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^-

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STEPS TO SOLVE LINEAR PROGRAMMING PROBLEMS

LINEAR PROGRAMMING: 
Linear programming (LP, or linear optimization) can be defined as a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships.
Generally, there are two type of problems:

1. MINIMIZATION PROBLEMS : ex cost minimization, operation time minimization etc.
2. MAXIMIZATION PROBLEMS: ex profit maximization, sales maximization etc.

LIMITATION:

       A LP problem with only 2 decision variables can be solved using graphical method.

STEP 1: Formulate the data given in the question.

A) List out the decision variables :
                                                  The variables for which the problems are asking solution for are called as decision variables.

B) Define the OBJECTIVE FUNCTION :
                                                             It is a linear function of the decision variables expressing the objective of the decision-maker.
C) List out the constraints :
                                            This are nothing but the limitations that a problem holds. Also define non- negativity constraints.

STEP 2:  Plot the graph according to the equations formed for the constraints.

STEP 3: Shade/ mark the regions based on the given conditions.
             
          case 1: equations with (>=  some value) condition are shaded away from the origin.
        case 2: equations with (<= some value) condition are shaded towards the origin.

STEP 4: Look for common region i.e the feasible region or region covered by all the constraint equations.

STEP 5: Find the corner points coordinates of the common/feasible region.

STEP 6: Put the corner point values in the objective function 

STEP 7: Compare values of objective function calculated using step 6.

STEP 8: 
IN MINIMIZATION PROBLEM: Consider the coordinates of the point which gives minimum objective function value.

IN MAXIMIZATION PROBLEM: Consider the coordinates of the point which gives maximum objective function value.

THE COORDINATE POINTS ARE THE SOLUTION TO THE PROBLEM.

EXAMPLE:
London Tourister is a manufacturer of travelling bags and its distributor has agreed to buy all the bags it produces for next 3 months. The bags have to undergo 4 processes as shown below.

Department	Standard bag (hrs/bag)	Deluxe bag (hrs/bag)
Cutting & dyeing	7/10	1
Sewing	½	5/6
Finishing	1	2/3
Inspection & Packaging	1/10	1/4

LT's manager has analyzed the constraints in production and estimates that 630hrs for cutting & dyeing, 600hrs for sewing, 708hrs for finishing, 135hrs for inspection & packaging will be available for production of traveling bags during the next 3 months. The profit contribution of each std. Bag is Rs.10 and rs.9 for each deluxe bag. Formulate the problem to determine the no. Of std. Bags and no. Of deluxe bags LThas to mfg in order to maximize the profit contribution.

Solution:
STEP 1: Formulate the data given in the question.
A) List out the decision variables :

1. S= No. Of standard bags 
2. D= No. Of deluxe bags

Lines from question:Formulate the problem to determine the no. Of std. Bags and no. Of deluxe bags LThas to mfg.

B) Define the OBJECTIVE FUNCTION : 
 _{10S +9D}

lines from question specifying objective function: In order to maximize the profit contribution.
                                                            

C) list out the constraints :

                                    

    7/10S + D <= 630

    1/2S + 5/6D <= 600

     S + 2/3D <= 708

    1/10S + 1/4D <= 135 ; 

    
    S,D >= 0 (NON NEGATIVITY CONSTRAINTS)

Lines from question : LT's manager has analyzed the constraints in production and estimates that 630hrs for cutting & dyeing, 600hrs for sewing, 708hrs for finishing, 135hrs for inspection & packaging 

AND THE VALUES FROM THE TABLE.


FOLLOW STEP 2 to STEP 8: 

ANSWER = (S=540, D=252)
TOTAL PROFIT = 7668 

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STEPS TO SOLVE TRANSPORTATION PROBLEMS.

AIM : To choose a suitable transportation route in order to incur minimum transportation cost while fulfilling demand.

STEP 1: Check the type of problem is it balanced or unbalanced.

STEP 2: If unbalanced, balance it by adding demand excess or supply excess column and row respectively. Otherwise continue with 3rd step.

STEP 3: Find basic feasible solution through any of the three methods

1. NORTH WEST CORNER METHOD
2. LEAST COST METHOD
3. VOGEL'S APPROXIMATION METHOD

STEP 4 : Check for degeneracy (no. Of basic variables = m+n-1)

STEP 5 : Remove degeneracy.

STEP 6 : Calculate optimum  feasible solution using Modi method.

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