# Optimization of Production Scheduling System

Umeh Maryrose, Sunday P Aguh, Nwakoby John Oguejiofor

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Umeh Maryrose, Sunday P Aguh, Nwakoby John Oguejiofor. Optimization of Production Scheduling System. Journal of Scientific and Engineering Research, 2017, 4 (9). hal-05107914

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Research Article

ISSN: 2394-2630

CODEN(USA): JSERBR

# Optimization of Production Scheduling System

Umeh Maryrose N.<fnref n="1" />, Aguh Sunday P.<fnref n="2" />, Nwakoby John Oguejiofor<fnref n="2" />

**Abstract** In manufacturing industries production scheduling is always a key to the company's economic growth and profitability. It defines the appropriate timing system for production. It also establishes just-in-time system of the company under study. In this research work, the researcher makes use of 36x75x40 size of foam product which was analyzed using response surface, integer programming and linear programming optimization tools to optimize the production output of the product. The result shows that maximum production output of the product using response surface, integer programming and linear programming were 1854 units, 2160.0 units and 2172.7 units respectively over any given monthly production. The results were recommended to the case company for optimum use in scheduling their monthly production output.

**Keywords** Scheduling, response surface, integer programming, linear programming optimization, Production and regression

## Introduction

Manufacturing is very critical to economic growth, prosperity and a higher standard of living. It is a catalyst for industrial and economic development. Its satisfaction of individual, communities and nations by manufacturing things in workshops by utilizing men, materials, machines, money and methods [1].

Essentially, manufacturing can be simply defined as value addition processes by which raw materials of low utility and value to its inadequate material properties and poor irregular size, shape and finish are converted into high utility and valued product with definite dimensions, forms, and finish imparting some functional ability by utilizing resources [2]. The resources could be people, machines, computers and/or organized integration of one or more of the above mentioned [3]. To realize higher efficiency, there must be optimal allocation of these resources to activities (scheduling).

Critical areas like cost, time, quality, and flexibility need to be optimized.

Optimization is finding an alternative with the most cost effective or highest achievable performance under the given constraints, by maximizing desired factors and minimizing undesired ones. One of the tools of optimization is scheduling.

Scheduling is the process of arranging, controlling and optimizing work and workloads in a production or manufacturing process. It is used to allocate plant and machinery resources, plan human resources, plan production processes and purchase materials [4].

It is an important tool for production, engineering and in sciences, where it can have a major impact on the productivity of a process. In manufacturing, the purpose of scheduling is to minimize the production time and costs, by telling a production facility when to make, with which staff, and on which equipment. Production scheduling aims to maximize the efficiency of the operation and reduce costs [5].

Wilson (2000a) provides an overview of manufacturing management and notes how modern manufacturing organizations developed from the mills and workshops and projects of the past. Unfortunately, neither of these

<figure><img src="image_3.png" /></figure>

<footnote n="1">¹Department of Computer Engineering, Nnamdi Azikiwe University, Awka, Anambra State, Nigeria</footnote>

<footnote n="2">²Industrial and Production Engineering Department, Nnamdi Azikiwe University, Awka, Anambra State, Nigeria</footnote>
excellent sources discusses the scheduling function in detail [6]. Hopp and Spearman (1996) also provide a general overview of manufacturing in America since the First Industrial Revolution [7]. McKay (2003) provides a historical overview of the key concepts behind the practices that manufacturing firms have adopted in modern times, highlighting, for instance, how the ideas of just-in-time (though not the term) were well-known in the early twentieth century [8].

According to Wight (1984), the two key problems in production scheduling are, "priorities" and "capacity." Wight defines scheduling as "establishing the timing for performing a task" and observes that, in manufacturing, there are multiple types of scheduling, including the detailed scheduling of a shop order that shows when each operation must start and complete [9].

Cox et al. (1992) also define detailed scheduling as "the actual assignment of starting and/or completion dates to operations or groups of operations to show when these must be done if the manufacturing order is to be completed on time." They note that this is also known as operations scheduling, order scheduling, and shop scheduling which this research is concerned about [10]. The computer based scheduling can help manufacturers improve on time delivery, respond quickly to customer orders and create realistic schedules, but success requires using finite scheduling techniques and integrating them with other manufacturing planning systems [11]. This research investigate the minimization of the makespan via scheduling

The aim of the study is to develop an optimal time scheduling system that will be more suitable in foam manufacturing Industry.

The analysis of the research work were based on the case company data collected over a given period of three years. The data was analyzed and optimized using integer programming model, linear programming model and response surface optimization model. The models were applied to the data in other to obtain the maximum quantity and maximum time scheduling system in the production industry. However, the data collected is a size of foam produced in the case company. Product F is 36X75X40 size of foam produced.

