Linear Programming Technique
for
Cotton Mixing
Pavani
Harsha
MB 30 - Quantitative Methods in Decisions
Prof.Jaideep Naidu
LPT for Cotton Mixing
Acknowledgements

Prof. Jaideep Naidu, Philadelphia University

Mr. Thavasi Vijayakumar, Spinning Manager,
Pt. Gokak, Indonesia
LPT for Cotton Mixing
Introduction:
Quantitative methods used in Textile industry:
•Linear programming technique - Cotton mixing
- Scheduling
•Forecasting in apparel industry - Seasonal Forecasting
•Inventory control – Various production stages
•CPM/ PERT - two or more simultaneous projects on time
•Transportation technique – Raw materials & finished goods
LPT for Cotton Mixing
LPT for Cotton Mixing:
•Why is cotton mixed?
Balancing Cost and Quality
Achieve best quality yarn at lowest production Cost
•Why cotton quality varies?
Natural fiber
Uneven and non-uniform
•How are Cost & Quality related?
As Quality increases cost increases
LPT for Cotton Mixing
Is Cotton Cost a big factor?
Around 60 % of total production cost
Cost Breakdown in a Typical Textile Mill (%)
Fiber Cost
Capital Cost
Labor Cost
Operating Cost
http://www.cottoninc.com/EFSConference/homepage.cfm?page=1099
LPT for Cotton Mixing
Properties of Raw Materials that affect the
final product:
•Length
•Strength
•Maturity Coefficient
•Fineness – Micronaire
Why is LPT used?
•Minimize Total cost
•Maximize quality
LPT for Cotton Mixing
Formulation of LPT model:
Let
C1, C2, C3,….Cn be the costs of ‘n’ cottons
P1, P2, P3,…..Pn be the Percentages of each cotton to be mixed
L1, L2, L3,….Ln be the lengths of each cotton
S1, S2, S3,….Sn be the strengths of each cotton
M1, M2, M3,….Mn be the maturity coefficients of each cotton
F1, F2, F3,….Fn be the micronaire value of each cotton
LPT for Cotton Mixing
Objective Function:
Min Z = (C1*P1) + (C2*P2) + (C3*P3) + … + (Cn*Pn)
S.T. Constraints
L1*P1 + L2*P2 + L3*P3 + … + Ln*Pn  Lr
S1*P1 + S2*P2 + S3*P3 + … + Sn*Pn  Sr
M1*P1 + M2*P2 + M3*P3 + … + Mn*Pn  Mr
F1*P1 + F2*P2 + F3*P3 + … + Fn*Pn  Fr
P1 + P2 + P3 + … + Pn = 1
P1, P2, P3,….. Pn  0
Right Hand Side values are obtained from given set of Norms
LPT for Cotton Mixing
Example:
Aim:
To manufacture 10 Tex cotton yarn
Required properties for the raw material:
•Length: 31.5mm – 34mm
•Strength: 20gpt – 23gpt
•Maturity Coefficient: 80% - 83%
•Micronaire Value: 3.6 – 3.9
The values shown above are examples and do not represent any cottons as such.
LPT for Cotton Mixing
Properties of Cottons available and their Costs:
Properties
Length (mm)
Strength (gpt)
Maturity Coefficient (%)
Micronaire
1
33
24
83
3.5
Cost per lb (US $)
2.05
Cottons
2
3
31
30
20.5
19
80.2
79.8
3.85
3.9
1.70
Norms
32
21.5
82
3.7
1.66
The values shown above are examples and do not represent any cottons as such.
Objective Function:
LPT for Cotton Mixing
Min Z = (2.05*P1) + (1.70*P2) + (1.66*P3)
S.T. Constraints:
33*P1 + 31*P2 + 30*P3  32
24*P1 + 20.5*P2 + 19*P3  21.5
0.83*P1 + 0.802*P2 + 0.798*P3  0.82
3.5*P1 + 3.85*P2 + 3.9*P3  3.7
P1 + P2 + P3 = 1
Non-Negativity Constraints: P1, P2, P3  0
P1, P2, P3 values are obtained by solving this LP Model using
SIMPLEX method (Microsoft Excel can be used)
LPT for Cotton Mixing
Results:
Objective Function Value: Min Z = 1.925
Cotton Percentage to be Mixed (%)
1
64.3
2
35.7
3
0
Conclusion:
• LPT can be efficiently used in cotton mixing
• LPT eliminates wastage of raw materials and hence reduces
the total Production Cost.