Suppose that a lumberyard has a supply of 10-ft boards, which are cut into 3-ft, 4-ft, and 5-ft boards according to customer dem
and. The 10-ft boards can be cut into several sensible patterns. each in such a way that there the leftover material is less than 3-ft. The lumberyard just received an order for 90 3-ft boards, 60 4-ft boards. and 60 5-ft boards. a. Determine all sensible patterns the lumberyard may use to cut the 10-ft boards.b. The lumberyard would like to use as few 10-ft boards as possible in meeting the demand. Write out a mathematical formulation for the problem as an integer linear program ILP), and setup/solve this problem using Microsoft Excel solver. What is the optimal number of each pattern, and what is the minimum number of boards to cut?
2 answers:
Answer :
Some answers are attached below
Explanation:
a) Based on the information provided, below are the all possible sensible patterns the lumberyard may use to cut the 10-ft boards -
Pattern Number 3-ft boards 4-ft boards 5-ft boards Waste (<3-ft)
1 3 0 0 1
2 2 1 0 0
3 1 0 1 2
4 0 1 1 1
5 0 2 0 2
6 0 0 2 0
b) ILP Formulation
Objective Function
Minimize Number of 10 ft-boards to be cut
Z = x1+x2+x3+x4+x5+x6 where xi is the number of boards for pattern i
subject to constraints -
3x1+2x2+x3>=90 ----> 3-ft boards
x2+x4+2x5>=60 ----->4-ft boards
x3+x4+2x6>=60 ----->5-ft boards
Xi>=0 ---->Non-negativity constraint
Below is the excel solver formulation and solution -
The optimal number of Pattern 1,5,6 =0, Pattern 2,3 and 4 = 30
The minimum number of 10 ft-boards to be cut = 90
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Answer:
attached is the diagram
Explanation:
Each extra worker produces an extra unit of output, is said to be the marginal production of an extra worker employed
marginal production :
change in total production / change in labor = ΔTp / ΔL
Average production = Tp / L
Tp = total production , L = number of labor
To draw the Total product of labor , average product labor and marginal product labor curves starting from zero labor
0 worker : Total product = 0, average product labor = 0 , marginal = 0
1 worker : Total product = 1, average product = 1 , marginal = 0
2 worker : Total product = 2, average product = 1, marginal = 1
3 workers: total product = 3 average product = 1, marginal = 1
4 workers: Total product = 4, average product = 1, marginal = 1
5 workers : Total product = 5 average product = 1, marginal = 1
6 workers : total product = 6 average product = 1 , marginal = 1
7 workers : total product = 7 , average product = 0.85, marginal = 0
8 workers : total product = 8, average product = 0.75 marginal = 0