## Response surface optimization for scheduling of product F

### Response Surface Regression: Yield versus W1, W2, W3, W4, W5, W6, W7, W8

<table><caption>Table 1: Estimated Regression Coefficients for Yield</caption><thead><tr><th>Term</th><th>Coef</th><th>SE Coef</th><th>T P</th></tr></thead><tbody><tr><td>Constant</td><td>5163.14</td><td>0.000000</td><td>*</td></tr><tr><td>W1</td><td>478.94</td><td>0.000000</td><td>*</td></tr><tr><td>W2</td><td>390.83</td><td>0.000000</td><td>*</td></tr><tr><td>W3</td><td>615.71</td><td>0.000000</td><td>*</td></tr><tr><td>W4</td><td>575.61</td><td>0.000000</td><td>*</td></tr><tr><td>W5</td><td>278.77</td><td>0.000000</td><td>*</td></tr><tr><td>W6</td><td>345.19</td><td>0.000000</td><td>*</td></tr><tr><td>W7</td><td>315.31</td><td>0.000000</td><td>*</td></tr><tr><td>W8</td><td>308.70</td><td>0.000000</td><td>*</td></tr><tr><td>W1*W1</td><td>0.00</td><td>0.000000</td><td>*</td></tr><tr><td>W2*W2</td><td>0.00</td><td>0.000000</td><td>*</td></tr><tr><td>W3*W3</td><td>0.00</td><td>0.000000</td><td>*</td></tr><tr><td>W4*W4</td><td>-0.00</td><td>0.000000</td><td>*</td></tr><tr><td>W5*W5</td><td>-0.00</td><td>0.000000</td><td>*</td></tr><tr><td>W6*W6</td><td>-0.00</td><td>0.000000</td><td>*</td></tr><tr><td>W7*W7</td><td>-0.00</td><td>0.000000</td><td>*</td></tr><tr><td>W8*W8</td><td>0.00</td><td>0.000000</td><td>*</td></tr><tr><td>W1*W2</td><td>-0.00</td><td>0.000000</td><td>*</td></tr><tr><td>W1*W3</td><td>0.00</td><td>0.000000</td><td>*</td></tr><tr><td>W1*W4</td><td>-0.00</td><td>0.000000</td><td>*</td></tr><tr><td>W1*W5</td><td>0.00</td><td>0.000000</td><td>*</td></tr><tr><td>W1*W6</td><td>-0.00</td><td>0.000000</td><td>*</td></tr><tr><td>W1*W7</td><td>0.00</td><td>0.000000</td><td>*</td></tr><tr><td>W1*W8</td><td>-0.00</td><td>0.000000</td><td>*</td></tr><tr><td>W2*W3</td><td>-0.00</td><td>0.000000</td><td>*</td></tr></tbody></table>

<figure><img src="image_4.png" /></figure>
<table><caption>Table 2: Analysis of Variance for Yield</caption><tr><td>W2*W4</td><td>0.00</td><td>0.000000</td><td>*</td><td>*</td></tr><tr><td colspan="5">R-Sq = 100.00% R-Sq(adj) = 100.00%</td></tr></table>

<table><caption>Table 3: Residuals in Analysis of Variance</caption><thead><tr><th>Source</th><th>DF</th><th>Seq SS</th><th>Adj SS</th><th>Adj MS</th><th>F P</th></tr></thead><tbody><tr><td>Regression</td><td>25</td><td>33669948</td><td>33669948</td><td>1346798</td><td>*</td></tr><tr><td>Linear</td><td>8</td><td>33669948</td><td>2751180</td><td>343898</td><td>*</td></tr><tr><td>W1</td><td>1</td><td>4614041</td><td>143</td><td>143</td><td>*</td></tr><tr><td>W2</td><td>1</td><td>15162256</td><td>140</td><td>140</td><td>*</td></tr><tr><td>W3</td><td>1</td><td>6978921</td><td>84</td><td>84</td><td>*</td></tr><tr><td>W4</td><td>1</td><td>492189</td><td>67</td><td>67</td><td>*</td></tr><tr><td>W5</td><td>1</td><td>4722590</td><td>17</td><td>17</td><td>*</td></tr><tr><td>W6</td><td>1</td><td>1145438</td><td>26</td><td>26</td><td>*</td></tr><tr><td>W7</td><td>1</td><td>465969</td><td>31</td><td>31</td><td>*</td></tr><tr><td>W8</td><td>1</td><td>88545</td><td>6</td><td>6</td><td>*</td></tr><tr><td>Square</td><td>8</td><td>0</td><td>0</td><td>0</td><td>*</td></tr><tr><td>W1*W1</td><td>1</td><td>0</td><td>0</td><td>0</td><td>*</td></tr><tr><td>W2*W2</td><td>1</td><td>0</td><td>0</td><td>0</td><td>*</td></tr><tr><td>W3*W3</td><td>1</td><td>0</td><td>0</td><td>0</td><td>*</td></tr><tr><td>W4*W4</td><td>1</td><td>0</td><td>0</td><td>0</td><td>*</td></tr><tr><td>W5*W5</td><td>1</td><td>0</td><td>0</td><td>0</td><td>*</td></tr><tr><td>W6*W6</td><td>1</td><td>0</td><td>0</td><td>0</td><td>*</td></tr><tr><td>W7*W7</td><td>1</td><td>0</td><td>0</td><td>0</td><td>*</td></tr><tr><td>W8*W8</td><td>1</td><td>0</td><td>0</td><td>0</td><td>*</td></tr><tr><td>Interaction</td><td>9</td><td>0</td><td>0</td><td>0</td><td>*</td></tr><tr><td>W1*W2</td><td>1</td><td>0</td><td>0</td><td>0</td><td>*</td></tr><tr><td>W1*W3</td><td>1</td><td>0</td><td>0</td><td>0</td><td>*</td></tr><tr><td>W1*W4</td><td>1</td><td>0</td><td>0</td><td>0</td><td>*</td></tr><tr><td>W1*W5</td><td>1</td><td>0</td><td>0</td><td>0</td><td>*</td></tr><tr><td>W1*W6</td><td>1</td><td>0</td><td>0</td><td>0</td><td>*</td></tr><tr><td>W1*W7</td><td>1</td><td>0</td><td>0</td><td>0</td><td>*</td></tr><tr><td>W1*W8</td><td>1</td><td>0</td><td>0</td><td>0</td><td>*</td></tr><tr><td>W2*W3</td><td>1</td><td>0</td><td>0</td><td>0</td><td>*</td></tr><tr><td>W2*W4</td><td>1</td><td>0</td><td>0</td><td>0</td><td>*</td></tr><tr><td>Residual Error</td><td>10</td><td>0</td><td>0</td><td>0</td><td></td></tr><tr><td>Pure Error</td><td>10</td><td>0</td><td>0</td><td>0</td><td></td></tr><tr><td>Total</td><td>35</td><td>33669948</td><td></td><td></td><td></td></tr></tbody></table>

