STEPS TO SOLVE GOAL PROGRAMMING

GOAL PROGRAMMING: Management by multiple objectives & also referred as multi objective decision making technique.

3 basic things to remember:  
a) Multiple objectives
b) Goal constraints
c) System constraints

GOAL PROGRAMMING FORMULATION:

1) Define decision variables

2) Decide the priority level of each goal.

3) Decide your most important priority & put it in a sequence according to the importance you have given & use deviation variables ( d_i ⁺ & d_i ^-) acting as surplus & slack variables.

4) Then define the system constraints ( the main constraints as used in linear programming).

5) Write objective function: add all the deviation variables which are not of use to minimize the objective function.

6) Solve all the constraints and get the minimized objective.

Example: Conceptual product is a computer company that produces cp400 & cp500 computers. The computers uses different mother boards produced in abundant supply by the company, but uses the same cases & disk drivers. The cp400 model uses 2 floppy disk drives & no zip disk drive whereas cp500 model uses 1 floppy disk drive & 1 zip disk drive.

           Disk drives & cases are bought from the vendors. There are 1000 floppy disk drives, 500 zip disk drives & 600 cases available to conceptual product on a weekly basis. It takes 1 hour to produce cp400 & its profit is 2000 rs. And it takes 1 & half hour to produce cp500 & its profit is 5000 rs.

SOLUTION: 

The company has 4 goals:

Priority 1: Meet a state contract of 200 cp400 machine weekly (goal 1)

Priority 2: Make at least 500 total computers weekly (goal 2)

Priority 3: Make atleast 2500000 rs weekly. (goal 3)

Priority 4: Use no more than 500 man hour per week (goal 4)

Decision variables:

X_1: No. Of cp400 computer produced weekly.

X_2: No. Of cp500 computers produced weekly.

D_i^-: Amount the right hand side of goal is deficient.

D_i⁺: Amount the right hand side of goal is exceeded.

Functional constraints:

2x₁+ x₂ ≤ 1000

X₂≤ 500

X₁ + x₂ ≤ 600

Goals:

1)    200 cp400 computers weekly:

X₁- d₁⁺ +d₁^-= 200

2)     500 total computers weekly:

X₁+x₂-d₂⁺+d₂ ^-=500

3)2500000 profit:

.2x₁ + .5x₂- d₃⁺ + d₃^-=2500000

3)    400 total man hours weekly:

X₁+ 1.5x₂-d₄⁺+d₄^- =400

Non- negativity constraints:

X₁,x₂,d_i⁺,d_i^-≥0 for all goals.

Objective function:

Priority 1: Minimize the amount the state contract is note met: min d₁^-

Priority 2: Minimize the number under 500 computers produced weekly : min d₂^-

Priority 3: Minimize the amount under 2500000 rs. Produced weekly. : min d₃^-

Priority 4: Minimize the man hours over 400 used weekly: min d₄⁺

Formulation summary:

Min. D₁^-+ d₂^-+ d₃^-+d₄⁺

s.t.c.

2x₁+x₂≤1000

x₂≤500

x₁+x₂≤600

x₁ –d₁⁺+d₁^-=200

x₁+x₂ –d₂⁺ +d₂^-=500

.2x₁+.5x₂-d₃⁺+d₃^-=250

X₁+1.5x₂-d₄⁺+d₄^-=400

X₁,x₂,d₁⁺,d₁^-,d₂⁺,d₂^-,d₃⁺,d₃^-,d₄⁺,d₄^-