A furniture copmany produces charis and tables from two reso

A furniture copmany produces charis and tables from two resources-labor and wood. The company has 125 hours of labor and 45 board ft of wood available each day. Demand for chairs is limited to 5 per day. Each chair requires 7 hours of labor and 3.5 board ft of wood whereas a table requires 14 hours of labor and 7 board ft of wood. The profit derived from each chair is $325 and from each table, $120. The compnay want to determine the number of chairs and table to produce each day in order to maximize profit. Formulte a linera programmng model for this probloem. How much labor and wood will be unused if the optimal number of chairs and tables are produced?

There are 5 vertex and I think are 7x1+14x2<=125 hours; 3.5x1+7x2 <= 45 ; $325x1+120x1=profit ; (5)7x1+14x2<=125 ; (5)3.5x1+14x2<=45 What am I doing wrong? Thanks

Solution

x1 is the number of chairs

x2 is the number of tables

x1<=5 .....................(1)

7x1+114x2<=125 ...................(2)

3.5x1+7x2<=45 ..................(3)

x1>=0, x2>=0 .........................(5)

Besides this, x1 and x2 are both integers.

Solving the above-

equation (2) is useless when we have equation (3), as (3)=>(2)

When we maximize 325x1+120x2 using equations (1) to (5)

We get optimal point to be (5,3.93). Now this cannot be the solution as y-cordinate is not an an integer.

So, we can check for points (5,3) and (4,4)

We can thus conclude that (5,3) is the solution.

Now, labour used =7*5+14*3=77 and wood used is 3.5*5+7*3=38.5

Hence, labour unused is 48 and wood unused is 6.5

Note- we could have directly said this because a chair earns more profit than a table but consumes less labour and wood. So, maximum number of chairs possible will be produced (5 in this case). Now, labour used is 35 and wood used is 17.5.

So, we are left with 90 hours of labour and 27.5 board ft of wood.

90/14= 6.42 So, labour is sufficient to produce 6 tables

27.5/7=3.92 So, wood is sufficient for 3 tables.

Hence, 3 tables will be produced.

Again, we can calculate the unused labour and wood in the same way.