macy's_presentation
TRANSCRIPT
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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.