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**Linear Programming Technique for Cotton Mixing**

Pavani Harsha MB 30 - Quantitative Methods in Decisions Prof.Jaideep Naidu

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**Acknowledgements Prof. Jaideep Naidu, Philadelphia University**

LPT for Cotton Mixing Acknowledgements Prof. Jaideep Naidu, Philadelphia University Mr. Thavasi Vijayakumar, Spinning Manager, Pt. Gokak, Indonesia

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**Introduction: LPT for Cotton Mixing**

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

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**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

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**Is Cotton Cost a big factor? Around 60 % of total production cost**

LPT for Cotton Mixing Is Cotton Cost a big factor? Around 60 % of total production cost

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**Properties of Raw Materials that affect the final product:**

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

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**Formulation of LPT model:**

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

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**Objective Function: S.T. Constraints LPT for Cotton Mixing**

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

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**Example: Aim: Required properties for the raw material:**

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.

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**Maturity Coefficient (%)**

LPT for Cotton Mixing Properties of Cottons available and their Costs: Properties Cottons Norms 1 2 3 Length (mm) Strength (gpt) Maturity Coefficient (%) Micronaire 33 24 83 3.5 31 20.5 80.2 3.85 30 19 79.8 3.9 32 21.5 82 3.7 Cost per lb (US $) 2.05 1.70 1.66 The values shown above are examples and do not represent any cottons as such.

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**Non-Negativity Constraints: P1, P2, P3 0**

LPT for Cotton Mixing Objective Function: Min Z = (2.05*P1) + (1.70*P2) + (1.66*P3) S.T. Constraints: 33*P1 + 31*P2 + 30*P3 32 24*P *P2 + 19*P3 21.5 0.83*P *P *P3 0.82 3.5*P *P *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)

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**Percentage to be Mixed (%)**

LPT for Cotton Mixing Results: Objective Function Value: Min Z = 1.925 Cotton Percentage to be Mixed (%) 1 64.3 2 35.7 3 Conclusion: LPT can be efficiently used in cotton mixing LPT eliminates wastage of raw materials and hence reduces the total Production Cost.

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