In addressing this issue, I believe that the leadership style that best fits this situation is GII. With this method, you as a manager share the problem with you subordinates. You ask their input on the situation and attempt to reach an agreement about a way to solve the problem.
As the manager in this situation, I feel it is necessary to increase group participation. Without group participation, you have no use for GII because it is based on gathering a bunch of information and coming up with the best possible, most accepted decision.
There are many factors to take into consideration when choosing this particular managing style: the first a foremost being that the group actually participates in solving the problem. Even though the division manager and I agree that there is a problem with production, this concern needs to be felt throughout the group as a whole. A good way to get your subordinates involved in offering up solutions to the production problem would be to offer then incentives. Set production goals for your subordinates and see if they can reach them. If and when these goals are met, you can reward with incentives. The incentives can range anywhere from monetary incentives to extra vacation time. By not putting a limit on what you are willing to offer, you do not exclude any employee to work hard to reach their goals. Because if extra time off is important to one employee it may not be important to another. By putting extra incentives on the table, you should increase the need/want for your subordinates to contribute and help solve the problem at hand.
Another good thing about implementing GII is that your subordinates get a chance to be involved in the decision making process. When employees are involved in making a decision it is often easier to put into action the decision that was made. There is sometimes a higher acceptance of the decision when it comes form the g
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