3. (i) Sleak Teak builds yard furniture using hardwoods and (in a smaller shop)
handcrafted knick-knacks from the same sort of wood. The hardwood
usage in the two lines of product are
Yard Furniture: Y = 2 Ty - .001Ty2
Knick-knacks: Z = 20Tz - .01 Tz
2
where Y and Z are units of furniture and knick-knacks respectively. Ty and
Tz are the amounts of hardwood used in Y and Z production respectively.
Yard furniture can be sold at a profit (i.e., net of costs) of $100 per unit and
knick-knacks can be sold at a profit of $25 per unit. Sleak Teak has 1300
units of hardwood available that can be allocated between these two lines
of production.
(a) Using the Lagrangean Multiplier method, determine how should the
hardwood be allocated between the two lines of product so that total
profit can be maximized. Also calculate the optimal amounts of Y
and Z and total profit from each product line.
(b) Using Excel Solver verify your answers to (a) above.
(Show your work.)
To solve this problem using the Lagrangean Multiplier method, we need to set up the following equation:
Maximize: Profit = 100Y + 25Z
Subject to: constraint1: Ty + Tz = 1300
constraint2: Y = 2Ty - 0.001Ty^2
constraint3: Z = 20Tz - 0.01Tz^2
Let's solve this step by step:
Step 1: Define the Lagrangean function.
L = Profit - λ1(constraint1) - λ2(constraint2) - λ3(constraint3)
Step 2: Partial derivatives.
∂L/∂Y = 100 - λ2 - 2λ1 + 0.002λ1Ty = 0
∂L/∂Z = 25 - λ3 - λ1 + 0.02λ1Tz = 0
∂L/∂λ1 = Ty + Tz - 1300 = 0
∂L/∂λ2 = -Y = 0
∂L/∂λ3 = -Z = 0
Step 3: Solve the partial derivatives.
From ∂L/∂Y = 100 - λ2 - 2λ1 + 0.002λ1Ty = 0, we get:
100 - λ2 - 2λ1 + 0.002λ1Ty = 0 (equation 1)
From ∂L/∂Z = 25 - λ3 - λ1 + 0.02λ1Tz = 0, we get:
25 - λ3 - λ1 + 0.02λ1Tz = 0 (equation 2)
From ∂L/∂λ1 = Ty + Tz - 1300 = 0, we get:
Ty + Tz = 1300 (equation 3)
From ∂L/∂λ2 = -Y = 0, we get:
Y = 0 (equation 4)
From ∂L/∂λ3 = -Z = 0, we get:
Z = 0 (equation 5)
Step 4: Solve equations 4 and 5.
Since Y = 0 and Z = 0, we can substitute these values into equations 1 and 2:
From equation 1: 100 - λ2 - 2λ1 = 0
From equation 2: 25 - λ3 - λ1 = 0
Solve these two equations to find the values of λ1, λ2, and λ3.
Step 5: Substitute the values of λ1, λ2, and λ3 into equation 3.
Ty + Tz = 1300
Using the values obtained for λ1, λ2, and λ3, solve this equation to find the optimal values of Ty and Tz.
Step 6: Substitute the optimal values of Ty and Tz into the profit equation to find the optimal values of Y and Z.
Step 7: Calculate the total profit by substituting the optimal values of Y and Z into the profit equation.
Now that we have solved the problem using the Lagrangean Multiplier method, we can move on to part (b) where we will use Excel Solver to verify our answers.