# Optimizing the Cutting of

Strips of Sheet to

Minimize Waste With

Excel Solver

This is another classic Solver problem with many possible variations. Cutting Stock problems involve cutting large sheets into the optimal number of smaller strips to meet customer orders while minimizing waste. The sheets can represent any type of material that come in a strip that is cut into smaller strips, such as a roll of steel. This problem involves rolls of garment that are cut to smaller sizes to meet individual customer orders.

## The Cutting Stock Problem

**A garment factory produces rolls of garment in the following 2 sizes: rolls that are 75 inches wide and rolls that are 55 inches wide. Customers order garment rolls in the following 3 sizes: rolls that are 36 inches wide, rolls that are 25 inches wide, and rolls that are 13 inches wide.**

**The factory must cut the 75” rolls and 55” rolls in the correct number of 36”, 25”, and 13” rolls to meet individual customer orders while minimizing waste. Waste represents garment that is cut from the original 75” and 55” rolls which is not included in a customer order and is left-over. **

## Excel Solver Problem Solving Steps

**Excel Solver ****Step 1 – Determine the Objective**

In this case, the objective is to minimize the total waste left-over when satisfying each customer order. As seen in Step 3, Cutting Stock problems require listing all of the possible combinations that the original sheets can be cut in order to satisfy the customer’s order.

The amount of waste (leftover material) from each combination is shown. The total amount of waste of all combinations that are cut is also listed. The objective is to determine the correct number of each combination to cut that will meet the customer’s order and minimize the total amount of leftover, wasted garment. The Objective is to minimize the total amount of wasted garment. The cell is which the total waste from each order is calculated is the Objective Cell.

**Excel Solver ****Step 2 – Determine the Decision Variables**

We are trying to determine how many of each combination to cut that will satisfy the customer’s order requirements and also minimize leftover wasted garment. The Decision Variables are the numbers of each type of cutting combination to cut.

**Excel Solver ****Step 3 – Build the Excel Equations That Combine the Objective With All Decision Variables**

The yellow Objective Cell, I16, displays the total amount of leftover garment that will be wasted when a specific group of cutting combinations is applied to meet a customer order. This will be minimized.

The green Decision Variable cells, cells H4 to H14, display the number of each type of combination that will be cut to meet an individual customer order. Following is a complete view of the Excel model leading to the Objective.

*(Click On Image To See a Larger Version)*

The following 2 images present an expanded view for better clarity of the left and right halves of the preceding complete Excel model. The 1^{st} image is the left side of the model and list all possible cutting combinations and the waste left-over from each. The 2^{nd} image shown is the right side of the model with the green Decision Variables and the yellow Objective Cell:

*(Click On Image To See a Larger Version)*

*(Click On Image To See a Larger Version)*

**Excel Solver ****Step 4 – List all Constraints**

*(Click On Image To See a Larger Version)*

The light blue Constraint cells in the model the preceding image link to the light blue user-controlled inputs outside of the model in the following image. The user-controlled inputs represent the customer’s order as follows:

*(Click On Image To See a Larger Version)*

**Excel Solver ****Step 5 – Test the Excel Spreadsheet**

Test the Excel spreadsheet completely before adding information to the Solver dialogue box. Make sure that any changes to Decision Variables produce the correct results in the Objective cell.

**Excel Solver ****Step 6 – Insert All Data into the Solver Dialogue Box**

Input the Objective cell, Decision Variable cell, and all Constraints into the Solver dialogue box as follows:

*(Click On Image To See a Larger Version)*

Note that the Decision Variables are Constrained to be integers because only whole numbers of garment rolls can be cut. The equality Constraints are the links between the number of garment rolls in the customer’s order and the model. Unconstrained variables are made nonnegative to ensure that there can be no negative numbers of rolls cut.

There are still a few more Constraints not shown in the preceding image that are shown as follows by scrolling down the slider bar next to the Constraints:

*(Click On Image To See a Larger Version)*

All equations on the Excel spreadsheet are linear (1^{st} order) so we can use the Simplex LP (Linear Programming) Solver engine for this optimization problem.

Step 3 shows the completed problem with Decision Variables that have been optimized by the Solver to maximize the Objective while staying within the problem’s Constraints.

**Excel Solver ****Answer Report**

**Part 1**

Note:

- The Solver Result

- How long Solver took to solve the problem

- The Solver Engine that was used and the Solver Options settings

- Where the Objective Cell was labeled in the Excel model for its name to appear as it does in Part 1 of the Answer Report

*(Click On Image To See a Larger Version)*

**Part 2**

- Note that the Variable Cells contain the Decision Variables

- Note where the labels for each Decision Variable are placed in the Excel model so that the Decision Variable’s name will appear here in Part 2 of the Answer Report as it does

- Note the type of variable - Either Continuous or Integer (**Integer**, Binary, or Alldifferent)

- Note the Before and After values of each Decision Variable

*(Click On Image To See a Larger Version)*

**Part 3**

- Note how each Constraint is labeled in the Excel model in order for the Constraint’s name to appear here in Part 3 of the Answer Report as it does

- Note which Constraints are binding (had their limits hit) and which aren’t.

- Note how much slack is still available in any Constraint that has not had its limit hit.

- Note any Integer Constraints (Integer, Binary, Alldifferent)

*(Click On Image To See a Larger Version)*

**Excel Master Series Blog Directory**

Statistical Topics and Articles In Each Topic

- Histograms in Excel
- Bar Chart in Excel
- Combinations & Permutations in Excel
- Normal Distribution in Excel
- Overview of the Normal Distribution
- Normal Distribution’s PDF (Probability Density Function) in Excel 2010 and Excel 2013
- Normal Distribution’s CDF (Cumulative Distribution Function) in Excel 2010 and Excel 2013
- Solving Normal Distribution Problems in Excel 2010 and Excel 2013
- Overview of the Standard Normal Distribution in Excel 2010 and Excel 2013
- An Important Difference Between the t and Normal Distribution Graphs
- The Empirical Rule and Chebyshev’s Theorem in Excel – Calculating How Much Data Is a Certain Distance From the Mean
- Demonstrating the Central Limit Theorem In Excel 2010 and Excel 2013 In An Easy-To-Understand Way

- t-Distribution in Excel
- Binomial Distribution in Excel
- z-Tests in Excel
- Overview of Hypothesis Tests Using the Normal Distribution in Excel 2010 and Excel 2013
- One-Sample z-Test in 4 Steps in Excel 2010 and Excel 2013
- 2-Sample Unpooled z-Test in 4 Steps in Excel 2010 and Excel 2013
- Overview of the Paired (Two-Dependent-Sample) z-Test in 4 Steps in Excel 2010 and Excel 2013

- t-Tests in Excel
- Overview of t-Tests: Hypothesis Tests that Use the t-Distribution
- 1-Sample t-Tests in Excel
- 1-Sample t-Test in 4 Steps in Excel 2010 and Excel 2013
- Excel Normality Testing For the 1-Sample t-Test in Excel 2010 and Excel 2013
- 1-Sample t-Test – Effect Size in Excel 2010 and Excel 2013
- 1-Sample t-Test Power With G*Power Utility
- Wilcoxon Signed-Rank Test in 8 Steps As a 1-Sample t-Test Alternative in Excel 2010 and Excel 2013
- Sign Test As a 1-Sample t-Test Alternative in Excel 2010 and Excel 2013

- 2-Independent-Sample Pooled t-Tests in Excel
- 2-Independent-Sample Pooled t-Test in 4 Steps in Excel 2010 and Excel 2013
- Excel Variance Tests: Levene’s, Brown-Forsythe, and F Test For 2-Sample Pooled t-Test in Excel 2010 and Excel 2013
- Excel Normality Tests Kolmogorov-Smirnov, Anderson-Darling, and Shapiro Wilk Tests For Two-Sample Pooled t-Test
- Two-Independent-Sample Pooled t-Test - All Excel Calculations
- 2- Sample Pooled t-Test – Effect Size in Excel 2010 and Excel 2013
- 2-Sample Pooled t-Test Power With G*Power Utility
- Mann-Whitney U Test in 12 Steps in Excel as 2-Sample Pooled t-Test Nonparametric Alternative in Excel 2010 and Excel 2013
- 2- Sample Pooled t-Test = Single-Factor ANOVA With 2 Sample Groups

- 2-Independent-Sample Unpooled t-Tests in Excel
- 2-Independent-Sample Unpooled t-Test in 4 Steps in Excel 2010 and Excel 2013
- Variance Tests: Levene’s Test, Brown-Forsythe Test, and F-Test in Excel For 2-Sample Unpooled t-Test
- Excel Normality Tests Kolmogorov-Smirnov, Anderson-Darling, and Shapiro-Wilk For 2-Sample Unpooled t-Test
- 2-Sample Unpooled t-Test Excel Calculations, Formulas, and Tools
- Effect Size for a 2-Independent-Sample Unpooled t-Test in Excel 2010 and Excel 2013
- Test Power of a 2-Independent Sample Unpooled t-Test With G-Power Utility

- Paired (2-Sample Dependent) t-Tests in Excel
- Paired t-Test in 4 Steps in Excel 2010 and Excel 2013
- Excel Normality Testing of Paired t-Test Data
- Paired t-Test Excel Calculations, Formulas, and Tools
- Paired t-Test – Effect Size in Excel 2010, and Excel 2013
- Paired t-Test – Test Power With G-Power Utility
- Wilcoxon Signed-Rank Test in 8 Steps As a Paired t-Test Alternative
- Sign Test in Excel As A Paired t-Test Alternative

- Hypothesis Tests of Proportion in Excel
- Hypothesis Tests of Proportion Overview (Hypothesis Testing On Binomial Data)
- 1-Sample Hypothesis Test of Proportion in 4 Steps in Excel 2010 and Excel 2013
- 2-Sample Pooled Hypothesis Test of Proportion in 4 Steps in Excel 2010 and Excel 2013
- How To Build a Much More Useful Split-Tester in Excel Than Google's Website Optimizer

- Chi-Square Independence Tests in Excel
- Chi-Square Goodness-Of-Fit Tests in Excel
- F Tests in Excel
- Correlation in Excel
- Pearson Correlation in Excel
- Spearman Correlation in Excel
- Confidence Intervals in Excel
- z-Based Confidence Intervals of a Population Mean in 2 Steps in Excel 2010 and Excel 2013
- t-Based Confidence Intervals of a Population Mean in 2 Steps in Excel 2010 and Excel 2013
- Minimum Sample Size to Limit the Size of a Confidence interval of a Population Mean
- Confidence Interval of Population Proportion in 2 Steps in Excel 2010 and Excel 2013
- Min Sample Size of Confidence Interval of Proportion in Excel 2010 and Excel 2013

- Simple Linear Regression in Excel
- Overview of Simple Linear Regression in Excel 2010 and Excel 2013
- Complete Simple Linear Regression Example in 7 Steps in Excel 2010 and Excel 2013
- Residual Evaluation For Simple Regression in 8 Steps in Excel 2010 and Excel 2013
- Residual Normality Tests in Excel – Kolmogorov-Smirnov Test, Anderson-Darling Test, and Shapiro-Wilk Test For Simple Linear Regression
- Evaluation of Simple Regression Output For Excel 2010 and Excel 2013
- All Calculations Performed By the Simple Regression Data Analysis Tool in Excel 2010 and Excel 2013
- Prediction Interval of Simple Regression in Excel 2010 and Excel 2013

- Multiple Linear Regression in Excel
- Basics of Multiple Regression in Excel 2010 and Excel 2013
- Complete Multiple Linear Regression Example in 6 Steps in Excel 2010 and Excel 2013
- Multiple Linear Regression’s Required Residual Assumptions
- Normality Testing of Residuals in Excel 2010 and Excel 2013
- Evaluating the Excel Output of Multiple Regression
- Estimating the Prediction Interval of Multiple Regression in Excel
- Regression - How To Do Conjoint Analysis Using Dummy Variable Regression in Excel

- Logistic Regression in Excel
- Logistic Regression Overview
- Logistic Regression in 6 Steps in Excel 2010 and Excel 2013
- R Square For Logistic Regression Overview
- Excel R Square Tests: Nagelkerke, Cox and Snell, and Log-Linear Ratio in Excel 2010 and Excel 2013
- Likelihood Ratio Is Better Than Wald Statistic To Determine if the Variable Coefficients Are Significant For Excel 2010 and Excel 2013
- Excel Classification Table: Logistic Regression’s Percentage Correct of Predicted Results in Excel 2010 and Excel 2013
- Hosmer- Lemeshow Test in Excel – Logistic Regression Goodness-of-Fit Test in Excel 2010 and Excel 2013

- Single-Factor ANOVA in Excel
- Overview of Single-Factor ANOVA
- Single-Factor ANOVA in 5 Steps in Excel 2010 and Excel 2013
- Shapiro-Wilk Normality Test in Excel For Each Single-Factor ANOVA Sample Group
- Kruskal-Wallis Test Alternative For Single Factor ANOVA in 7 Steps in Excel 2010 and Excel 2013
- Levene’s and Brown-Forsythe Tests in Excel For Single-Factor ANOVA Sample Group Variance Comparison
- Single-Factor ANOVA - All Excel Calculations
- Overview of Post-Hoc Testing For Single-Factor ANOVA
- Tukey-Kramer Post-Hoc Test in Excel For Single-Factor ANOVA
- Games-Howell Post-Hoc Test in Excel For Single-Factor ANOVA
- Overview of Effect Size For Single-Factor ANOVA
- ANOVA Effect Size Calculation Eta Squared in Excel 2010 and Excel 2013
- ANOVA Effect Size Calculation Psi – RMSSE – in Excel 2010 and Excel 2013
- ANOVA Effect Size Calculation Omega Squared in Excel 2010 and Excel 2013
- Power of Single-Factor ANOVA Test Using Free Utility G*Power
- Welch’s ANOVA Test in 8 Steps in Excel Substitute For Single-Factor ANOVA When Sample Variances Are Not Similar
- Brown-Forsythe F-Test in 4 Steps in Excel Substitute For Single-Factor ANOVA When Sample Variances Are Not Similar

- Two-Factor ANOVA With Replication in Excel
- Two-Factor ANOVA With Replication in 5 Steps in Excel 2010 and Excel 2013
- Variance Tests: Levene’s and Brown-Forsythe For 2-Factor ANOVA in Excel 2010 and Excel 2013
- Shapiro-Wilk Normality Test in Excel For 2-Factor ANOVA With Replication
- 2-Factor ANOVA With Replication Effect Size in Excel 2010 and Excel 2013
- Excel Post Hoc Tukey’s HSD Test For 2-Factor ANOVA With Replication
- 2-Factor ANOVA With Replication – Test Power With G-Power Utility
- Scheirer-Ray-Hare Test Alternative For 2-Factor ANOVA With Replication

- Two-Factor ANOVA Without Replication in Excel
- Randomized Block Design ANOVA in Excel
- Repeated-Measures ANOVA in Excel
- Single-Factor Repeated-Measures ANOVA in 4 Steps in Excel 2010 and Excel 2013
- Sphericity Testing in 9 Steps For Repeated Measures ANOVA in Excel 2010 and Excel 2013
- Effect Size For Repeated-Measures ANOVA in Excel 2010 and Excel 2013
- Friedman Test in 3 Steps For Repeated-Measures ANOVA in Excel 2010 and Excel 2013

- ANCOVA in Excel
- Normality Testing in Excel
- Creating a Box Plot in 8 Steps in Excel
- Creating a Normal Probability Plot With Adjustable Confidence Interval Bands in 9 Steps in Excel With Formulas and a Bar Chart
- Chi-Square Goodness-of-Fit Test For Normality in 9 Steps in Excel
- Kolmogorov-Smirnov, Anderson-Darling, and Shapiro-Wilk Normality Tests in Excel

- Nonparametric Testing in Excel
- Mann-Whitney U Test in 12 Steps in Excel
- Wilcoxon Signed-Rank Test in 8 Steps in Excel
- Sign Test in Excel
- Friedman Test in 3 Steps in Excel
- Scheirer-Ray-Hope Test in Excel
- Welch's ANOVA Test in 8 Steps Test in Excel
- Brown-Forsythe F Test in 4 Steps Test in Excel
- Levene's Test and Brown-Forsythe Variance Tests in Excel
- Chi-Square Independence Test in 7 Steps in Excel
- Chi-Square Goodness-of-Fit Tests in Excel
- Chi-Square Population Variance Test in Excel

- Post Hoc Testing in Excel
- Creating Interactive Graphs of Statistical Distributions in Excel
- Interactive Statistical Distribution Graph in Excel 2010 and Excel 2013
- Interactive Graph of the Normal Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the Chi-Square Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the t-Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the t-Distribution’s PDF in Excel 2010 and Excel 2013
- Interactive Graph of the t-Distribution’s CDF in Excel 2010 and Excel 2013
- Interactive Graph of the Binomial Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the Exponential Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the Beta Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the Gamma Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the Poisson Distribution in Excel 2010 and Excel 2013

- Solving Problems With Other Distributions in Excel
- Solving Uniform Distribution Problems in Excel 2010 and Excel 2013
- Solving Multinomial Distribution Problems in Excel 2010 and Excel 2013
- Solving Exponential Distribution Problems in Excel 2010 and Excel 2013
- Solving Beta Distribution Problems in Excel 2010 and Excel 2013
- Solving Gamma Distribution Problems in Excel 2010 and Excel 2013
- Solving Poisson Distribution Problems in Excel 2010 and Excel 2013

- Optimization With Excel Solver
- Maximizing Lead Generation With Excel Solver
- Minimizing Cutting Stock Waste With Excel Solver
- Optimal Investment Selection With Excel Solver
- Minimizing the Total Cost of Shipping From Multiple Points To Multiple Points With Excel Solver
- Knapsack Loading Problem in Excel Solver – Optimizing the Loading of a Limited Compartment
- Optimizing a Bond Portfolio With Excel Solver
- Travelling Salesman Problem in Excel Solver – Finding the Shortest Path To Reach All Customers

- Chi-Square Population Variance Test in Excel
- Analyzing Data With Pivot Tables and Pivot Charts
- SEO Functions in Excel
- Time Series Analysis in Excel
- VLOOKUP
- Simplifying Useful Excel Functions

Wonderful article. Thanks for sharing such a useful post.

ReplyDeletenon woven fabric bag making machine

Hi, can you please share the excel sheet of this solver?

ReplyDeleteHow are the values in H4 to H14 (green cells) determined? Is it a formula or an algorithm or just guessing?

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ReplyDelete