Extra Credit

Problem 3 Page 199

Highland Appliances

Group 8

Ricardo Bernal

Hans Domingo

Ismael Reyes Jr.

November 15, 2011

IE 416

Dr. Parisay

Problem Statement:

Summary:

Purchase Cost

Sales Profit

Sq yd of storage space

TVs $300 $150 3 sq ydVCRs $200 $100 1 sq yd

Goals : Max 20,000 Min 11,000 Max 200

Goal 1(P1): Maximum of $20,000 can be spent on purchasing.Goal 2(P2): At least $11,000 in profits.Goal 3(P3): Inventory should not be more than 200 sq yd

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Parisay’s comments are in red.

Formulation:

X1 = Number of TVs to be kept as inventory X2 = Number of VCRs to be kept as inventory

Subject To (S.T.):

300X1 + 200X2 ≤ 20,000 Dollars

150X1 + 100X2 ≥ 11,000 Dollars

3X1 + 1X2 ≤ 200 sq yd

X1, X2 ≥ 0

We can try to solve at this point using Linear Programming. To do so we need to define an imaginary O.F. as below. This OF will have no effect on the problem and will just facilitate using Simplex method (and WinQSB) to find the solution

Objective Function (O.F.): Z = 0X1 + 0X2

Input to WinQSB software:

Output from WinQSB:

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Notice that there is no solution for this problem. Using GP can assist to have some solution.

Goal Programming:

Order of importance: P1>> P2 >> P3

OF:

Purchase Sales profit Space

Z1=P1 (S1+ )

Z2=P2 (S2− )

Z3=P3 (S3+ )

ST:

Maximum spent Sales profit goal Inventory space

300 X1+200 X2+S1−−S1

+=20000150 X1+100 X 2+S2

−−S2+=11000

3 X1+1X 2+S3−−S3

+=200

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Input to WinQSB software:

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Output from WinQSB:

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Summary Solution:

X1=66.7, X2=0

Z1=0, Z2=1000, Z3=0

Analysis of output:

X1=66.7>>> It means we will stock 66.7 TVs in inventory.

X2=0>>> It means we will stock zero VCR’s in inventory.

Z1=0>>> This is related to our expenditure which will not exceed our limit, which is $20,000.

Z2=1000>>> This means we will be short $1,000 from our profit goal of $11,000 which is not desirable. This results in S2

-=1000.

Z3=0>>> This means there’s no undesirable situation when it comes to space capacity. We will use exactly the 200 sq yd of space available.

Report:

Based on the information that was given, we need to purchase 66.7 TVs and no VCRs. It will result in a profit of $10,000.We will spend $20,000 which is within our given budget, and also use up 200 sq yards which is also within our space budget. However, we will be $1,000 short of our desired profit which is $11,000.

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Alternative Optimal Solution:

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Summary Solution:

X1=0, X2=100Z1=0, Z2=1,000, Z3=0

Analysis of output:

X1=0, which means we will have zero TVs in stock.

X2=100, which means we will have one-hundred VCRs in stock.

Z1=0, which means we will not exceed our spending limit of $20,000, which is desirable.

Z2=1,000, which means we will be $1,000 short of our goal in profits; which means we will make $10,000 in profits. This results in S2

-=1000, which is not desirable.

Z3=0, which means we will not exceed our space capacity, which is desirable. S3-=100, which

means we will use 100 sq yd less of space.

Report:

Based on the information that was given, we need to purchase no TV’s and one-hundred VCR’s. It will result in a profit of $10,000.We will spend $20,000 which is within our given budget, and also use only 100 sq yards which is also within our space limitation. However, we will be $1,000 short of our desired profit which is $11,000.

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How would the preemptive goal formulation be modified if Highland’s goal were to have a profit of exactly $11,000?

Goal 1(P1): Maximum of $20,000 can be spent on purchasing.Goal 2(P2): Exactly $11,000 in profits.Goal 3(P3): Inventory should not be more than 200 sq yd

Formulation:

X1 = Number of TVs to be kept as inventory X2 = Number of VCRs to be kept as inventory

Goal Programming:

Order of importance: P1>> P2 >> P3

OF for approach 1: (notice Z2, in fact we do not need it)

Purchase Sales profit Space

Z1=P1 (S1+ )

Z2=P2

Z3=P3 (S3+ )

ST:

Maximum spent Sales profit goal Inventory space

300 X1+200 X2+S1−−S1

+=20000150 X1+100 X 2=110003 X1+1X 2+S3

−−S3+=200

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Input to WinQSB software:

Output from WinQSB:

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Summary of solution:

X1=60, X2=20Z1=0, Z2=0, Z3=0

Analysis of output:

X1=60, which means we will stock sixty TVs in our inventory.

