Question

27. Six Sigma is a well-recognized quality management system that employs improvement projects to improve business processes to reduce product and service defects, reduce costs, and increase profits. Each project has a “champion” from upper management who is responsible for the proj-ect’s success. The project leader is known as a “black belt,” and project team members are called “green belts.” Simsun, a manufacturer of cell phone parts and components, has eight improvement projects under consideration, and six champions, 10 black belts, and 25 green belts available to be assigned to the projects. The following table shows for each of the projects the number of green belts required, the percent reduction in defects expected, the cost savings (in $ millions), and the profit improvement (in $ millions).

Project	Green Belts	Defect Reduction (%)	Cost Savings ($ M)	Profit Improvements ($ M)
1	3	0.97	1.32	1.9
2	4	1.35	1.16	2.3
3	7	1.16	0.95	3.6
4	2	0.82	0.88	2.7
5	6	1.02	1.05	4.1
6	5	0.86	1.26	3.8
7	4	1.15	1.13	5.3
8	6	0.96	0.85	4.8

The company wants to select the projects that will result in the greatest profit improvement while achieving a defect reduction of at least 5 percent and cost savings of at least $6.3 million. Formu-late and solve an integer programming model for this problem.

Answer 1

Project	Green belts	Defect redn	Cost savings	Profit improvement
1	3	0.97	1.32	1.9
2	4	1.35	1.16	2.3
3	7	1.16	0.95	3.6
4	2	0.82	0.88	2.7
5	6	1.02	1.05	4.1
6	5	0.86	1.26	3.8
7	4	1.15	1.13	5.3
8	6	0.96	0.85	4.8

Let yi be the varialbe that project i is selected; yi is binary that is it takes values 0 or 1 only

So total profit is 1.9y1 + 2.3 y2 + 3.6y3 + 2.7y4 + 4.1y5 + 3.8y6 + 5.3y7 + 4.8y8 that needs to be maximized

Green belts available are 25

i.e 13y1 + 4 y2 + 7y3 + 2y4 + 6y5 + 5y6 + 4y7 + 6y8 <= 25

Defect reduction of atleast 5%

0.97y1 + 1.35 y2 + 1.16y3 + 0.82y4 + 1.02y5 + 0.86y6 + 1.15y7 + 0.96y8 >= 5

cost savings of atleast 6.3

1.32y1 + 1.16 y2 + 0.95y3 + 0.88y4 + 1.05y5 + 1.26y6 + 1.13y7 + 0.85y8 >= 6.3

The above is formulated in excel as shown:

The values of yi are taken as 1 as starting value. The excel solver output is as shown:

The solution obtained is as shown:

The above solution satisfies all conditions with projects selected being 1,2,4,5,7 and 8