OSCM 230 Fall 2013 Management Science Linear Programming Case Study: Parket Sisters 9/25/2013,...

OSCM 230 Fall 2013Management Science

Linear Programming Case Study:Parket Sisters

9/25/2013, 9/30/2013 1

Linear Programming Case Study:

Parket Sisters

Professor Dong, Washington University in St. Louis, MO



2

Problem Setup

MAX: 3.0 X1 + 3.0 X2 + 5.0 X3

Subject to: Plastic: 1.2 X1 + 1.7 X2 + 1.2 X3 1,000Chrome: 0.8 X1 + 0 X2 + 2.3 X3 1,200Stainless steel: 2.0 X1 + 3.0 X2 + 4.5 X3 2,000

X1 , X2 , X3 0X1 = number of ballpoint pensX2 = number of mechanical pencilsX3 = number of fountain pens

MAX: 3.0 X1 + 3.0 X2 + 5.0 X3

Subject to: Plastic: 1.2 X1 + 1.7 X2 + 1.2 X3 1,000Chrome: 0.8 X1 + 0 X2 + 2.3 X3 1,200Stainless steel: 2.0 X1 + 3.0 X2 + 4.5 X3 2,000

X1 , X2 , X3 0X1 = number of ballpoint pensX2 = number of mechanical pencilsX3 = number of fountain pens

Cell FormulaF6 =SUMPRODUCT(C6:E6,C4:E4)

F9 = SUMPRODUCT(C9:E9,$C$4:$E$4)

Copy F9 to F10:F11

Cell FormulaF6 =SUMPRODUCT(C6:E6,C4:E4)

F9 = SUMPRODUCT(C9:E9,$C$4:$E$4)

Copy F9 to F10:F11



3

Case Questions

Weekly optimal mix? Weekly optimal net profit?Answer:

3. Is the optimal solution unique?Answer:

4. What is the marginal value of one more unit of chrome? Of plastic?Answer:

Ballpoint = 700, Mechanical = 0, Fountain =133.3Profit = 2766.67

• Check if there is a 0 for the reduced cost of a product we do not produce

marginal value of Chrome=SPchrome=$0marginal value of Plastic=SPplastic=$1.167



4

Case Questions

5. Should Parket Sister accept an offer of additional 500 ounces of stainless steel for $0.60 per ounce more than its ordinary price?Answer:

• Because 500<555.56 (allowable increase for stainless steel), we can use SPss. The additional unit cost 0.60 <0.80=SPss. Yes, accept the offer.

9/25/2013, 9/30/2013




5

6. If accept the offer noted in question 5, what is the new product mix and new optimal profit?Answer:

• New profit = old profit +(0.8-0.6)*500 = 2766.67+100=2866.67• To solve for the new product mix, reformulate

LP Max 3x1 +3x2+5x3 -500*0.6s.t. Plastic: 1.2 X1 + 1.7 X2 + 1.2 X3 1,000Chrome: 0.8 X1+ 0 X2 + 2.3 X3 1,200SS: 2.0 X1 + 3.0X2+4.5 X3 2,500

X1 , X2 , X3 0

Max 3x1 +3x2+5x3 -500*0.6s.t. Plastic: 1.2 X1 + 1.7 X2 + 1.2 X3 1,000Chrome: 0.8 X1+ 0 X2 + 2.3 X3 1,200SS: 2.0 X1 + 3.0X2+4.5 X3 2,500

X1 , X2 , X3 0

9/25/2013, 9/30/2013




6

Case Questions7. Should Parket Sister accept the offer of additional plastic at a price

of $1.00 over its usual cost of $5.00 per ounce in lot sizes 500 ounces? Answer:

• Since allowable increases of plastic constraint is 200, we can use the shadow price to evaluate the contribution of the first 200 oz.

• Since 1.00<1.167, accept the first 200 oz. • Need to reformulate LP to evaluate the next 300 oz.

8. An offer of additional plastic at a price of $1.00 over its usual cost of $5.00 per ounce in lot sizes 100 ounces. How many lots (if any) should Parket Sisters buy? What is the optimal product mix if the company buys these lots, and what is the optimal profit?Answer:

• We can answer for the first 200 oz: new profit =2766.67+200*(1.167-1)=2800.• To find the optimal number of lots of additional plastic to purchase, we need to

reformulate LP by introducing a new decision variable y =# of additional lots of plastic to purchase. The new objective function becomes

MAX 3x1 +3x2 +5x3 -100 y We also need to change the plastic constraint (other constraints remain the same) to 1.2 X1 + 1.7 X2 + 1.2 X3 1,000+100y



7

Case Questions

9. Parket Sisters has an opportunity to sell some of its plastic for $6.50 per ounce to another company, who wants to buy 300 ounces of plastic. Should Parket Sister sell? What happens to the product mix and overall profit if it does sell?

10. The chrome supplier can send only 1000 ounces of chrome this week instead of 1200 ounces. What will happen to the optimal product mix and optimal profit?

9/25/2013, 9/30/2013




8

Case Questions

11. A new design of the mechanical pencil requires 1.1 ounces of plastic, 2.0 ounces of chrome, and 2.0 ounces of stainless steel. If the company can sell one of these pencils at a net profit of $3.00, should it approve the new design?

12. If the per unit profit on ballpoint pens decreases to $2.50, what is the optimal product mix and the total profit?

9/25/2013, 9/30/2013




9

Case Questions

13. A new felt tip pen requires 1.8 ounces of plastic, 0.5 ounces of chrome, and 1.3 ounces of stainless steel. What profit must this product generate in order to make it worthwhile to produce?

14. What must be the minimum per-unit profit of mechanical pencils be in order to make them worthwhile to produce?Answer:

• We use reduced cost of mechanical pencil, -1.38 to answer this question.

• The mechanical pencil should have a per-unit profit of at least $4.38 ($3.00 plus $1.38).

9/25/2013, 9/30/2013




10

15. Management believes that the company should produce at least 20 mechanical pencils per week to round out its product line. What effect would this have on overall profit?

16. If the profit on a fountain pen is $6.75 instead of $5.00, what is the optimal product mix and optimal profit?

9/25/2013, 9/30/2013


OSCM 230 Fall 2013 Management Science Linear Programming Case Study: Parket Sisters 9/25/2013,...

Documents

Transcript of OSCM 230 Fall 2013 Management Science Linear Programming Case Study: Parket Sisters 9/25/2013,...