X2=20 which means we will stock twenty VCRs in our inventory.

Z1=0, this is related to our limit budget(max of $20,000) for purchasing equipment this means we will spend more than $20,000.If we plug in X1 and X2 into constraint one we get 300X1+200X2=300(60)+200(20)=22,000. We spend $2,000 over our expenditure limit of $20,000 which is undesirable. You can find 2000 just by looking at deviation variable.

Z2=0 which means our deviation variable for profit tell us our profits will be exactly $11,000 but since we will spend a total of $22,000, our final profits will be in fact $9,000.

Z3= 0 which means we will use exactly the 200 sq yd of space available.

Report:

Based on the given information, our profit will be exactly $11,000 if we purchase sixty TVs and twenty VCRs .However, this will only be possible if we spend $2,000 more than the given $20,000 budget. We used all the space.

OF for approach 2: (notice Z2 has both deviations)

OF:

Purchase Sales profit Space

Z1=P1 (S1+ )

Z2=P2 (S2−)+P2(S 2+)

Z3=P3 (S3+ )

ST:

Maximum spent Sales profit goal Inventory space

300 X1+200 X2+S1−−S1

+=20000150 X1+100 X 2+S2

−−S2+=11000

3 X1+1X 2+S3−−S3

+=200

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Solve it to check if you get the same answer.

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Sensitivity Analysis (Changing order of priorities):

By reordering the priorities assigned to the goals, many solutions can be generated. From

among these solutions, the manager can choose a solution that best fits his preference. Our

original scenario is:

Goal 1(P1): Maximum of $20,000 can be spent on purchasing.Goal 2(P2): At least $11,000 in profits.Goal 3(P3): Inventory should not be more than 200 sq yd

After reordering the priorities we came up with five different combinations. The table below

shows the results of our findings.

                        Priorities                                                                                                               Optimal Solution                                                            

                        Second                                                                             Deviation

Highest                       Highest                       Lowest                                       X1           X2                             Z1                     Z2                     Z3            

S1+             S2-              S3+         66.7      0                  0            1000          0

S1+ S3+         S2-          66.7      0                  0            0            1000

S2-         S1+                S3+ 60      20                 0            2000             0

S2- S3+         S1+         60      20                  0            0            2000

S3+ S2-         S1+         60      20                  0            0            2000

S3+ S1+         S2-          66.7      0                  0            0            1000

Report to Manager:

Dear Manager,

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As we know how important maximizing profits are to you, allow us to present you the two most optimal recommendations for the allocation of TV’s, VCR’s and space used to meet your goals of profits, expenditure, and space available.

For our first optimal solution, our maximum profit will be $10,000. This is accomplished by stocking 66.7 TV’s and no VCR’s in inventory. Our spending will be $20,000 and 200 Sq. yd. of space will be used.

For our second optimal solution, our maximum profit will be $10,000. This is accomplished by having no TV’s and stocking 100 VCR’s in inventory. Our spending will be $20,000, and 100 Sq. yd. of space will be used.

Lastly, if we are to state our profit goal to be exactly $11,000, a spending of $22,000 will have to occur, which means our profits in reality will be $9,000. Below is a summary table of our solutions mentioned so you can visualize any decision that best interests you. Each row represents one full decision and its consequences.

Furthermore, if in the future you decide to rearrange the priorities of our goals, according to a specific point of view, we recommend the following change:

PrioritiesHighes

t 2nd HighestLowes

t Space Expenditure Profit

TV VCRExpenditur

e Profit Space66 0 20,000 10,000 200

We recommend making space our highest priority, expenditure second highest, and profits as our lowest. With this, we will need to purchase sixty-six TV’s and zero VCR’s which will yield a profit of $10,000, expenditure $20,000, and use up 200 Sq. Yd of storage space. We recommend this because we get the highest profit while staying within the allotted space and the given budget.

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TV VCR Cost Profit Space66 0 $20,000 $10,000 200

0 100 $20,000 $10,000 100

60 20 $22,000 $11,000 200

AppendixSensitivity Analysis:

Input for original goals scenario:

Output for original goals scenario:

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Input for reordered goals scenario 1:

Output for reordered goals scenario 1:

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Input for reordered goals scenario 2:

Output for reordered goals scenario 2:

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Input for reordered goals scenario 3:

Output for reordered goals scenario 3:

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Input for reordered goals scenario 4:

Output for reordered goals scenario 4:

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