Example of Goal Programming. - A Product Mix Problem.

 

                                                                 Data of a fertilizer company is shown in the Table below.

Item

Tons required to make one ton of
       Hi-ph                   Lo-ph

Maximum amount of item available daily(tons)

RM 1

RM 2

RM 3

          2                          1

          1                          1

          1                          0

1500

1200

500

Net profit ($)  per ton made

          15                        10  

 

     RM 1,2,3 are the raw materials required to make two products Hi-ph and Lo-ph.

In this problem there are two decision variables

                          x1 =  the tons of Hi-ph made per day.

                           x2 =  the tons of Lo-ph made per day.

 

Now the LP formulation of the fertilizer product mix problem is given as

 

                  Maximize            z(x)  =  15x1 + 10x2                                        Item

                   Subject to                        2x1   +   x2     ≤    1500                      RM 1

                                                            x1    +   x2     ≤    1200                      RM 2

                                                            x1                  ≤      500                      RM 3

                                                            x1 ,      x2      ≥     0

Now this is a single objective problem of maximizing z(x). If there are more objective to be consider like Market shares, public's perception of the company. So there may be more objectives like

                        c1(x)  =  z(x)  =  15x1 + 10x2

                        c2(x)  =  300x1 + 175x2

                         c3(x)  =  300x1.

 all c1(x), c2(x), c3(x) are to be maximized. Then it is multiobjective problem.

 

Consider the multiobjective problem of the fertilizer manufacturer in which we will measure the net daily profit by

                            c1(x) =  (15x1 + 10x2)    net daily profit in dollars

                            c2(x)  =  (x1 + x2 )tons    the market share, the total tonnage of fertilizers sold daily by the company

                             c3(x)  =  x1tons               the tonnage of Hi-ph sold by the company.

                             All these objective functions to be maximized.

                             Company has decided to set a goal    g1 = $13,000, g2 = 1150 tons and  g3 = 400tons for  c1(x), c2(x) and c3(x) respectively.

                              The penalty coefficient associated with shortfalls in these goals are required to be 0.5, 0.3, 0.2 respectively.

 

                  With the given data the goal programming formulation of these problem is

                   Minimize                                  0.5u1-          +   0.3u2-            +  0.2 u3-    

                   Subject to    15x1 + 10x2    +   u1+ - u1-                                                               =    13,000

                                          x1  +    x2                                     +    u2+ - u2-                                       =    1150                                        

                                                       x1                                                                                         +      u3+ - u3-               =    400

                                        2x1   +    x2                                                                                      ≤    1500

                                          x1    +    x2                                                                                     ≤    1200

                                          x1                                                                                                   ≤      500

                                          x1 ,         x2  ,     u1+ , u1-,        u2+ , u2-,              u3+ , u3-           ≥     0

 

An optimum solution of this problem is x = ( x1 ,   x2 )T   =  (350 , 800 )T   . x  attains the goal set for net daily profit, the total fertilizer tonnage sold daily; but falls short of the goal on the Hi-ph tonnage sold daily by 50 tons.

 

Goal Programming