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 demand. 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?

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 Number3-ft boards4-ft boards5-ft boardsWaste (<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