# interpret reduced cost of all binary variables

I have prepared and run a Linear Programming model in SAS. I have some questions about the output that I can’t find answers to, and am hoping that someone can help.

My model contains the decision variables X1, X2 …………..X12, and all are binary. The model minimizes cost of inputs on a weekly basis over a period of 7 weeks.

There are constraints that restrict use of specific inputs to no m ore than in 2 consecutive weeks, for minimum levels of effectiveness, and others. An example constraint for week 1 was

0.598X1-1 +0.585X2-1+0.435 X3-1+0.372X4-1 +0.479X5-1+0.63X6-1+0.678X7-1+0.648X8-1+0.631X9-1 +0.48X10-1+0.144X11-1+0.207X12-1 ≥ 0.5

Same as constraints for week2 …..week 7. X1-1 indicates pesticide 1 applied in week 1.

The objective function: MinCost= C ∑(i=1)^12▒∑(j=1)^7 X(i-j)

In the solution, the variable summary from the SAS output is

                            Variable Summary


Variable name Status type pric Activity Reduced cost

X1-1 Binary 6.98 1 1.12

X1-6 Binary 6.98 1 5.69

X7-7 Binary 17.01 1 11.15

X8-3 Binary 12.95 1 7.09

X8-4 Binary 12.95 1 7.09

X9-2 Binary 1.44 1 4.57

X9-5 Basi Binary 1.44 1 0

I picked the variables from the summary output that had the value ‘1' in the “activity column”. Only one of the variables has status of ‘basic’, the others apparently being non-basic by default. When a variable is a basic variable, it is in the optimal solution, and its reduced cost is o. However, only one of these reduced costs is 0.

My question is

• does this output suggest an error in the program?

• if no, is this just the way that SAS presents output from the LP model?

• how do I explain and interpret these non-zero reduced costs for what appear to be basic variables?

Thanks for the help!

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You might have a better chance to get an answer if you improve the formatting. Especially the weird gray box character in the objective function is rather off-putting. – joriki Oct 1 '12 at 21:14