case study: application of linear programming for

CASE STUDY: APPLICATION OF LINEAR PROGRAMMING FOR OPTIMISING THE

PRODUCT MIX PROBLEM IN M.K. KNITWEARS INDUSTRY

Sangeeta Gupta1, Deepti Gupta

2, Vishal Chauhan

3 & Mansi Rana

4

1,3,4 Department of Mathematics, Sharda University, Greater Noida, Uttar Pradesh, India.

2College of Engineering and Technology, Bhubaneswar, India.

[email protected] , [email protected]

Abstract: The industry has made effective management decision-making techniques possible through

surveys and the efficient use of sources and assets. In this article, we consider the problem of

optimizing the range of products when planning production and formulate the problem of M.K.

Knitting industry (Punjab). Get the optimal solution of linear programming modal.M.K. Knitwears

industry, (Punjab) produce Leather, Flash, D50, and Washing Jackets. The model was solved

using MATLAB software.The research result shows that the company's profit can be

increased by 66.89%.

Keywords: Linear programming, MATLAB, Optimal,Product mix.

Introduction:Worldwide the companieshave facedthe problem on optimization of utilization of

limited resources.For worldwide extension and competition in businesses, industrialists must uplift

their operations and practices.Linear Programming is a problem-solving approach helps managers to

take best decisions. According to Shaheen and Ahmad (2015) linear programming is the best method

for regulating an optimal solution among all the alternatives to meet a specified objective function by

various constraints and restrictions. Ezema, B.I and Amakoml, U. (2012) worked on optimizing profit

with the linear programming model for Golden Plastic Industry Limited.JonnalaSubbaReddy ,

M.Bhavani, G.Kartheek , J. Venkata Somi Reddy (2018) did the study on optimization of a product

mix in a paper mill.

Kellerer, H., &Strusevich, V.A. (2016) devolved the model for optimizing the half-product

and related quadratic Boolean functions: approximation and scheduling applications. Fagoyinbo, I. S.,

&Ajibode, I. A. (2010) worked on application of linear programming techniques in the effective use

of resources for staff training. Workie, G. (2017) studied in apparel Industry by applied the

optimization problem of product mix and Linear programming applications. A.I.Iheagwara, J.Opara,

J.I. Lebechi and P.A. Esemokumo (2014) used linear programming problem on Niger Mills Company

PLC Calabar. W.B.Yahya, M.K.Garba, S.O.Ige and A.E. Adeyosoye (2012)did the work on profit

maximization in a product mix company using linear programming.

When defining the product mix, M.K. Knitwears industry has been faced discrepancy. The problem

arises from the incompetent use of resources, which makes it difficult to ensure the optimal range of

products for maximum profit that also meet customer needs. Thus, M.K. Knitwears industry be

required to go through operations research techniques to enhance best resource utilization that would

Journal of Shanghai Jiaotong University

Volume 16, Issue 9, September - 2020

ISSN:1007-1172

https://shjtdxxb-e.cn/ Page No: 841

result in optimal product mix and total profit. Thus, this paper pivot on product mix determination

based on efficient resource utilization for the industry. The issue addressed here was to determine the

product mix for optimal profit with available resources, using the linear programming technique, that

the how much company should produce Leather, Flash, D50, and Washing Jackets. The objective of

the study was to suggest linear programming as a decision tool to determine the optimal product mix

for maximum profit with available resources.

Methodology:

The data collection procedure was quantitative in nature. The amount of resources used per

unit of product is discussed below via tabulation:

Table 1: Resources needed per unit of product.

PRODUCT

(JACKETS)

RESOURCES USED PER UNIT PRODUCT

FABRICS THREAD LABOR OVERHEAD CUTTING SEWING FINISHING

(meter) (Rs.) (Rs.) (Rs.) (min.) (min.) (min.)

LEATHER 3.5 5 200 50 15 80 16

FLASH 3.9 6 140 50 13 50 22

D50 3 5 150 60 18 60 20

WASHING 3.15 5 180 50 21 120 25

Seven constraints have been identifiedas quantities of - fabrics, thread, labor, overhead,

cutting, sewing and finishing. The amount of resources held for a month is discussed below

via tabulation:



ISSN:1007-1172


Table 2: Average monthly resources held in quantity/value terms (in Rs.)

RESOURCE TYPE MEASURING

UNIT

HELD

VALUE

FABRICS meter 32705

THREADS Rs. 43812

LABOR Rs. 1367801

OVERHEAD Rs. 619079

CUTTING min. 121000

SEWING min. 909800

FINISHING min. 160400

The demand and profit earned from each product during a month is discussedbelow via

tabulation:

Table 3: Demand and profit earned.

LEATHER

JACKET

FLASH

JACKET

D50

JACKET

WASHING

JACKET

DEMAND 1200 1800 1286 1070

PROFIT 680 850 470 930

MODEL FORMULATION

In formulating a given decision problem in mathematical form, one should tryto

understand the problem carefully. While understanding the problem, the decision maker



ISSN:1007-1172


may decide that the model consist of linear relationships representing the firm objectives

and resource constraints. However, the way of approaching the problem is same for

decision making problems, but the complexity of the problem may differ.

