The Dakota Furniture Company manufactures desks, tables and chairs. The manufact

Question

The Dakota Furniture Company manufactures desks, tables and chairs. The manufacture of each type of furniture requires lumber and two types of skilled labor: finishing and carpentry. The amount of each resource needed to make each type of furniture is given in the table below. Desk Table Kesource Chair Lumber (bd ft) inishing hrs Carpentry hrs 1.5 0.5 1.5 At present, 48 board feet of lumber, 20 finishing hours, and 8 carpentry hours are available. A desk sells for $60, a table for $30, and a chair for $20. Dakota believes that the demand for desks and chairs is unlimited, but at most five tables can be sold. The linear programming model that can be used to maximize the company's total profits is given below: Let, XI = # Desks, X2 = # Tables, X3 = # Chairs, Max 60x1 30x2+20x3 S.T 8x1 + 6x2 + x3

Explanation / Answer

1. Dual price for finishing hours. Interpret the value

Dual price for finishing hours. -$ 10

One additional unit of finishing hour is worth around $10

2. If carpenters are wiling to work overtime, what is the most that they should be paid?

The dual value/shadow price of carpentry is $10 per hour. i.e. One additional unit of carpentry is worth around $10.

So, this is the maximum they should be paid i.e. $10 per hour

3. How much should will to pay for extra lumber?

Lumber has a surplus of 24 feet left behind unused. Dual price is also $ 0 .ie Additional unit of lumber is not any worth.

So, extra lumber is worth $0

4. Dual price for finishing hours. - $ 10

One additional unit of finishing hour is worth around $10

With optimal solution,

Current revenue = 60*2+30*0 +20*8= $280

Current available finishing hours =20

For 21 hours,

Additional unit of finishing hours available,

21-20=1 hour

Amount revenue added= worth of one additional unit* number of units added = $ 10*1 = $10

New revenue = 280+10=$290

For 23 hours,

Additional unit of finishing hours available,

23-20=3 hours

Amount revenue added= worth of one additional unit* number of units added = $ 10*3 = $30

New revenue = 280+30=$310

5. For variable       X2-table, the upper bound for co-efficient (price) is 35 $.Increasing up to $ 35 will not cause any change in optimal solution

When the value of the co-efficient of the variable is increased or decreased between the bounds, the optimal solution does not change.

Hence there will be no change of optimal solution if the price of table changes to $34

6. For variable       X3-chair , the upper bound and lower bounds for co-efficient (price) is $22.5 and 15$. When the value of the co-efficient of the variable is increased or decreased between the bounds, the optimal solution does not change.

Hence there will be no change of optimal solution if the price of table changes to $18

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