Using Weighted Criteria to Make Decisions
Date: 07/18/2008 at 19:37:32 From: AS Subject: Finding a result by assigning weights Hypothetical Example: I want to award my employee for highest attendance and learning new skills. I have 3 employees who worked the following days in a month: Employee 1: 10 days Employee 2: 15 days Employee 3: 20 days Employee 1: Learned 2 new skills Employee 2: Learned 2 new skills Employee 3: Learned 1 new skill Weighting: 50% for attendance and 50% for new skills learned therefore best employee is employee 3 Employee 1 score is .5(10)+.5(2) = 6.5 Employee 2 score is .5(15)+.5(2)= 8.5 Employee 3 score is .5(20)+ .5(1)= 10.5 Therefore Employee 3 gets that best employee award. I somehow feel this is not the right method of calculation. Am I able to calculate averages as above where skill and attendance are apples and oranges (different)? I looked up weighted average but that only shows calculating weights for one type of quantity.
Date: 07/19/2008 at 22:34:47 From: Doctor Wilko Subject: Re: Finding a result by assigning weights Hi AS, Thanks for writing to Dr. Math! This is a very practical question in business and government. Often a decision maker has to prioritize many alternatives (picking which employee gets the quarterly award), where the candidates can be ranked by multiple criteria (attendance record, skills learned, etc.). If you were prioritizing your employees strictly based on attendance, Employee 3 gets the reward because he has the highest attendance. But, you actually have two criteria to rank your employees against: attendance record and skills learned. Now it's not so clear if Employee 3 still deserves the award. He does have the highest attendance, but he only learned one new skill. Employee 2 on the other hand learned two new skills, but worked five less days. Now it's getting a little fuzzy on which employee should be rewarded. Now suppose you have 20 employees and there are four or five criteria to pick an award winner; now the selection is almost impossible to do fairly. This is where the decision maker needs some objective method to help him make a fair decision. Your decision is to pick the award winner from the three employees, based on two criteria, attendance record and skills learned. If we use a decision tree to model your problem, it looks as follows: Attendance Skills Days Learned / E1 10 2 / / Decision ----- E2 15 2 \ \ \ E3 20 1 There are two main components we'll look at to model your decision. 1. You have to measure all criteria on similar numerical scales (to compare apples to apples), and 2. You have to assign your importance (weights) to those criteria that will be used to rank the employees. Once you have these two components, you can calculate a weighted average for each employee. This final weighted average will allow you to rank your employees in order to determine who gets the reward. Let's do these two steps now. =================================================== 1. Measure all Criteria on Similar Numerical Scales =================================================== The first thing we'll do is use the same scale to measure both criteria, say a 0-100 scale, where 0 is the worst or least desired outcome for each criterion and 100 is the best or most desired outcome for each criterion. ------------------------ Criteria #1, Attendance: ------------------------ Assume for attendance, 5 days is the worst outcome and 30 days is the best outcome for your employees. 5 days of attendance gets mapped to the score 0 and 30 days of attendance gets mapped to a score of 100. To get the scores for the intermediate days, you could use "proportional scoring" as one technique. For example, Employee 1 is 20% of the way from the lowest to the highest value [(10-5)/(30-5)=0.20, and 100*0.20 = 20%], so Employee 1 gets a score of 20 for his 10 days of attendance. Likewise, Employee 2's attendance gets mapped to a score of 40, and Employee 3's attendance gets mapped to a score of 60. Remember, we're trying to map days of attendance onto a scale from 0-100. The mapping looks as follows: E1, 10 days --> score of 20 E2, 15 days --> score of 40 E3, 20 days --> score of 60 --------------------------- Criteria #2, Skills Learned: --------------------------- Now, let's map the number of skills learned onto the same 0-100 scale using the same proportional weighting technique. First set your endpoints of the scale. 0 skills learned gets a score of 0 and say 4 skills learned gets a score of 100. Now, using proportional weighting the mapping looks as follows: E1, 2 skills --> score of 50 E2, 2 skills --> score of 50 E3, 1 skill --> score of 25 Are you seeing how we can compare apples to apples? I have taken two completely different criteria and put them on the same scale so comparing them makes sense. Now, if I re-draw my decision tree from above, it looks like: Attendance Skills Score Score / E1 20 50 / / Decision ----- E2 40 50 \ \ \ E3 60 25 ======================= 2. Weight the Criteria ======================= Next, you have to decide what's more important between attendance record and skills learned. Is a score of 50 on the attendance scale the same as a score of 50 on the skills scale? It might be, but this is where you as the decision maker assign your importance to the criteria. Importance is assigned to the criteria through weights. These weights will allow you to calculate a weighted average using the two criteria to get an overall score for each employee. Then you can rank the employees by the overall score to determine who gets the award. It is your call concerning the relative importance of the two criteria. It is important however to consider the ranges of the criteria. The weights should reflect the relative value of going from best to worst on each scale. For example, if improving attendance from 5 days to 30 is three times more important than going from zero skills learned to four skills learned, then this implies you weight attendance = 0.75 and skills = 0.25. But to go with your proposal that the criteria are equal, you'd weight attendance = 0.50 and skills = 0.50. Now with your weights chosen, you can finally calculate the weighted averages to get the overall score of each employee. The decision tree will look as follows: 0.50 0.50 Attendance Skills Overall Score Score Score / E1 20 50 = .50(20)+.50(50) = 35 / / Decision ----- E2 40 50 = .50(40)+.50(50) = 45 \ \ \ E3 60 25 = .50(60)+.50(25) = 42.5 Now you can rank your employees according to the final weighted average to see who shall receive the award: E2 = 45 (highest overall score) E3 = 42.5 E1 = 35 (lowest overall score) I'd say before accepting this as the final answer, especially when you are building the model the first time, you'd step back and look at the scales you constructed and the weights you applied to the criteria. This is called "Sensitivity Analysis". It is easy to see that if Skills was weighted higher, Employee 3 may have come out the winner. Also, if the Skills score was a little higher than 25, Employee 3 might have received the highest overall score. Just be aware of this as you construct your scales for the criteria and as you assign weights to the criteria. Model building is usually an iterative process, but once you've tested your model and you feel confident it has captured your priorities, you can use it objectively to rank all your employees to see who gets the award. I think this is where the value of this method lies; in the end you've built a transparent, standardized, and repeatable process for selecting employees for the award. If you're interested in this topic, try looking up articles related to decision analysis, value-focused thinking, and multiple-objective decision making to name a few. Does this help? Please write back if you still have questions. - Doctor Wilko, The Math Forum http://mathforum.org/dr.math/
Search the Dr. Math Library:
Ask Dr. MathTM
© 1994-2015 The Math Forum