<table><thead><tr><th>Obs</th><th>StdOrder</th><th>Yield</th><th>Fit</th><th>SE Fit</th><th>Residual</th></tr></thead><tbody><tr><td>1</td><td>1</td><td>6643.290</td><td>6643.290</td><td>0.000</td><td>0.000</td></tr><tr><td>2</td><td>2</td><td>6745.690</td><td>6745.690</td><td>0.000</td><td>0.000</td></tr><tr><td>3</td><td>3</td><td>6061.110</td><td>6061.110</td><td>0.000</td><td>0.000</td></tr><tr><td>4</td><td>4</td><td>4749.040</td><td>4749.040</td><td>0.000</td><td>-0.000</td></tr><tr><td>5</td><td>5</td><td>4900.190</td><td>4900.190</td><td>0.000</td><td>0.000</td></tr><tr><td>6</td><td>6</td><td>3151.770</td><td>3151.770</td><td>0.000</td><td>-0.000</td></tr><tr><td>7</td><td>7</td><td>6164.920</td><td>6164.920</td><td>0.000</td><td>0.000</td></tr><tr><td>8</td><td>8</td><td>6717.970</td><td>6717.970</td><td>0.000</td><td>-0.000</td></tr><tr><td>9</td><td>9</td><td>8350.040</td><td>8350.040</td><td>0.000</td><td>0.000</td></tr><tr><td>10</td><td>10</td><td>6756.480</td><td>6756.480</td><td>0.000</td><td>0.000</td></tr><tr><td>11</td><td>11</td><td>5418.720</td><td>5418.720</td><td>0.000</td><td>0.000</td></tr><tr><td>12</td><td>12</td><td>7336.500</td><td>7336.500</td><td>0.000</td><td>-0.000</td></tr><tr><td>13</td><td>13</td><td>4861.340</td><td>4861.340</td><td>0.000</td><td>0.000</td></tr><tr><td>14</td><td>14</td><td>4850.710</td><td>4850.710</td><td>0.000</td><td>0.000</td></tr><tr><td>15</td><td>15</td><td>6300.980</td><td>6300.980</td><td>0.000</td><td>-0.000</td></tr><tr><td>16</td><td>16</td><td>5565.220</td><td>5565.220</td><td>0.000</td><td>0.000</td></tr><tr><td>17</td><td>17</td><td>5100.540</td><td>5100.540</td><td>0.000</td><td>0.000</td></tr></tbody></table>

<figure><img src="image_5.png" /></figure>
<table><caption>Table 4: Predicted Response for New Design Points Using Model for Yield</caption><tr><td>18</td><td>18</td><td>5954.800</td><td>5954.800</td><td>0.000</td><td>0.000</td></tr><tr><td>19</td><td>19</td><td>6616.660</td><td>6616.660</td><td>0.000</td><td>0.000</td></tr><tr><td>20</td><td>20</td><td>6198.640</td><td>6198.640</td><td>0.000</td><td>0.000</td></tr><tr><td>21</td><td>21</td><td>4868.060</td><td>4868.060</td><td>0.000</td><td>0.000</td></tr><tr><td>22</td><td>22</td><td>5129.650</td><td>5129.650</td><td>0.000</td><td>0.000</td></tr><tr><td>23</td><td>23</td><td>6357.060</td><td>6357.060</td><td>0.000</td><td>0.000</td></tr><tr><td>24</td><td>24</td><td>4858.080</td><td>4858.080</td><td>0.000</td><td>-0.000</td></tr><tr><td>25</td><td>25</td><td>4861.340</td><td>4861.340</td><td>0.000</td><td>0.000</td></tr><tr><td>26</td><td>26</td><td>4850.710</td><td>4850.710</td><td>0.000</td><td>0.000</td></tr><tr><td>27</td><td>27</td><td>6300.980</td><td>6300.980</td><td>0.000</td><td>-0.000</td></tr><tr><td>28</td><td>28</td><td>5565.220</td><td>5565.220</td><td>0.000</td><td>0.000</td></tr><tr><td>29</td><td>29</td><td>6057.420</td><td>6057.420</td><td>0.000</td><td>0.000</td></tr><tr><td>30</td><td>30</td><td>4838.560</td><td>4838.560</td><td>0.000</td><td>0.000</td></tr><tr><td>31</td><td>31</td><td>6616.660</td><td>6616.660</td><td>0.000</td><td>0.000</td></tr><tr><td>32</td><td>32</td><td>6198.640</td><td>6198.640</td><td>0.000</td><td>0.000</td></tr><tr><td>33</td><td>33</td><td>4868.060</td><td>4868.060</td><td>0.000</td><td>0.000</td></tr><tr><td>34</td><td>34</td><td>5129.650</td><td>5129.650</td><td>0.000</td><td>0.000</td></tr><tr><td>35</td><td>35</td><td>6357.060</td><td>6357.060</td><td>0.000</td><td>0.000</td></tr><tr><td>36</td><td>36</td><td>4858.080</td><td>4858.080</td><td>0.000</td><td>-0.000</td></tr></table>

