective trade spend.

We provide a comprehensive toolkit to optimize the parameters of existing trade promotions, identify new promotion opportunities, and jointly optimize promotion calendars for entire categories. This ensures that the trade promotion optimization process truly helps to improve the business performance of the company, as opposed to only boosting selected metrics and negatively impacting related products and customer experiences.

We build tools that optimize promotion calendars for several months ahead, but we also develop components that can dynamically optimize and personalize promotions for micro-moment marketing. This enables the integration of smart price promotion management with online product catalogs, search engines, point of sale systems, and more. We ensure that all the data coming in your trade promotion system drive an increase in your sales.

We put considerable emphasis on data completeness and data quality. Our solutions for trade promotion optimization usually integrate with third-party data providers or include web scraping components to collect competitor pricing, information about the customer base, macroeconomic data, and more. Comprehensive data collection from your business sales along with market data from other companies helps to provide long-term sustainability and business growth.

Would you like to learn more about the economic and algorithmic foundations of trade promotion optimization? We have published a 500-page book on enterprise data science that is available for free download, and there is an entire chapter on price and promotion management in it.

## Industries

For retail customers, we create trade promotion optimization solutions that analyze promotion calendars, detect promotions that drive negative profits and need to be turned off, identify new promotion opportunities, and maximize the overall performance of a promotion calendar. The retailers we work with have a proven track record of optimizing promotion strategies using our TPO.

Our decision support tools help manufacturers to optimize their B2B trade promotion planning and execution, ensure trade promotion consistency across retail partners, and directly integrate with digital channels and marketplaces such as Amazon.

We help technology companies and video game publishers to understand their markets and the structure of the demand to optimize prices and promotions. We have extensive experience in handling international sales data and implementing trade promotion optimization for multiple countries, which is typically the case for technology and video game companies.

## Two Popular Network Optimization Solutions

In today’s economic atmosphere, IT resources can be stretched thin. Luckily, there are several types of solutions that can help your team with network optimization. This includes network optimization tools that empower you to in-house your optimization efforts, and network optimization managed services that do the heavy lifting for you.

### Network Performance Monitoring Software

Network performance monitoring software is a type of software that helps you track the performance of your network. This type of software enables you to optimize your enterprise network across devices supplied by a host of different vendors.

Enterprise network monitoring software can automate network device discovery and provide intuitive workflows that make it easy to identify when faults arise. Responsive dashboards let you quickly assess your network health or drill down to the device level to quickly and fix network problems.

Entuity Software™ supports network infrastructure monitoring for thousands of devices out of the box across hundreds of vendors. And, Entuity has been consistently recognized as leader in the Network Monitoring & Management category by G2.

### Network Management as a Service

Network management as a service (NMaaS) provides managed engineer assistance for optimizing your network. Network management services can help you with a variety of tasks, including monitoring performance, identifying issues, and implementing solutions.

NMaaS providers can help streamline your network management processes by taking care of hybrid environments, supporting new networking technologies, and ensuring outstanding end-user experiences. Some of the best network infrastructure management providers, like ParkView Managed Services™, are backed by an Enterprise Operations Center (EOC) that ensures a smooth deployment with a tailored onboarding plan that includes custom set-ups for network device management, custom dashboards and reports that cater to your organizational goals.

Authorship:

The importance of network performance cannot be overemphasized. A reliable, efficient network can reduce the stress of your day-to-day duties, help you focus more on strategic IT initiatives with your team, and avoid those 2 a.m. network failure calls that could ruin your week (or year). By monitoring your network and implementing the right solutions, you can keep your business running smoothly and avoid costly network downtime.

## Pricing optimization plaftorm

Measuring the efficiency of pricing actions and decisions is a difficult problem because customer behavior is very sophisticated: customers can switch between substitutable items, stockpile products at reduced prices, and so on. The true outcomes of price optimization can be reliably measured only after all these factors are estimated using statistical analysis. Our pricing optimization platform can provide you with a competitive edge by precisely measuring various economic and behavioral factors.

Predictive modeling is quite commonly used in price optimization software nowadays. However, the understandability and explainability of demand models remains a challenge for many. We invested years of research to develop efficient representations and visualizations of diagnostic information that can be easily comprehended by business users in both B2C and B2B industries.