Linear programming consists of followingparameters:

1. Decision variables or the unknown variables.

2. The objective function, which is to be maximized or minimized.

3. Constraints, which defines the limitation to the type of data.

Mathematical formulation:-

The procedure consists of following steps:-

1. The given situation should be studied to find the key decision.

2. Identify the variables and define them as �� (j=1, 2,…, n).

3. State the feasible alternatives: ��≥0, for all j.

4. Identify the constraints and express them as linear inequalities or equations,

Left hand side of which are linear function of decision variables

5. Identify the objective function and express it as linear function of decision

variables.

General form of LPP

In general, if C = (��,��,…,��) is a tuple or set of real numbers, then the function F of

real variables, y = (��, ��,…, ��) is defined by,

f (y) = ��+ �� +…+ ��

is known as a linear function.

If g is a linear function and b = (��,��,…,��) is a tuple or set of real numbers, then

g(y) = b is called a linear equation, whereas g(y)(≤, ≥) bis called a linear inequality.

A linear programming problem (LPP) is one which optimizes (maximizes or minimizes) a

linear function subject to a finite collection of linear constraints.

Formally, any LPP having � decision variables can be written in the following form:-

Optimize,

= � ��

��

Subject to ∑ �� ≤, =, ≥��

{��≥0,i=(1,2,…,n),j=(1,2,…,n)}

where �� , �� , �� are constants.



ISSN:1007-1172


The function being optimized subject to the given constraints is referred to as the

objective function. The restrictions or the sources which defines a limitation to the type of

data are referred to as the constraints. The n constraints are called functional (or

structural) constraints. The ��≥0, j=(1, 2,…, n) restrictions are called non-negativity

constraints(or conditions) and the aim is to find these. A feasible solution is a solution for

which all the constraints are satisfied, otherwise infeasible solution. Hence, the optimal

solution is referred to as the feasible solution where the objective function reaches its

maximum or minimum value.

RESULT AND DISCUSSION

The information collected from the company consideringthe demands and other data

provides estimate for LPP model variables. To set up the model, the first step is to define

the decision variables on the number of products to be produced were set.

Let, �� = number of leather jacket

�� = number of flash jacket

�� = number of D50 jacket

�� = number of washing jacket

Z= total profit during the month

��= 680�� + 850�� + 470�� + 930��

Subject to,

3.5�� + 3.9�� + 3�� + 3.15�� ≤ 32705(fabric)

5�� + 6�� + 5�� + 5�� ≤ 43812 (thread)

200�� + 140�� + 150�� + 180�� ≤ 1367801(labor)

50�� + 50�� + 360�� + 50�� ≤ 619079(overhead)

15�� + 13�� + 18�� + 21�� ≤ 121000(cutting time)

80�� + 50�� + 60�� + 120�� ≤ 909800 (sewing time)

16�� + 22�� + 20�� + 25�� ≤ 160400 (finishing time)

�� ≥ 1200, �� ≥ 1800, �� ≥ 1286, �� ≥ 1070 (customer orders)

MODEL SOLUTION



ISSN:1007-1172


Algorithm: To generate the MATLAB programme for finding the solution of above linear

programming problem.

MATLAB coding in script file given below:

% solve the LPP in MATLAB

% f is the objective function (but in a minimal form),

% A, Aeq, Beq are the matrices,

% b is the limited resource vector,

% ub and lb are the upper and lower bound respectively (demand

vector),

% x is the vector of number of respective products,

% fval is the value of the objective function.

f=[-680 -850 -470 -930];% objective function in minimization.

A=[3.5 3.9 3 3.15;5 6 5 5;200 140 150 180;50 50 60 50;15 13 18

21;80 50 60 120;16 22 20 25]; % constrint matrix

b=[32705 43812 1367801 619079 121000 909800 160400];% limited

resoursevactor.

Aeq=[];beq=[];

ub=[]; lb=[1200 1800 1286 1070];% demand vector

format bank

[x,fval,exitflag,output]= linprog(f,A,b,Aeq,beq,lb,ub)

MATLAB Results:



ISSN:1007-1172


Here we observed that the difference between the actual production and the solutionof

linear programming by using MATLAB in table 3 is considerable.

In this case, the profit of the company could be improved by 66.89% using linear

programming. Monthly consumption by LPP techniques and customer order production

RESOUR

CES

HELD

PER

MONTH

MONTHLY RESOURCES

CONSUMPTION

PERCENTAGE (%)

OF USAGE

CUSTOMER

ORDER LPP

CUSTOMER

ORDER LPP

TYPE UNIT VALUE

FABRICS meter 32705 18448.5 27750.378 56.40880599 84.85056719

THREAD Rs. 43812 28580 42547.22 65.23326942 97.11316534

LABOR Rs. 1367801 877500 1277796.8 64.1540692 93.4197884

OVERHEAD Rs. 619079 280660 404493 45.33508648 65.3378648

CUTTING min. 121000 87018 121000.06 71.91570248 100.0000496

SEWING min. 909800 391560 542175.2 43.03803034 59.59278962

FINISHING min. 160400 111270 160400.08 69.37032419 100.0000499

As a result, it shows that by using of LPP we can improve resource (fabrics, thread, labor,

overhead, cutting, sewing, finishing) utilization approximately 85%, 97%, 93%, 65%,

100%, 60% and 100% respectively. Also, the graph shows the inefficient use of resources

since most resources are kept idle.



ISSN:1007-1172


Comparison of customer order and LPP production resources utilization

CONCLUSION

The resource utilization was mentioned as the major constraint. The profits

comparison between the actual production and the production using LPP models

shows the considerable differences. So, we can conclude that M.K. Knitwears

industry should use the quantitative research methods of Linear Programming to

determine their optimal product mix. The profit of the company can be improved

from Rs.3945520 per month to Rs.5898911.97 per month.

0

10

20

30

40

50

60

70

80

90

100

CUSTOMER ORDER

LPP

Customer orders LPP model

x1 1200 2092.74

x2 1800 3383.92

x3 1286 1286

x4 1070 1070

Max. Z = 3945520 5898911.97

difference= 1953391.97



ISSN:1007-1172


References:

1. A.I.Iheagwara, J.Opara, J.I. Lebechi and P.A. Esemokumo, “Application of Linear

Programming Problem on Niger Mills Company PLC Calabar”, International Journal

of Innovation and Research in Educational Sciences (IJIRES), 1(2),(2014), pp7-12.

2. A. R. Mohammed and S. S. Kassem, "Product Mix Optimization Scenarios: A Case

Study for Decision Support Using Linear Programming Approach," 2020

International Conference on Innovative Trends in Communication and Computer

Engineering (ITCE), Aswan, Egypt,(2020), pp. 50-55.

3. Anieting, A.E., Ezugwu, V.O., & Ologun, S., “Application of linear programming

technique in the determination of optimum production capacity”, IOSR Journal of

Mathematics (IOSR-JM), 5(6), (2013) pp 62-65.

4. Ekwonwune, E.N., &Edebatu, D.C., “ Application of linear programming algorithm

in the optimization of financial portfolio of Golden Guinea Breweries Plc, Nigeria”,

Open Journal of Modeling and Simulation, 4, (2016), pp 93-101.

5. Ezema, B.I and Amakoml, U., “ Optimizing Profit with the Linear Programming

Model: A Focus on Golden Plastic Industry Limited, Enugu, Nigeria”,

Interdisciplinary Journal of Research in Business, 2 ( 2),(2012), pp 37–49.

6. Fagoyinbo, I. S., &Ajibode, I. A., “ Application of linear programming techniques in

the effective use of resources for staff training”, Journal of Emerging Trends in

Engineering and Applied Sciences (JETEAS), 1(2),(2010), pp 127-132.

7. Hadidi, L., Moawad, O., “The product-mix problem for multiple production lines in

sequenced stages: a case study in the steel industry”, Int J Adv Manuf Technol 88,

(2017), pp 1495–1504.

8. JonnalaSubbaReddy , M.Bhavani, G.Kartheek , J. Venkata Somi Reddy ,

“optimization of a product mix in a paper mill – a case study”, International Journal of

Advance Engineering and Research Development Volume 5, Issue 04, (2018), pp 1-6.

9. Kellerer, H., &Strusevich, V.A., “Optimizing the half-product and related quadratic

Boolean functions: approximation and scheduling applications”, Annals of

Operations Research, 240,(2016), pp 39–94.

10. N. P.&Iwok, I.A., “Application of Linear Programming for Optimal Use of Raw

Materials in Bakery Akpan”, International Journal of Mathematics and Statistics

Invention (IJMSI), 4(8),(2016), pp 51-57.

11. P. J. Nikumbh , S. K. Mukhopadhyay, Bijan Sarkar and Ajoy Kumar Dutta, “Rough

Sets Based Product Mix Analysis”, International Journal of Advancements in

Technology, Vol 2, No 3 (July 2011), pp 382-399.



ISSN:1007-1172


12. Shaheen, S. and Ahmad, T., “Linear Programming Based Optimum Resource

Utilization for manufacturing of Electronic Toys”. International Research Journal of

Engineering and Technology (IRJET), 2 (1),(2015), pp 261–264.

13. W.B.Yahya, M.K.Garba, S.O.Ige and A.E. Adeyosoye, “Profit Maximization In A

Product Mix Company Using Linear Programming”, European Journal of Business

and Management, 4(17), 2012, 126-131.

14. Workie, G., “The optimization problem of product mix and Linear programming

applications: Study in apparel Industry”. Open Science Journal, Vol. 2(2)(2017), pp 1-

11.



ISSN:1007-1172


case study: application of linear programming for

Documents