<table><thead><tr><th>Point</th><th>Fit</th><th>SE Fit</th><th>95% CI</th><th>95% PI</th></tr></thead><tbody><tr><td>1</td><td>6643.29</td><td>0</td><td>(6643.29, 6643.29)</td><td>(6643.29, 6643.29)</td></tr><tr><td>2</td><td>6745.69</td><td>0</td><td>(6745.69, 6745.69)</td><td>(6745.69, 6745.69)</td></tr><tr><td>3</td><td>6061.11</td><td>0</td><td>(6061.11, 6061.11)</td><td>(6061.11, 6061.11)</td></tr><tr><td>4</td><td>4749.04</td><td>0</td><td>(4749.04, 4749.04)</td><td>(4749.04, 4749.04)</td></tr><tr><td>5</td><td>4900.19</td><td>0</td><td>(4900.19, 4900.19)</td><td>(4900.19, 4900.19)</td></tr><tr><td>6</td><td>3151.77</td><td>0</td><td>(3151.77, 3151.77)</td><td>(3151.77, 3151.77)</td></tr><tr><td>7</td><td>6164.92</td><td>0</td><td>(6164.92, 6164.92)</td><td>(6164.92, 6164.92)</td></tr><tr><td>8</td><td>6717.97</td><td>0</td><td>(6717.97, 6717.97)</td><td>(6717.97, 6717.97)</td></tr><tr><td>9</td><td>8350.04</td><td>0</td><td>(8350.04, 8350.04)</td><td>(8350.04, 8350.04)</td></tr><tr><td>10</td><td>6756.48</td><td>0</td><td>(6756.48, 6756.48)</td><td>(6756.48, 6756.48)</td></tr><tr><td>11</td><td>5418.72</td><td>0</td><td>(5418.72, 5418.72)</td><td>(5418.72, 5418.72)</td></tr><tr><td>12</td><td>7336.50</td><td>0</td><td>(7336.50, 7336.50)</td><td>(7336.50, 7336.50)</td></tr><tr><td>13</td><td>4861.34</td><td>0</td><td>(4861.34, 4861.34)</td><td>(4861.34, 4861.34)</td></tr><tr><td>14</td><td>4850.71</td><td>0</td><td>(4850.71, 4850.71)</td><td>(4850.71, 4850.71)</td></tr><tr><td>15</td><td>6300.98</td><td>0</td><td>(6300.98, 6300.98)</td><td>(6300.98, 6300.98)</td></tr><tr><td>16</td><td>5565.22</td><td>0</td><td>(5565.22, 5565.22)</td><td>(5565.22, 5565.22)</td></tr><tr><td>17</td><td>5100.54</td><td>0</td><td>(5100.54, 5100.54)</td><td>(5100.54, 5100.54)</td></tr><tr><td>18</td><td>5954.80</td><td>0</td><td>(5954.80, 5954.80)</td><td>(5954.80, 5954.80)</td></tr><tr><td>19</td><td>6616.66</td><td>0</td><td>(6616.66, 6616.66)</td><td>(6616.66, 6616.66)</td></tr><tr><td>20</td><td>6198.64</td><td>0</td><td>(6198.64, 6198.64)</td><td>(6198.64, 6198.64)</td></tr><tr><td>21</td><td>4868.06</td><td>0</td><td>(4868.06, 4868.06)</td><td>(4868.06, 4868.06)</td></tr><tr><td>22</td><td>5129.65</td><td>0</td><td>(5129.65, 5129.65)</td><td>(5129.65, 5129.65)</td></tr><tr><td>23</td><td>6357.06</td><td>0</td><td>(6357.06, 6357.06)</td><td>(6357.06, 6357.06)</td></tr><tr><td>24</td><td>4858.08</td><td>0</td><td>(4858.08, 4858.08)</td><td>(4858.08, 4858.08)</td></tr><tr><td>25</td><td>4861.34</td><td>0</td><td>(4861.34, 4861.34)</td><td>(4861.34, 4861.34)</td></tr><tr><td>26</td><td>4850.71</td><td>0</td><td>(4850.71, 4850.71)</td><td>(4850.71, 4850.71)</td></tr><tr><td>27</td><td>6300.98</td><td>0</td><td>(6300.98, 6300.98)</td><td>(6300.98, 6300.98)</td></tr><tr><td>28</td><td>5565.22</td><td>0</td><td>(5565.22, 5565.22)</td><td>(5565.22, 5565.22)</td></tr><tr><td>29</td><td>6057.42</td><td>0</td><td>(6057.42, 6057.42)</td><td>(6057.42, 6057.42)</td></tr><tr><td>30</td><td>4838.56</td><td>0</td><td>(4838.56, 4838.56)</td><td>(4838.56, 4838.56)</td></tr><tr><td>31</td><td>6616.66</td><td>0</td><td>(6616.66, 6616.66)</td><td>(6616.66, 6616.66)</td></tr><tr><td>32</td><td>6198.64</td><td>0</td><td>(6198.64, 6198.64)</td><td>(6198.64, 6198.64)</td></tr><tr><td>33</td><td>4868.06</td><td>0</td><td>(4868.06, 4868.06)</td><td>(4868.06, 4868.06)</td></tr><tr><td>34</td><td>5129.65</td><td>0</td><td>(5129.65, 5129.65)</td><td>(5129.65, 5129.65)</td></tr><tr><td>35</td><td>6357.06</td><td>0</td><td>(6357.06, 6357.06)</td><td>(6357.06, 6357.06)</td></tr><tr><td>36</td><td>4858.08</td><td>0</td><td>(4858.08, 4858.08)</td><td>(4858.08, 4858.08)</td></tr></tbody></table>