## Network Optimization Metrics & KPIs

Although network traffic optimization is often viewed as a complex and daunting task, the main goal is actually very simple: to improve network performance. While there are many ways to achieve this, the most important factor is which metrics you track.

### Packet Loss

Packet loss is one of the most important metrics for network optimization. Loss can degrade network performance in many ways. This will result in slower response times, reduced bandwidth, and increased latency.

There are several factors that can cause packet loss, including hardware failure, software issues, and congestion. To optimize your network, you need to identify the source of the packet loss and take steps to mitigate it. Common resolutions include hardware refreshes or upgrades, expanding available bandwidth, and QoS prioritization. Fortunately for network admins like yourself, network fault management software platforms can help you pinpoint precise hardware failures that may be causing packet loss within your network.

### Latency

Latency is another important metric for network optimization. It measures the time it takes for a packet to travel from its source to its destination. Latency can be affected by many factors, including network, storage, or server hardware failures, software faults, and sub-optimal networking configuration.

High latency can lead to issues including dropped connections, choppy audio and video, and delayed response times. To reduce latency, you need to identify the source of the issue and take steps to mitigate it. If hardware failures are behind your latency issues, network topology mapping software can help you pinpoint the device-level geographic location of your culprit.

### Bandwidth

Bandwidth is another important metric for network traffic optimization. It measures the amount of data that can be transferred between two points in a period of time. Bandwidth can be affected by hardware, software, and networking configuration.

High bandwidth can lead to issues like increased latency, dropped connections, and poor audio and video quality over your network. To reduce bandwidth utilization, you need to identify the source of the issue and take steps to mitigate it. Common causes of enterprise bandwidth strain include malware, sporadic application updates, and even content streaming/social media applications that are resource-heavy but not business-critical.

### Availability

Availability is the final metric for network traffic optimization. It measures the uptime of your network, or the amount of time that it is available for use. Availability can be affected by several factors, including software, hardware, and the configuration of your network.

It’s important for your workforce and any external end users to have uninterrupted access to the resources and applications they need. To realize high availability, you must identify the source of your outage and put redundancies or prioritizations in place to maintain uptime.

## 5 Network Optimization Techniques and Best Practices

This technique helps distribute traffic evenly across a network, which can help prevent congestion and ensure optimal performance. This ensures that no one server is overutilized during high traffic periods and is instead smoothed among your broader server pool.

### 2. QoS Prioritization

QoS prioritization allows you to prioritize certain types of packets sent across your network. This includes prioritizing mission-critical packets like video conferencing VoIP packets over server backups that don’t need to happen in real-time. This can help ensure that critical data is always transferred smoothly and efficiently.

### 4. Leveraging an SD-WAN

Software-defined wide area networks (SD-WANs) can help improve network performance by dynamically routing traffic. SD-WAN leverages a virtualized, centralized control function to direct your network traffic across a wide area network. This eliminates the need to backhaul data to a central source if your organization has multiple campuses.

## Two Popular Network Optimization Solutions

In today’s economic atmosphere, IT resources can be stretched thin. Luckily, there are several types of solutions that can help your team with network optimization. This includes network optimization tools that empower you to in-house your optimization efforts, and network optimization managed services that do the heavy lifting for you.

### Network Performance Monitoring Software

Network performance monitoring software is a type of software that helps you track the performance of your network. This type of software enables you to optimize your enterprise network across devices supplied by a host of different vendors.

Enterprise network monitoring software can automate network device discovery and provide intuitive workflows that make it easy to identify when faults arise. Responsive dashboards let you quickly assess your network health or drill down to the device level to quickly and fix network problems.

Entuity Software™ supports network infrastructure monitoring for thousands of devices out of the box across hundreds of vendors. And, Entuity has been consistently recognized as leader in the Network Monitoring & Management category by G2.

### Network Management as a Service

Network management as a service (NMaaS) provides managed engineer assistance for optimizing your network. Network management services can help you with a variety of tasks, including monitoring performance, identifying issues, and implementing solutions.

NMaaS providers can help streamline your network management processes by taking care of hybrid environments, supporting new networking technologies, and ensuring outstanding end-user experiences. Some of the best network infrastructure management providers, like ParkView Managed Services™, are backed by an Enterprise Operations Center (EOC) that ensures a smooth deployment with a tailored onboarding plan that includes custom set-ups for network device management, custom dashboards and reports that cater to your organizational goals.

Authorship: