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Google OR Tools: Iterative Assignment Problem

This question is a Google OR-Tools specific implementation of recommendation from a previous question .

In short, the movie theater problem encompasses assigning viewers to seats such that the distance between viewers is maximized, however, the distance between viewers and fire exits is minimized. I've attempted to solve using OR-Tools, specifically the SCIP solver: Google Colab (reference for full code/details.)

The expert of interest is below:

Both attempt 1 and attempt 2 fail but for different reasons: Attempt multiplies two integer variables, which violates the assumption of linearity. Attempt 2 fails because the conditional logic is not accepted.

How can these objective functions be restated such that seat-seat distance penalty is only applied if both seat_i and seat_j equal 1.

• mixed-integer-programming
• linear-programming
• googlecolab

• 1 $\begingroup$ Edited on the other question on linearizing x's $\endgroup$ –  Sutanu Majumdar Commented Dec 1, 2022 at 23:41

Basically as earlier Linearize the obj Introduce $z_{i,j} \in\ {0,1} \ \forall i,j \in\ seats \ S$

Additional constraints $z_{i,j} \le x_i$ $z_{i,j} \le x_j$ $ x_i + x_j - 1 \le z_{i,j}$

In the objective part Viewer Dist, replace $x_ix_j$ with $z_{i,j}$

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Index     Assignment problem     Hungarian algorithm     Solve online    

Solve an assignment problem online

Fill in the cost matrix of an assignment problem and click on 'Solve'. The optimal assignment will be determined and a step by step explanation of the hungarian algorithm will be given.

Fill in the cost matrix ( random cost matrix ):

Don't show the steps of the Hungarian algorithm Maximize the total cost

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Solving Assignment Problem using Linear Programming in Python

Learn how to use Python PuLP to solve Assignment problems using Linear Programming.

In earlier articles, we have seen various applications of Linear programming such as transportation, transshipment problem, Cargo Loading problem, and shift-scheduling problem. Now In this tutorial, we will focus on another model that comes under the class of linear programming model known as the Assignment problem. Its objective function is similar to transportation problems. Here we minimize the objective function time or cost of manufacturing the products by allocating one job to one machine.

If we want to solve the maximization problem assignment problem then we subtract all the elements of the matrix from the highest element in the matrix or multiply the entire matrix by –1 and continue with the procedure. For solving the assignment problem, we use the Assignment technique or Hungarian method, or Flood’s technique.

The transportation problem is a special case of the linear programming model and the assignment problem is a special case of transportation problem, therefore it is also a special case of the linear programming problem.

In this tutorial, we are going to cover the following topics:

Assignment Problem

A problem that requires pairing two sets of items given a set of paired costs or profit in such a way that the total cost of the pairings is minimized or maximized. The assignment problem is a special case of linear programming.

For example, an operation manager needs to assign four jobs to four machines. The project manager needs to assign four projects to four staff members. Similarly, the marketing manager needs to assign the 4 salespersons to 4 territories. The manager’s goal is to minimize the total time or cost.

Problem Formulation

A manager has prepared a table that shows the cost of performing each of four jobs by each of four employees. The manager has stated his goal is to develop a set of job assignments that will minimize the total cost of getting all 4 jobs.  

Initialize LP Model

In this step, we will import all the classes and functions of pulp module and create a Minimization LP problem using LpProblem class.

Define Decision Variable

In this step, we will define the decision variables. In our problem, we have two variable lists: workers and jobs. Let’s create them using  LpVariable.dicts()  class.  LpVariable.dicts()  used with Python’s list comprehension.  LpVariable.dicts()  will take the following four values:

• First, prefix name of what this variable represents.
• Second is the list of all the variables.
• Third is the lower bound on this variable.
• Fourth variable is the upper bound.
• Fourth is essentially the type of data (discrete or continuous). The options for the fourth parameter are  LpContinuous  or  LpInteger .

Let’s first create a list route for the route between warehouse and project site and create the decision variables using LpVariable.dicts() the method.

Define Objective Function

In this step, we will define the minimum objective function by adding it to the LpProblem  object. lpSum(vector)is used here to define multiple linear expressions. It also used list comprehension to add multiple variables.

Define the Constraints

Here, we are adding two types of constraints: Each job can be assigned to only one employee constraint and Each employee can be assigned to only one job. We have added the 2 constraints defined in the problem by adding them to the LpProblem  object.

Solve Model

In this step, we will solve the LP problem by calling solve() method. We can print the final value by using the following for loop.

From the above results, we can infer that Worker-1 will be assigned to Job-1, Worker-2 will be assigned to job-3, Worker-3 will be assigned to Job-2, and Worker-4 will assign with job-4.

In this article, we have learned about Assignment problems, Problem Formulation, and implementation using the python PuLp library. We have solved the Assignment problem using a Linear programming problem in Python. Of course, this is just a simple case study, we can add more constraints to it and make it more complicated. You can also run other case studies on Cargo Loading problems , Staff scheduling problems . In upcoming articles, we will write more on different optimization problems such as transshipment problem, balanced diet problem. You can revise the basics of mathematical concepts in  this article  and learn about Linear Programming  in this article .

• Solving Blending Problem in Python using Gurobi
• Transshipment Problem in Python Using PuLP

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Assignment problem

The problem of optimally assigning $ m $ individuals to $ m $ jobs. It can be formulated as a linear programming problem that is a special case of the transport problem :

maximize $ \sum _ {i,j } c _ {ij } x _ {ij } $

$$ \sum _ { j } x _ {ij } = a _ {i} , i = 1 \dots m $$

(origins or supply),

$$ \sum _ { i } x _ {ij } = b _ {j} , j = 1 \dots n $$

(destinations or demand), where $ x _ {ij } \geq 0 $ and $ \sum a _ {i} = \sum b _ {j} $, which is called the balance condition. The assignment problem arises when $ m = n $ and all $ a _ {i} $ and $ b _ {j} $ are $ 1 $.

If all $ a _ {i} $ and $ b _ {j} $ in the transposed problem are integers, then there is an optimal solution for which all $ x _ {ij } $ are integers (Dantzig's theorem on integral solutions of the transport problem).

In the assignment problem, for such a solution $ x _ {ij } $ is either zero or one; $ x _ {ij } = 1 $ means that person $ i $ is assigned to job $ j $; the weight $ c _ {ij } $ is the utility of person $ i $ assigned to job $ j $.

The special structure of the transport problem and the assignment problem makes it possible to use algorithms that are more efficient than the simplex method . Some of these use the Hungarian method (see, e.g., [a5] , [a1] , Chapt. 7), which is based on the König–Egervary theorem (see König theorem ), the method of potentials (see [a1] , [a2] ), the out-of-kilter algorithm (see, e.g., [a3] ) or the transportation simplex method.

In turn, the transportation problem is a special case of the network optimization problem.

A totally different assignment problem is the pole assignment problem in control theory.

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How to Solve the Assignment Problem: A Complete Guide

Table of Contents

Assignment problem is a special type of linear programming problem that deals with assigning a number of resources to an equal number of tasks in the most efficient way. The goal is to minimize the total cost of assignments while ensuring that each task is assigned to only one resource and each resource is assigned to only one task. In this blog, we will discuss the solution of the assignment problem using the Hungarian method, which is a popular algorithm for solving the problem.

Understanding the Assignment Problem

Before we dive into the solution, it is important to understand the problem itself. In the assignment problem, we have a matrix of costs, where each row represents a resource and each column represents a task. The objective is to assign each resource to a task in such a way that the total cost of assignments is minimized. However, there are certain constraints that need to be satisfied – each resource can be assigned to only one task and each task can be assigned to only one resource.

Solving the Assignment Problem

There are various methods for solving the assignment problem, including the Hungarian method, the brute force method, and the auction algorithm. Here, we will focus on the steps involved in solving the assignment problem using the Hungarian method, which is the most commonly used and efficient method.

Step 1: Set up the cost matrix

The first step in solving the assignment problem is to set up the cost matrix, which represents the cost of assigning a task to an agent. The matrix should be square and have the same number of rows and columns as the number of tasks and agents, respectively.

Step 2: Subtract the smallest element from each row and column

To simplify the calculations, we need to reduce the size of the cost matrix by subtracting the smallest element from each row and column. This step is called matrix reduction.

Step 3: Cover all zeros with the minimum number of lines

The next step is to cover all zeros in the matrix with the minimum number of horizontal and vertical lines. This step is called matrix covering.

Step 4: Test for optimality and adjust the matrix

To test for optimality, we need to calculate the minimum number of lines required to cover all zeros in the matrix. If the number of lines equals the number of rows or columns, the solution is optimal. If not, we need to adjust the matrix and repeat steps 3 and 4 until we get an optimal solution.

Step 5: Assign the tasks to the agents

The final step is to assign the tasks to the agents based on the optimal solution obtained in step 4. This will give us the most cost-effective or profit-maximizing assignment.

Solution of the Assignment Problem using the Hungarian Method

The Hungarian method is an algorithm that uses a step-by-step approach to find the optimal assignment. The algorithm consists of the following steps:

• Subtract the smallest entry in each row from all the entries of the row.
• Subtract the smallest entry in each column from all the entries of the column.
• Draw the minimum number of lines to cover all zeros in the matrix. If the number of lines drawn is equal to the number of rows, we have an optimal solution. If not, go to step 4.
• Determine the smallest entry not covered by any line. Subtract it from all uncovered entries and add it to all entries covered by two lines. Go to step 3.

The above steps are repeated until an optimal solution is obtained. The optimal solution will have all zeros covered by the minimum number of lines. The assignments can be made by selecting the rows and columns with a single zero in the final matrix.

Applications of the Assignment Problem

The assignment problem has various applications in different fields, including computer science, economics, logistics, and management. In this section, we will provide some examples of how the assignment problem is used in real-life situations.

Applications in Computer Science

The assignment problem can be used in computer science to allocate resources to different tasks, such as allocating memory to processes or assigning threads to processors.

Applications in Economics

The assignment problem can be used in economics to allocate resources to different agents, such as allocating workers to jobs or assigning projects to contractors.

Applications in Logistics

The assignment problem can be used in logistics to allocate resources to different activities, such as allocating vehicles to routes or assigning warehouses to customers.

Applications in Management

The assignment problem can be used in management to allocate resources to different projects, such as allocating employees to tasks or assigning budgets to departments.

Let’s consider the following scenario: a manager needs to assign three employees to three different tasks. Each employee has different skills, and each task requires specific skills. The manager wants to minimize the total time it takes to complete all the tasks. The skills and the time required for each task are given in the table below:

The assignment problem is to determine which employee should be assigned to which task to minimize the total time required. To solve this problem, we can use the Hungarian method, which we discussed in the previous blog.

Using the Hungarian method, we first subtract the smallest entry in each row from all the entries of the row:

Next, we subtract the smallest entry in each column from all the entries of the column:

We draw the minimum number of lines to cover all the zeros in the matrix, which in this case is three:

Since the number of lines is equal to the number of rows, we have an optimal solution. The assignments can be made by selecting the rows and columns with a single zero in the final matrix. In this case, the optimal assignments are:

• Emp 1 to Task 3
• Emp 2 to Task 2
• Emp 3 to Task 1

This assignment results in a total time of 9 units.

I hope this example helps you better understand the assignment problem and how to solve it using the Hungarian method.

Solving the assignment problem may seem daunting, but with the right approach, it can be a straightforward process. By following the steps outlined in this guide, you can confidently tackle any assignment problem that comes your way.

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Operations Research

1 Operations Research-An Overview

• History of O.R.
• Approach, Techniques and Tools
• Phases and Processes of O.R. Study
• Typical Applications of O.R
• Limitations of Operations Research
• Models in Operations Research
• O.R. in real world

2 Linear Programming: Formulation and Graphical Method

• General formulation of Linear Programming Problem
• Optimisation Models
• Basics of Graphic Method
• Important steps to draw graph
• Multiple, Unbounded Solution and Infeasible Problems
• Solving Linear Programming Graphically Using Computer
• Application of Linear Programming in Business and Industry

3 Linear Programming-Simplex Method

• Principle of Simplex Method
• Computational aspect of Simplex Method
• Simplex Method with several Decision Variables
• Two Phase and M-method
• Multiple Solution, Unbounded Solution and Infeasible Problem
• Sensitivity Analysis
• Dual Linear Programming Problem

4 Transportation Problem

• Basic Feasible Solution of a Transportation Problem
• Modified Distribution Method
• Stepping Stone Method
• Unbalanced Transportation Problem
• Degenerate Transportation Problem
• Transhipment Problem
• Maximisation in a Transportation Problem

5 Assignment Problem

• Solution of the Assignment Problem
• Unbalanced Assignment Problem
• Problem with some Infeasible Assignments
• Maximisation in an Assignment Problem
• Crew Assignment Problem

6 Application of Excel Solver to Solve LPP

• Building Excel model for solving LP: An Illustrative Example

7 Goal Programming

• Concepts of goal programming
• Goal programming model formulation
• Graphical method of goal programming
• The simplex method of goal programming
• Using Excel Solver to Solve Goal Programming Models
• Application areas of goal programming

8 Integer Programming

• Some Integer Programming Formulation Techniques
• Binary Representation of General Integer Variables
• Unimodularity
• Cutting Plane Method
• Branch and Bound Method
• Solver Solution

9 Dynamic Programming

• Dynamic Programming Methodology: An Example
• Definitions and Notations
• Dynamic Programming Applications

10 Non-Linear Programming

• Solution of a Non-linear Programming Problem
• Convex and Concave Functions
• Kuhn-Tucker Conditions for Constrained Optimisation
• Quadratic Programming
• Separable Programming
• NLP Models with Solver

11 Introduction to game theory and its Applications

• Important terms in Game Theory
• Saddle points
• Mixed strategies: Games without saddle points
• 2 x n games
• Exploiting an opponent’s mistakes

12 Monte Carlo Simulation

• Reasons for using simulation
• Monte Carlo simulation
• Limitations of simulation
• Steps in the simulation process
• Some practical applications of simulation
• Two typical examples of hand-computed simulation
• Computer simulation

13 Queueing Models

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• Notations and Symbols
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• The M/M/I System
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Assignment Problem: Meaning, Methods and Variations | Operations Research

After reading this article you will learn about:- 1. Meaning of Assignment Problem 2. Definition of Assignment Problem 3. Mathematical Formulation 4. Hungarian Method 5. Variations.

Meaning of Assignment Problem:

An assignment problem is a particular case of transportation problem where the objective is to assign a number of resources to an equal number of activities so as to minimise total cost or maximize total profit of allocation.

The problem of assignment arises because available resources such as men, machines etc. have varying degrees of efficiency for performing different activities, therefore, cost, profit or loss of performing the different activities is different.

Thus, the problem is “How should the assignments be made so as to optimize the given objective”. Some of the problem where the assignment technique may be useful are assignment of workers to machines, salesman to different sales areas.

Definition of Assignment Problem:

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Suppose there are n jobs to be performed and n persons are available for doing these jobs. Assume that each person can do each job at a term, though with varying degree of efficiency, let c ij be the cost if the i-th person is assigned to the j-th job. The problem is to find an assignment (which job should be assigned to which person one on-one basis) So that the total cost of performing all jobs is minimum, problem of this kind are known as assignment problem.

The assignment problem can be stated in the form of n x n cost matrix C real members as given in the following table:

1. To formulate this assignment problem , answer the following three questions.

a. What are the decisions to be made? For this problem, we need Excel to find out which person to assign to which task (Yes=1, No=0). For example, if we assign Person 1 to Task 1, cell C10 equals 1. If not, cell C10 equals 0.

b. What are the constraints on these decisions? Each person can only do one task (Supply=1). Each task only needs one person (Demand=1).

c. What is the overall measure of performance for these decisions? The overall measure of performance is the total cost of the assignment, so the objective is to minimize this quantity.

2. To make the model easier to understand, create the following named ranges .

3. Insert the following functions.

Explanation: The SUM functions calculate the number of tasks assigned to a person and the number of persons assigned to a task. Total Cost equals the sumproduct of Cost and Assignment.

Trial and Error

With this formulation, it becomes easy to analyze any trial solution.

For example, if we assign Person 1 to Task 1, Person 2 to task 2 and Person 3 to Task 3, Tasks Assigned equals Supply and Persons Assigned equals Demand. This solution has a total cost of 147.

It is not necessary to use trial and error. We shall describe next how the Excel Solver can be used to quickly find the optimal solution.

Solve the Model

To find the optimal solution, execute the following steps.

1. On the Data tab, in the Analyze group, click Solver.

Note: can't find the Solver button? Click here to load the Solver add-in .

Enter the solver parameters (read on). The result should be consistent with the picture below.

You have the choice of typing the range names or clicking on the cells in the spreadsheet.

2. Enter TotalCost for the Objective.

3. Click Min.

4. Enter Assignment for the Changing Variable Cells.

5. Click Add to enter the following constraint.

Note: binary variables are either 0 or 1.

6. Click Add to enter the following constraint.

7. Click Add to enter the following constraint.

8. Check 'Make Unconstrained Variables Non-Negative' and select 'Simplex LP'.

9. Finally, click Solve.

The optimal solution:

Conclusion: it is optimal to assign Person 1 to task 2, Person 2 to Task 3 and Person 3 to Task 1. This solution gives the minimum cost of 129. All constraints are satisfied.

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Google OR-TOOLS VRP Problem with previous OR-TOOLS Assignment problem

Im trying to solve a 2 step problem, where in the first one I run an assignment model which calculates the best option that optimize the pick up and deliveries arcs between nodes because not all the vehicles can transport the same products and other complications the problem has. The result of the first model are the arcs that serves as an input in the second VRP model as data['pickups_deliveries']. The next code is an easy example where the code works but a node cant be a delivery and also a pickup node at the same time. Which is what i need to solve.

This code works fine for simple graph assignment where each pickup node is just a pickup node and each delivery node is just a delivery node . But if a want a node to be pickup and delivery, I thought i can add this as another graph, for example, making node 14, former delivery node, also a pickup node for the arc[14,13]. I thought i could force one vehicle to go 16->14->13->12 by adding this to the data['pickups_deliveries'] but python collapse and stops working.

Mainly what I want to do is be able to add graphs where in one a node can be a pickup node and in another the same node can be a delivery one. Thanks and sorry for the extension.

You must duplicate the node and adapt your transit callback accordingly.

Then you could merge node id when post processing the solution assignment.

Another way is to hack the transit callback to do the mapping there so you have to recompute a new transit matrix. e.g. create a duplicate node 17 and 18 for node 13 and 14. so you can add the new P&D pair [18, 17]

in your transit callback:

and also change

• Thank you!! It really help. The problem that i have now is that one vehicle goes 16->14->13 and another that goes 13->12, and i was wondering if it is possible to force the complete route dependency 16->14->13->12 for one vehicle. In the business case i need to make a pickup on the delivery points 14 and 13 and that pick up can only be deliver to node 13, and 12 that have to be de immediate next nodes . I thought i could solve this by adding the vehicle capacity constrain=1 it but it didnt work. Thank you again –  Alamond13 Commented Aug 13, 2021 at 19:08

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Assignment with Teams of Workers

There are many versions of the assignment problem, which have additional constraints on the workers or tasks. In the next example, six workers are divided into two teams, and each team can perform at most two tasks.

The following sections present a Python program that solves this problem using the CP-SAT or the MIP solver. For a solution using the min cost flow solver, see the section Assignment with teams .

CP-SAT solution

First, let's take a look at the CP-SAT solution to the problem.

Import the libraries

The following code imports the required library.

Define the data

The following code creates the data for the program.

Create the model

The following code creates the model.

Create the variables

The following code creates an array of variables for the problem.

There is one variable for each pair of a worker and task. Note that the workers are numbered 0 - 5 , while the tasks are numbered 0 - 3 , unlike in the original example , in which all nodes had to be numbered differently, as required by the min cost flow solver.

Add the constraints

The following code creates the constraints for the program.

Create the objective

The following code creates the objective function.

The value of the objective function is the total cost over all variables that are assigned the value 1 by the solver.

Invoke the solver

The following code invokes the solver and displays the results.

Display the results

Now, we can print the solution.

Here's the output of the program.

The entire program

Here is the entire program.

MIP solution

Next, we describe a solution to this assignment problem using the MIP solver.

Declare the solver

The following code creates the solver.

Here is the output of the program.

Except as otherwise noted, the content of this page is licensed under the Creative Commons Attribution 4.0 License , and code samples are licensed under the Apache 2.0 License . For details, see the Google Developers Site Policies . Java is a registered trademark of Oracle and/or its affiliates.

Last updated 2023-01-02 UTC.

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Our three favorite things about the new Copilot+ PCs from Microsoft Surface: 1 – Designed for your everyday work and play

• Power through your day without a worry. The new Surface Laptop and Surface Pro are more powerful than ever with Snapdragon X Series Processors, providing faster performance and all-day battery life with a powerful Neural Processing Unit (NPU) for all-new AI experiences.
• Sleek design and colors that match your aesthetic. Thoughtfully designed with your everyday in mind, the thin, lightweight and ultraportable devices feature premium finishes. They come in four stunning colors – perfect for any style: classic Black, timeless Platinum, bold Sapphire, and the new and refreshing Dune [i] .
• Brighter, more immersive displays for ultimate viewing. We’re introducing a new OLED with HDR [ii] display to the new Surface Pro for a cinematic experience, and the Surface Laptop has a new HDR touchscreen display with razor-thin bezels. No matter what you watch or view, your content is going to look stunning.
• Everyday AI companion with the Copilot key. The Copilot app is just a click away with the Copilot key – one of the newest additions to Windows 11 keyboards on Copilot+ PCs.

2 – Exclusive AI experiences designed to empower creativity and productivity  

• Express your creativity with Cocreator [iii] . Whether a seasoned artist or new to design, Cocreator simplifies image creation and photo editing with easy text prompts and natural inking using a Slim Pen [iv] on Surface Pro or touch on Surface Laptop. Exclusive to Copilot+ PCs, Cocreator lets you bring your ideas to life, and it works alongside you to iteratively update the image in real time. Cocreator is available in Paint – the app you’ve grown to know and love.
• No matter where you are, Live Captions keeps you better connected [v] . Available on Windows, Live Captions can quickly translate any live or prerecorded audio into English – and in real time. Connecting with friends, family and colleagues just got easier, and you’ll never miss a beat when watching your favorite international movies or TV shows.
• New and enhanced audio and video effects bring new meaning to ”camera ready.” Both device cameras are powered by new features to Windows Studio Effects. Powered by an industry-leading NPU, they help improve lighting, ensure you appear clear and crisp on video, reduce background noise and offer creative filters so you can express yourself on camera. Built to automatically improve video calls, it’s like having a studio ring light and microphone right on your Windows PC! And the Surface Pro’s ultrawide field-of-view camera keeps you, or the whole family, in focus, even as you move around your space.
• Recall (preview) coming soon: For the solo-preneur who has too many working files and emails to maintain organization, Recall helps you quickly find things you have seen on your PC, keeping all documents, images, websites, instant messages, emails and apps right at your fingertips. This experience comes with built-in privacy and security controls.

Learn how to unlock the best of the new AI-powered features on your Copilot+ PC .

3 – The all-new Surface Pro Flex Keyboard [vi] unlocks new levels of flexibility  

Alongside the new Surface Pro, we are introducing the Surface Pro Flex Keyboard , unlocking powerful new levels of flexibility to effortlessly adapt to your work and play routines. Ready to attach to your Pro for the ultimate laptop set-up or detach for more flexibility and to support your creative workflows. It is built with extra carbon fiber layers for stability and has a larger, customizable haptic touchpad. With integrated pen storage, your Slim Pen is secure, charged and ready to go. Accessibility remains core to our approach, so we designed the new Surface Pro Flex Keyboard with a bold keyset option to reduce eye strain and assist people with low vision.

Discover, learn and buy with Microsoft Store

Shopping at Microsoft Store is all about ease and convenience. Whether the new Copilot+ PCs from Microsoft Surface, Copilot Pro, Xbox consoles and games, apps, movies and TV shows, we’ve got you covered. Don’t miss our top deals on your favorite TV shows like Rick & Morty: Seasons 1-7, Buffy The Vampire Slayer Complete Series, Sons of Anarchy: The Complete Box Set and so much more – available for up to 50% off for a limited time .

• Flexible payment options : Find a payment plan that works for you with options like PayPal Pay Later and Citizens Pay Line of Credit [vii] . It’s budgeting made easy.
• Online Trade-in Program : For a limited time, buy a new Copilot+ PC from Microsoft Surface and get extra cash back when you trade in an eligible device.
• Free and fast shipping with 60-day returns : Get your items quickly with 2–3-day shipping at no extra cost or minimum purchase required and enjoy the flexibility of 60-day returns on almost any physical product.
• 60-day price protection : Shop with confidence knowing you have 60 days of price protection from your delivery date. If the price drops or you find a lower price elsewhere, we’ll honor a one-time price adjustment.

You can also bet on Microsoft Store offering lots of great deals throughout the upcoming back-to-school season. Be sure to keep an eye on the deals page !

Available alongside Microsoft Surface today, are brand new Copilot+ PCs from the biggest brands: Acer , ASUS , Dell , HP , Lenovo and Samsung . Learn more from major PC manufacturers or visit leading retailers, including Best Buy .

[i] Colors available on selected models only. Available colors, sizes, finishes and processors may vary by store, market and configuration. 

[ii] HDR requires HDR content and enabling HDR in device settings.

[iii] Microsoft account required.

[iv] Surface Slim Pen sold separately.

[v] Currently supports translation for video and audio subtitles into English from 40+ languages. See  https://aka.ms/copilotpluspcs . 

[vi] Surface Pro Flex Keyboard sold separately.

[vii] With approval of Citizens Pay Line of Credit at 0% APR and 12- or 18-month term. Subject to individual credit approval. See the Citizens Pay Line of Credit Agreement  for full terms and conditions. Citizens Pay Line of Credit Account offered by Citizens Bank, N.A. ​