<figure><img src="image_6.png" /></figure>
## Response Optimization

### Parameters

Goal Lower Target Upper Weight Import

Yield Minimum 3450 3450 10000 1 1

### Global Solution

W1 = 20.24

W2 = 0

W3 = 21.08

W4 = 17.64

W5 = 429.52

W6 = 542.81

W7 = 432.18

W8 = 390.6

### Predicted Responses

Yield = 1854.07, desirability = 1.000000

Composite Desirability = 1.000000

<figure><img src="image_7.png" /><figcaption>Figure 1: Optimization Plot for Scheduling Product F</figcaption></figure>

## INTEGER PROGRAMMING B&B OUTPUT SUMMARY

Title: 36.75.40 Product

### FEASIBLE SOLUTION 1:

Objective Value = 1950 Solution found at iteration 12

x1: w1 = 0

x2: w2 = 0 x3: w3 = 0 x4: w4 = 0 x5: w5 = 0 x6: w6 = 1 x7: w7 = 0 x8: w8 = 6

### FEASIBLE SOLUTION 2:

<figure><img src="image_8.png" /></figure>
Objective Value = 1960 Solution found at iteration 73

x1: w1 = 0

x2: w2 = 0 x3: w3 = 0 x4: w4 = 0 x5: w5 = 0 x6: w6 = 0 x7: w7 = 7 x8: w8 = 0

OPTIMAL SOLUTION:

Objective Value = 2160 (MAX)

solution found at iteration 5277 Result verified at iteration 5287

x1: w1 = 1

x2: w2 = 1 x3: w3 = 1 x4: w4 = 1 x5: w5 = 1 x6: w6 = 1 x7: w7 = 1 x8: w8 = 1

LINEAR PROGRAMMING OUTPUT SUMMARY

Title: 36.75.40 Product

Final Iteration No.: 20

Objective Value = 2172.7

<table><thead><tr><th>Variable</th><th>Value</th><th>Obj Coeff</th><th>Obj Val Contrib</th></tr></thead><tbody><tr><td>x1: w1</td><td>1.08</td><td>280.00</td><td>302.68</td></tr><tr><td>x2: w2</td><td>1.04</td><td>280.00</td><td>291.16</td></tr><tr><td>x3: w3</td><td>0.00</td><td>250.00</td><td>0.00</td></tr><tr><td>x4: w4</td><td>1.07</td><td>260.00</td><td>279.39</td></tr><tr><td>x5: w5</td><td>0.94</td><td>260.00</td><td>243.30</td></tr><tr><td>x6: w6</td><td>1.04</td><td>270.00</td><td>280.47</td></tr><tr><td>x7: w7</td><td>0.48</td><td>280.00</td><td>133.79</td></tr><tr><td>x8: w8</td><td>2.29</td><td>280.00</td><td>641.92</td></tr></tbody></table>

<table><thead><tr><th>Constraint</th><th>RHS</th><th>Slack-/Surplus+</th></tr></thead><tbody><tr><td>1 (&lt;)</td><td>1508.000.00</td><td></td></tr><tr><td>2 (&lt;)</td><td>1523.0010.69-</td><td></td></tr><tr><td>3 (&lt;)</td><td>1368.0014.10-</td><td></td></tr><tr><td>4 (&lt;)</td><td>1064.0039.14-</td><td></td></tr><tr><td>5 (&lt;)</td><td>1116.006.27-</td><td></td></tr><tr><td>6 (&lt;)</td><td>1180.000.00</td><td></td></tr><tr><td>7 (&lt;)</td><td>1389.0018.70-</td><td></td></tr><tr><td>8 (&lt;)</td><td>1515.004.38-</td><td></td></tr><tr><td>9 (&lt;)</td><td>1501.0022.12-</td><td></td></tr><tr><td>10 (&lt;)</td><td>1526.003.84-</td><td></td></tr><tr><td>11 (&lt;)</td><td>1511.000.00</td><td></td></tr><tr><td>12 (&lt;)</td><td>1489.0021.52-</td><td></td></tr><tr><td>13 (&lt;)</td><td>1519.0022.33-</td><td></td></tr><tr><td>14 (&lt;)</td><td>1120.0026.81-</td><td></td></tr><tr><td>15 (&lt;)</td><td>1865.0030.48-</td><td></td></tr><tr><td>16 (&lt;)</td><td>1508.005.89-</td><td></td></tr><tr><td>17 (&lt;)</td><td>1273.0023.14-</td><td></td></tr><tr><td>18 (&lt;)</td><td>1049.0033.48-</td><td></td></tr><tr><td>19 (&lt;)</td><td>935.00 58.88-</td><td></td></tr><tr><td>20 (&lt;)</td><td>1312.00109.77-</td><td></td></tr><tr><td>21 (&lt;)</td><td>1147.0075.47-</td><td></td></tr><tr><td>22 (&lt;)</td><td>1102.0064.22-</td><td></td></tr><tr><td>23 (&lt;)</td><td>1363.0021.49-</td><td></td></tr><tr><td>24 (&lt;)</td><td>1040.0019.09-</td><td></td></tr><tr><td>25 (&lt;)</td><td>1140.0058.83-</td><td></td></tr><tr><td>26 (&lt;)</td><td>1096.0043.26-</td><td></td></tr><tr><td>27 (&lt;)</td><td>1423.000.00</td><td></td></tr><tr><td>28 (&lt;)</td><td>1262.0035.76-</td><td></td></tr></tbody></table>

<figure><img src="image_9.png" /></figure>
<table><tr><td>29 (&lt;)</td><td>1378.0039.12-</td></tr><tr><td>30 (&lt;)</td><td>1098.0013.39-</td></tr><tr><td>31 (&lt;)</td><td>1492.0027.49-</td></tr><tr><td>32 (&lt;)</td><td>1407.000.00</td></tr><tr><td>33 (&lt;)</td><td>1091.0010.22-</td></tr><tr><td>34 (&lt;)</td><td>1173.000.00</td></tr><tr><td>35 (&lt;)</td><td>1433.0030.49-</td></tr><tr><td>36 (&lt;)</td><td>1094.000.00</td></tr><tr><td>UB-x1w1</td><td>300.00 298.92-</td></tr><tr><td>UB-x2w2</td><td>300.00 298.96-</td></tr><tr><td>UB-x3w3</td><td>300.00 300.00-</td></tr><tr><td>UB-x4w4</td><td>300.00 298.93-</td></tr><tr><td>UB-x5w5</td><td>300.00 299.06-</td></tr><tr><td>UB-x6w6</td><td>300.00 298.96-</td></tr><tr><td>UB-x7w7</td><td>300.00 299.52-</td></tr><tr><td>UB-x8w8</td><td>300.00 297.71-</td></tr></table>

***Sensitivity Analysis***

<table><thead><tr><th>Variable</th><th>Current</th><th>Obj</th><th>CoeffMin</th><th>Obj</th><th>CoeffMax</th><th>Obj</th><th>CoeffReduced</th><th>Cost</th></tr></thead><tbody><tr><td>x1: w1</td><td>280.00</td><td></td><td>276.95</td><td></td><td>292.07</td><td></td><td>0.00</td><td></td></tr><tr><td>x2: w2</td><td>280.00</td><td></td><td>277.41</td><td></td><td>283.09</td><td></td><td>0.00</td><td></td></tr><tr><td>x3: w3</td><td>250.00</td><td></td><td>-infinity</td><td></td><td>262.70</td><td></td><td>12.70</td><td></td></tr><tr><td>x4: w4</td><td>260.00</td><td></td><td>256.75</td><td></td><td>264.19</td><td></td><td>0.00</td><td></td></tr><tr><td>x5: w5</td><td>260.00</td><td></td><td>258.42</td><td></td><td>262.43</td><td></td><td>0.00</td><td></td></tr><tr><td>x6: w6</td><td>270.00</td><td></td><td>263.21</td><td></td><td>272.19</td><td></td><td>0.00</td><td></td></tr><tr><td>x7: w7</td><td>280.00</td><td></td><td>278.08</td><td></td><td>281.33</td><td></td><td>0.00</td><td></td></tr><tr><td>x8: w8</td><td>280.00</td><td></td><td>278.50</td><td></td><td>281.33</td><td></td><td>0.00</td><td></td></tr><tr><td>Constraint</td><td>Current</td><td>RHS</td><td>Min</td><td>RHS</td><td>Max</td><td>RHS</td><td>Dual</td><td>Price</td></tr><tr><td>1 (&lt;)</td><td>1508.00</td><td></td><td>1503.69</td><td></td><td>1509.93</td><td></td><td>0.81</td><td></td></tr><tr><td>2 (&lt;)</td><td>1523.00</td><td></td><td>1512.31</td><td></td><td>infinity</td><td></td><td>0.00</td><td></td></tr><tr><td>3 (&lt;)</td><td>1368.00</td><td></td><td>1353.90</td><td></td><td>infinity</td><td></td><td>0.00</td><td></td></tr><tr><td>4 (&lt;)</td><td>1064.00</td><td></td><td>1024.86</td><td></td><td>infinity</td><td></td><td>0.00</td><td></td></tr><tr><td>5 (&lt;)</td><td>1116.00</td><td></td><td>1109.73</td><td></td><td>infinity</td><td></td><td>0.00</td><td></td></tr><tr><td>6 (&lt;)</td><td>1180.00</td><td></td><td>1166.88</td><td></td><td>1191.22</td><td></td><td>0.04</td><td></td></tr><tr><td>7 (&lt;)</td><td>1389.00</td><td></td><td>1370.30</td><td></td><td>infinity</td><td></td><td>0.00</td><td></td></tr><tr><td>8 (&lt;)</td><td>1515.00</td><td></td><td>1510.62</td><td></td><td>infinity</td><td></td><td>0.00</td><td></td></tr><tr><td>9 (&lt;)</td><td>1501.00</td><td></td><td>1478.88</td><td></td><td>infinity</td><td></td><td>0.00</td><td></td></tr><tr><td>10 (&lt;)</td><td>1526.00</td><td></td><td>1522.16</td><td></td><td>infinity</td><td></td><td>0.00</td><td></td></tr><tr><td>11 (&lt;)</td><td>1511.00</td><td></td><td>1508.76</td><td></td><td>1514.12</td><td></td><td>0.24</td><td></td></tr><tr><td>12 (&lt;)</td><td>1489.00</td><td></td><td>1467.48</td><td></td><td>infinity</td><td></td><td>0.00</td><td></td></tr><tr><td>13 (&lt;)</td><td>1519.00</td><td></td><td>1496.67</td><td></td><td>infinity</td><td></td><td>0.00</td><td></td></tr><tr><td>14 (&lt;)</td><td>1120.00</td><td></td><td>1093.19</td><td></td><td>infinity</td><td></td><td>0.00</td><td></td></tr><tr><td>15 (&lt;)</td><td>1865.00</td><td></td><td>1834.52</td><td></td><td>infinity</td><td></td><td>0.00</td><td></td></tr><tr><td>16 (&lt;)</td><td>1508.00</td><td></td><td>1502.11</td><td></td><td>infinity</td><td></td><td>0.00</td><td></td></tr><tr><td>17 (&lt;)</td><td>1273.00</td><td></td><td>1249.86</td><td></td><td>infinity</td><td></td><td>0.00</td><td></td></tr><tr><td>18 (&lt;)</td><td>1049.00</td><td></td><td>1015.52</td><td></td><td>infinity</td><td></td><td>0.00</td><td></td></tr><tr><td>19 (&lt;)</td><td>935.00</td><td></td><td>876.12</td><td></td><td>infinity</td><td></td><td>0.00</td><td></td></tr><tr><td>20 (&lt;)</td><td>1312.00</td><td></td><td>1202.23</td><td></td><td>infinity</td><td></td><td>0.00</td><td></td></tr><tr><td>21 (&lt;)</td><td>1147.00</td><td></td><td>1071.53</td><td></td><td>infinity</td><td></td><td>0.00</td><td></td></tr><tr><td>22 (&lt;)</td><td>1102.00</td><td></td><td>1037.78</td><td></td><td>infinity</td><td></td><td>0.00</td><td></td></tr></tbody></table>

<figure><img src="image_10.png" /></figure>
<table><tr><td>23 (&lt;)</td><td>1363.00</td><td>1341.51</td><td>infinity</td><td>0.00</td></tr><tr><td>24 (&lt;)</td><td>1040.00</td><td>1020.91</td><td>infinity</td><td>0.00</td></tr><tr><td>25 (&lt;)</td><td>1140.00</td><td>1081.17</td><td>infinity</td><td>0.00</td></tr><tr><td>26 (&lt;)</td><td>1096.00</td><td>1052.74</td><td>infinity</td><td>0.00</td></tr><tr><td>27 (&lt;)</td><td>1423.00</td><td>1415.35</td><td>1445.62</td><td>0.23</td></tr><tr><td>28 (&lt;)</td><td>1262.00</td><td>1226.24</td><td>infinity</td><td>0.00</td></tr><tr><td>29 (&lt;)</td><td>1378.00</td><td>1338.88</td><td>infinity</td><td>0.00</td></tr><tr><td>30 (&lt;)</td><td>1098.00</td><td>1084.61</td><td>infinity</td><td>0.00</td></tr><tr><td>31 (&lt;)</td><td>1492.00</td><td>1464.51</td><td>infinity</td><td>0.00</td></tr><tr><td>32 (&lt;)</td><td>1407.00</td><td>1400.56</td><td>1412.11</td><td>0.06</td></tr><tr><td>33 (&lt;)</td><td>1091.00</td><td>1080.78</td><td>infinity</td><td>0.00</td></tr><tr><td>34 (&lt;)</td><td>1173.00</td><td>1167.80</td><td>1179.58</td><td>0.04</td></tr><tr><td>35 (&lt;)</td><td>1433.00</td><td>1402.51</td><td>infinity</td><td>0.00</td></tr><tr><td>36 (&lt;)</td><td>1094.00</td><td>1087.54</td><td>1098.64</td><td>0.08</td></tr><tr><td>UB-x1</td><td>300.00</td><td>1.08</td><td>infinity</td><td>0.00</td></tr><tr><td>UB-x2</td><td>300.00</td><td>1.04</td><td>infinity</td><td>0.00</td></tr><tr><td>UB-x3</td><td>300.00</td><td>0.00</td><td>infinity</td><td>0.00</td></tr><tr><td>UB-x4</td><td>300.00</td><td>1.07</td><td>infinity</td><td>0.00</td></tr><tr><td>UB-x5</td><td>300.00</td><td>0.94</td><td>infinity</td><td>0.00</td></tr><tr><td>UB-x6</td><td>300.00</td><td>1.04</td><td>infinity</td><td>0.00</td></tr></table>

UB-x7300.000.48infinity0.00

UB-x8300.002.29infinity0.00

## Discussion

In product two (2), Response surface, integer programming and linear programming optimization tools were employed to optimize the production output of the 36x75x40 size of foam product. From the analysis, it shows the result of the maximum production output of 1854 units, 2160.0 units and 2172.7 units of the product respectively over any given monthly of production. In linear programming algorithm, it shows the slacks in the variables and also it performs the sensitivity analysis of the product. However, the linear programming optimum production was achieved in the seventeenth iterations while the integer programming maximum optimal solution was found at iteration 5277 and also the Result of this iteration was verified at iteration 5287. In linear programming algorithm, it shows the slacks in the variables and also it performs the sensitivity analysis of the product. However, the optimum production was achieved in the fourteenth iterations. Furthermore, the sensitivity analysis in linear programming develops the coefficients of the independent variables and also reduced cost at the optimum iteration.

In response surface model employed to optimize the production time scheduling of the 36x75x40 size of foam product shows the result of the optimum production time scheduling of 1854 units of the product over any given monthly of production. However, response surface method shows the coefficients of the independent variables and the analysis of variance (ANOVA) in the variables. It develops new design points for the variables. The response surface model shows the coefficient of relationship (R²) to be hundred percent (100%). However, the response optimization shows the composite desirability of achieving the optimum of 1854 units to be 100%. The response optimization also shows the optimization plot which contains the optimum value of the dependent variable (1854units) and the current response values of the independent variables at optimum. The response optimum analysis also reveals that the composite desirability of achieving the predicted optimum result is hundred percent (100%).

## Conclusion

In conclusion, the research work have really achieved the aim of the study which is to optimize the production time scheduling system in foam industry. The specific size of foam used is 36x75x40. From the discussion of

<figure><img src="image_11.png" /></figure>
the result, it shows that the optimum production of the foam size at every month runs at the optimal quantity of 1854 units using response surface, 2160 units using integer programme and 2173 units approximately using linear programme model. Having achieved the stated aim of the work, the results were recommended to the aforementioned case Company.

## References

[1]. Jeffrey W. Herrmann (2012). A History of Decision-Making Tools for Production Scheduling. *Department of Mechanical Engineering and Institute for Systems Research, University of Maryland, College Park, MD 20742, USA*.

[2]. Jagadeesha T, (2016) Assistant Professor, MED, National Institute of Technology, Calicut

[3]. Krishna Kumar C. and Bani K. Sinha, (1999)," Efficiency Based Production Planning and Control Models", European Journal of Operational Research, Vol. 117, pages 450-469.

[4]. Marcus V. Magalhaes and Nilay Shah, "Crude Oil Scheduling," *Foundations of Computer-Aided Operations (FOCAPO) 2003*, pp 323-325.

[5]. Zhenya Jia and Marianthi Ierapetritou, "Efficient Short-Term Scheduling of Refinery Operation Based on a Continuous Time Formulation," *Foundations of Computer-Aided Operations (FOCAPO) 2003*, pp 327-330

[6]. Wilson, James M., History of manufacturing management, in *Encyclopedia of Production and Manufacturing Management*, Paul M. Swamidass, ed., Kluwer Academic Publishers, Boston, 2000a.

[7]. Hopp, W.J., and Spearman, M.L., 1996, Factory Physics, Irwin/McGraw-Hill, Boston.

[8]. McKay, K.N., 2003, Historical survey of manufacturing control practices from a production research perspective, *International Journal of Production Research*, 41(3):411-426.

[9]. Wight, Oliver W., *Production and Inventory Management in the Computer Age*, Van Nostrand Reinhold Company, Inc., New York, 1984.

[10]. Cox, James F., John H. Blackstone, Jr., and Michael S. Spencer, editors, *APICS Dictionary*, American Production and Inventory Control Society, Falls Church, Virginia, 1992.

[11]. LaForge, R. Lawrence, and Christopher W. Craighead, *Manufacturing scheduling and supply chain integration: a survey of current practice*, American Production and Inventory Control Society, Falls Church, Virginia, 1998.

<figure><img src="image_12.png" /></figure>