Hi, I need to know how to formulate the goal programming model for this problem.

Question

Hi, I need to know how to formulate the goal programming model for this problem.
I would also like to know how to input this information into Excel and/or QM for Windows and how to obtain these optimal solutions for this goal programming problem:

x1 = 60 x2 = 28
x3 = 35 x4 = 17
d1+ = 3.4 d2+ = 7.62
d3? = 36,714
Priority 3 is the only one not achieved

The Barrett Textile Mill was checked by inspectors enforcing Occupational Safety and Health Administration (OSHA) codes. The inspectors found violations in four categories: hazardous materials, fire protection, hand-powered tools, and machine guarding. In each category the mill was not in 100% compliance. Each percentage point of increase in the compliance level in each category will reduce the frequency of accidents, decrease the accident cost per worker, and constitute progress toward satisfying the OSHA compliance level. However, achieving compliance does cost the mill money. The following table shows the benefits (in accident frequency and accident cost per worker) and the costs of a percentage point increase in compliance in each category Accident Frequency Reduction (accidents/105 hr. of exposure) Accident Cost/Worker Reduction Cost/Percentage Point Compliance Category Hazardous0.18 materials $1.21 $135 2. Fire 0.11 0.48 87 protection 3. Hand powered 0.17 tools 0.54 58 4. Machine 0.21 1.04 160 guarding To achieve 100% compliance in all four categories, the mill will have to increase compliance in hazardous materials by 60 percentage points (ie.. it is now at 40% compliance), in fire protection by 28 percentage points, in hand-powered tools by 35 percentage points, and in machine guarding by 17 percentage points. However, the management of the mill faces a dilemma, in that only $52,000 is available to spend on safety. Any larger expenditure could jeopardize the financial standing of the mill. Thus, management hopes to achieve a level of accident reduction and compliance that is within the company's budget limitation and that will satisfy OSHA authorities enough to temporarily delay punitive action. Therefore, management has established four goals, listed here in order of importance

Explanation / Answer

1.

Percentage point in hazardous material = x1

Percentage point in fire protection = x2

Percentage point in hand powered tool = x3

Percentage point in machine guarding = x4

Goal 1:

135*x1 + 87*x2 + 58*x3 +160*x4 = 52000 +d+1 - d-1

Where,

d+1   = Amount by which cost exceeds target value 52000

d-1 = Amount by which cost is less than target value 52000

Goal2:

x1 + x2 + x3 +x4 =100 + d+2 - d-2

Where,

d+2   = Amount by which compliance exceeds target value 100

d-2 = Amount by which compliance is less than target value 100

Goal 3:

0.18*x1 + 0.11*x2 + 0.17*x3 + 0.21*x4 = 20 +d+3 - d-3

Where,

d+3   = Amount by which total accident frequency reduction exceeds target value 20

d-3 = Amount by which total accident frequency reduction is less than target value 20

Goal 4:

1.21*x1 + 0.48*x2 + 0.54*x3 + 1.04*x4 = 115 +d+4 - d-4

Where,

d+4 = Amount by which total accident cost per worker exceeds target value 115

d-4 = Amount by which total accident cost per worker is less than target value 115

Min. d+1   

s.t.

135*x1 + 87*x2 + 58*x3 +160*x4 - d+1+ - d-1 = 52000

x1 + x2 + x3 +x4 - d+2 + d-2 =100

0.18*x1 + 0.11*x2 + 0.17*x3 + 0.21*x4 -d+3 + d-3 = 20

1.218x1 + 0.48*x2 + 0.54*x3 + 1.04*x4 -d+4 + d-4 = 115

2. Solver for goal programming is used in the same way as it is used for solving linear programming model except that we have to minimze the added variable.

Steps are:

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