2. Formulation
Decision variables: Unknown production quantities X1& X2are decision variables.
Objective function: 60X1 + 80X2 = G (G is the total cost per day which is to be minimized)
Constraints:
20X1+30X2≥ 900 ……….(1) Vitamin A Constraint (at-least) 40X1+30X2≥ 1200...(2) Vitamin B Constraint (at-least) X1, X2 ≥ 0………Non-negativity constraint
3. Standard form: The mathematical formulation suitable for simplex method is written in terms of decision variables, surplus variables, artificial variables and their coefficients. While X1& X2 are decision variables, S1 & S2are surplus variables introduced to established equality between LHS & RHS, and A1& A2 are artificial variable required as per simplex method. A1& A2serve as artificial slack variables.
Objective function: 60X1+ 80X2+ 0S1+0 S2+ A1+A2= G (G is the total cost per day which is to be minimized)
Constraints:
20X1+ 30X2- S1+ 0S2+A1+0A2 = 900 ……….(1) Vitamin A Constraint (at-least)
40X1+ 30X2+ 0S1- S2+ 0A1+A2= 1200..…….(2) Vitamin B Constraint (at-least)
X1, X2, S1, S2, A1, A2≥ 0………Non negativity constraint
4. Starting feasible solution: Put all non-slack variables equal to zero, then
X1=0, X2=0, S1=0, S2=0; A1=900, A2=1200.
5. Structuring of Simplex table: setup the simplex table as per the guide lines already laid down. Put the starting feasible solution in the first simplex table ST1.
Phase I:
Objective function: Maximize G* = 0X1 + 0X 2+ 0S1+0 S2- A1 - A2 to drive out A1& A2
The simplex table ST1is constructed as per the guidelines ST1
Contributions >>> 0 0 0 0 -1 -1 C
The above table doesn’t give optimal solution as all NER values are not non positive. As per the methodology first iteration is performed and next table ST2is written.
ST2.
Contributions >>> 0 0 0 0 -1 -1 C
The above table doesn’t give optimal solution as all NER values are not non positive. As per the methodology first iteration is performed and next table ST3 is written. As A1 & A2 are already driven out the columns of A1& A2are dropped.
ST3.
Contributions >>> 0 0 0 0
C
It can be seen that the solution in ST3 is optimal as all NER values are non-positive.
Phase II:
The solution in ST3 is now minimized in Phase II. Objective function is now changed by rewriting the coefficients of variables. The artificial variables are already driven out.
Objective function: Minimize G = 60X1+ 80X2+ 0S1+0 S2
ST4.
Contributions >>> 60 80 0 0 C
It can be seen that the solution in ST4 is optimal as all NER values are non-negative.
Learner may put the solution values of basic variables in the objective function and find the optimal minimum cost.
Objective function: 60X1 + 80X2 = G (G is the total cost per day which is to be minimized). Substituting the values of the decision variables,
60 X 15 + 80 X 20 =900+1600 = 2500. As the cost contributions are written in paise, the total minimum cost is Rs
25/-4.5 SENSITIVITY ANALYSIS OF OPTIMAL SOLUTION
Post optimality analysis is a comprehensive study of an optimal simplex solution to interpret its features in order to solve a business problem. The post optimality analysis consists of interpretation of the optimal solution and sensitivity analysis.
Interpretation of the optimal solution:
Basis of the optimal solution: the basic variables (unknown quantities) occupy the basis and solution values column gives the respective values of the variables. Any variable not in the basis (non-basic variable) is zero.
NER values: NER values of decision variables are opportunity costs. A negative opportunity cost for a decision variable in a maximization problem (when ∆ ═ C-Z) indicates the amount by which the total maximum profit (Z) would come down if a unit of this particular product is produced.
NER values of resources represented by slack variables/surplus variables are called Shadow prices or unit worth of resources. In a maximization problem (when ∆ ═ C-Z), negative NER values for slack variables indicate that the resource is scarce or just enough.
Special cases of optimal solution: this is discussed separately after sensitivity analysis
Sensitivity analysis:
Sensitivity analysis is testing the optimal solution to see within what range of changes in the environment it remains use full. The changes in the environment bring pressure on the basis of the optimal solution and solution values to take up new values. If the basis changes the optimal solution no longer is the same. If the basis remains the same and only the solution values change, the optimal solution holds good. The sensitivity analysis which is being discussed here considers following changes:
a. changes in the unit contributions or Cj values (changes in coefficients of decision variables in the objective function)
b. changes in limits on the resources (bivalues in the first solution) c. plan to introduce a new product
When the above changes take place in the market after the optimal solution is established, sensitivity analysis (also known as post optimality analysis) is performed.
In the current context, post optimality analysis which includes interpretation of the optimal solution is also discussed.
Specific business cases are taken up to discuss interpretation of the optimal solution, sensitivity analysis and special cases of the simplex solution.
4.6 STUDY OF SPECIAL CASES OF SIMPLEX SOLUTION
Following cases are said to be special cases of simplex solutions.
Each case has its own features which are important for its applicability to a business problem. These cases are discussed in previous chapter # 3 1. Alternate optima: Alternate optima to an optimal solution are other optimal solutions which also give the same optimal value of the objective function. An optimal solution which doesn’t have alternate optimum is called a unique optimal solution.
Test for uniqueness of an optimal solution: if one or more of non-basic variables have zero value in NER then one or more alternate solutions exist. To find the alternate solution, the column of the non-basic variable with zero value in NER is treated as key column and θ values are calculated. Key element is identified and the iteration is carried out to improve the solution. The optimal solution found after necessary iterations is an alternate one.
2. Unbounded solutions: when least non-negative replacement ratio (θ) is