Macy's_presentation

8/7/2019 Macy's_presentation

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Linear Programming Problemon Profit Maximization of

Executives:

Rajat KatariaChandan Gaddamanugu

Sushant DashRagesh Nair


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Macys, an internationally knownmanufacturer of Mens wear, producesfour varieties of Mens shirt.

One is an expensive, all-silk shirt, one isan all-polyester shirt, and two are blends

of polyester and cotton.


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The following table illustrates the cost andavailability (per monthly production

planning period) of the three materialsused in the production process:

Material Cost per meter

($)

Material

Available

Silk 20 5000

Polyester 7 4000

Cotton 10 4250


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Table (shown in the next slide) summarizesthe contract demand for each of the fourstyles of shirts, the selling price per shirt,and the fabric requirements of eachvariety.

OBJECTIVE:

1. Maximize its monthly profit.

2. Also need to keep in mind the constraintsinvolved while aiming for maximumprofitability.


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Variety ofShirts

SellingpricePer shirt

($)

MonthlyContractMinimu

m

MonthlyDemand

MaterialRequired per

shirt(meters)

MaterialRequirements

All Silk 52.99 400 800 2.5 100% silk

All Polyester 18.20 600 2.3 100%polyester

Poly-Cotton 25.00 2.8 50%

polyester-

50%

cottonPoly-Cotton 20.99 2.1 30%

polyester-

70%

cotton


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SOLUTION:Let

X1 = Number of all-silk shirts

X2 = Number of all-polyester shirtsX3 = Number ofBlend 1 (50% Polyester

+ 50% Cotton) shirts

X4 = Number ofBlend 2 (30%Polyester + 70% Cotton) shirts


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Determining the Profit Function:

1. For all-silk shirt (X1), each requires 2.5

meters of silk, at a cost of $20 per meter.Therefore, the cost per shirt is $50. Theselling price per silk shirt is $52.99, leaving

Profit of ($52.99 - $50.00 =) $2.99 per unit ofX1.

2. For all-polyester shirts (X2), each requires2.3 meters of polyester at a cost of $7 permeter. The cost per shirt is, therefore,$16.10.

Profit per unit of X2 is ($18.20 -$16.10 =)

$2.10.


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3. For poly-cotton blend 1 (X3), each shirtrequires 1.4 meters of polyester at $7 per

meter and 1.4 meters of cotton at $10 permeter, for a cost of $23.80.

Profit : $25.00 - $23.80 = $1.20 per shirt.

The profit is $1.20.4. For poly-cotton blend 2 (X4), each shirt

requires 0.63 meters of polyester at $7 per

meter and 1.33 meters of cotton at $10 permeter, for a cost of $19.11

Profit : $20.99 - $19.11 = $1.88 per shirt.T

he profit is $1.88.


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Hence, the profit formula that can be derivedfrom the above information is as follows:

We have to maximize this function to earnmaximum profit for the company.


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Constraints:

1. The silk material available for theproduction of shirts per month is 5000

meters.2. The Polyester material available for the

production of shirts per month is 4000meters.

3. The Cotton material available for theproduction of shirts per month is 4250meters.


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Also:

4. Macys should keep in mind thecontractual demands and hence produce aminimum number of shirts as per thecontract.

5. Macys should also produce according tothe demand in the market.


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The constraints are formulated as follows:

X1 >= 400 X1 = 600

2.5 X1


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Using Solver, we found the following

Solution.Silk

Shirts

Polyester

Shirts

Poly - Cotton

Blend 1

Poly - Cotton

Blend 2

Total

Limit

800

947.204968

9 0 2891.156463

Minimum 400 600

Maximum 800

Profit 2.99 2.1 1.2 1.88

9816.50

4584

Silk Required 2.5 0 0 0 2000 5000

Polyester

Required 0 2.3 1.4 0.63 4000 4000

Cotton

Required 0 0 1.4 1.47 4250 4250


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The Maximum Profitability is shown asfollows:

The no. of units to be produced (for eachtype of shirt) is shown as:

Cell Name Original Value Final Value

$C$5 Silk Shirts 0 800

$D$5 Polyester Shirts 0 947.2049689

$E$5 Poly - Cotton Blend 1 0 0

$F$5 Poly - Cotton Blend 1 0 2891.156463

Cell Name Original Value Final Value

$G$8 Profit Total 0 9816.504584


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The Constraint sheet gives the following

picture:Cell Name Cell Value Formula Status Slack

$G$9 Silk Required Total 2000 $G$9


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Final Shadow

Name Value Price

Silk Required Total 2000 0

Polyester Required Total 4000 0.913043478Cotton Required Total 4250 0.887607217

The Shadow price analysis shows that ifan extra metre of Polyester is utilized inthe Production then the Profitability jumpsup by $ 0.913 and for Cotton theProfitability jumps up by $0.887

But pumping an extra one meter of silkinto Production wont fetch any increase inthe Profitability.


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Final Shadow Constraint Allowable Allowable

Name Value Price R.H. Side Increase Decrease

Silk Required Total 2000 0 5000 1E+30 3000

Polyester Required Total 4000 0.913043478 4000 1E+30 798.5714286

Cotton Required Total 4250 0.887607217 4250 1863.333333 4250

The above table shows the number ofmeters that can be increased ordecreased from each of the clothing type

without having any impact on the shadowprice of each material.


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Final Reduced Objective Allowable Allowable

Name Value Cost Coefficient Increase Decrease

Silk Shirts 800 2.99 2.99 1E+30 2.99Polyester Shirts 947.2049689 0 2.1 4.763492063 2.1

Poly - Cotton Blend 1 0 -1.320910973 1.2 1.320910973 1E+30

Poly - Cotton Blend 2 2891.156463 0 1.88 1E+30 1.304782609

The above table shows the coefficients ofeach type of shirts which forms theobjective function of profitability.

The allowable increase/decrease showsthe permissible change in the values ofcoefficients without having any impact onthe main objective function.


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Conclusion

The objective is to maximize and adhering tothe constraints of silk material available,polyester material, cotton material,contractual demands and the demand inthe market. Using Linear Programmingmethodology for arriving to the conclusion,

keeping in mind the constraints, we havearrived to the conclusion that Macys willhave maximum profit using this approach.

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