Specify tolerance

After you set your tolerance for an answer, preview the question and do the problem as your students might. Try answering using one or two fewer decimal places than the output variables use to see how much the final answer varies. Test a few instances to determine the maximum required tolerance.

Creating an additional set of variables when rounding an intermediate calculation can cause the value of the final answer to differ widely. This problem often occurs when the final answer is a small number. For example, an answer that is supposed to be 1.24 can become 3.5 if an intermediate calculation is done slightly differently.

To create an additional set of variables, duplicate the intermediate calculation variable that is formatted to, say, 2 decimal places, and format the copy to only 1 decimal place. Then, duplicate the final answer output variable, substitute the new variable, and then make the new variable a second possible solution for the free response. When you do this, you can specify a smaller tolerance—something like 0.02 to 0.05.

If your question includes multiple interdependent responses, the tolerances you set for each response must take into account the tolerances allowed for contributing responses. So if your question includes two responses and the answer to the second response depends on the answer to the first, your tolerance for the second response must be large enough to encompass the tolerance you set for the first response. For example, if the answer to the first response is 10.5 and has a tolerance of .5, then 11 is accepted as a correct answer. You then need to make sure that the tolerance for the second response is large enough to accept 11 in the intermediate calculation.

To help you set tolerances for different types of questions, here are some of the tolerances typically used for publisher questions. You can adjust the recommendations to match the number of decimal places and other factors in your own questions:

• Operations and productivity: Productivity calculations normally have a tolerance of about 0.05. Percentage changes should be about 0.5 and above, depending on how much tolerance you allow for individual productivity calculations.
• Project management: Most of these calculations don’t require tolerances. Project lengths, slack times, total costs, and so on are normally simple addition or multiplication of whole numbers. Tolerances for variance and expected time should be about 0.1-0.2. Probability calculations should have a tolerance of about 0.05.
• Forecasting: Three-week moving averages, exponential smoothing, and so on usually need tolerances of about 0.2-0.3. MAD and MAPE need larger tolerances—somewhere between 0.5 and 1.
• Designing goods and services: Most of these calculations don’t require tolerances. If the question asks students to compare options and choose the best, make sure that the output variables exclude ties so that companies don't have the same costs/profits, and so on.
• Statistical process control: UCL and LCL usually have a tolerance around 0.02-0.03 (for X and R) and 0.01 (for P).
• Process strategy: Most break-even points don’t require tolerances, but you can specify a tolerance of 1 if it makes sense for your question. If the question asks students to compare options and choose the best, make sure that the output variables exclude ties so that companies don't have the same costs/profits, and so on.
• Capacity planning: Utilization rates and efficiency rates normally have tolerances of 0.1. Break-even units often don’t require tolerances, but you can specify a maximum tolerance of 1 if it makes sense for your question. Break-even dollars could require a large tolerance based on the tolerance for the units.
• Location strategies: Weighted distance scores are normally 0.05. Center of gravity coordinates should be about 0.1. If the question asks students to compare options and choose the best, make sure that the output variables exclude ties so that companies don't have the same scores.
• Layout strategies: Tolerances for theoretical and actual efficiency should be about 0.1. Make sure that your output variables are set up so that immediate predecessor relationships do not conflict. If you set up your output variables so that two activities can be logistically combined, make sure you account for that possibility in other areas of the problem.
• Work measurement: The normal time for a process should be 0.01 or 0.02. Standard times are usually about 0.05-0.2. (If you have to multiply by a unit of time to convert from, say, seconds to minutes, this tolerance may need to be larger). The calculation of necessary number of observations is very dependent on the rounding of intermediate calculations are rounded, so a tolerance of 2-5 might be necessary.
• Supply chain management: Tolerances for inventory turnover, weeks of supply, and weighted scores should be about 0.05. Make sure to check for ties in the output variables.
• Outsourcing as a supply chain strategy: Tolerance for weighted scores should be 0.1. If the question asks students to compare options and choose the best, make sure that the output variables exclude ties so that companies don't have the same scores.
• Inventory management: The tolerance required for inventory management depends on the size of the numbers. Make sure that your calculations for holding costs and ordering costs are large enough to reflect the tolerance you put on your EOQ. Also, if calculations require using the standard normal table, make sure these tolerances are large enough to incorporate a rounded table value as well as the exact value.
• Aggregate planning: Most of these calculations don’t require tolerances. However, be very careful when setting up your output variables. You might need to create arrays to deal with the many possibilities that affect such decisions as when to use overtime and when to hire or fire.
• MRP and ERP: Most of these calculations don’t require tolerances. However, be very careful when structuring tables.
• Short-term scheduling:Tolerance for average tardiness should be about 0.1; percentage utilization should be about 0.5. Make sure no ties occur when calculating optimal assignment to jobs.
• JIT and lean operations: Use a tolerance of 0.5 or 1 for the kanbans. Tolerances for setup costs and setup times should be about 0.2-0.5.
• Maintenance and reliability: Tolerances for percentages should be about 0.5. Tolerance for costs of maintenance policy can vary widely depending on the values in the question and the tolerance for the expected number of breakdowns.
• Decision-making tools: Tolerance for EMV should be about 1 or 2, depending on the values in the calculation. Watch out for tied values.
• Linear programming: After you set the tolerances for optimal X and Y, make sure the optimal value for Z has a tolerance large enough to incorporate these tolerances. If the numbers for X and Y work out well, a tolerance of 0.1 should work for Z. If these values are an intersection point, you might need to use a tolerance of 0.5. When setting up the variables, make sure that none of the constraints is parallel to the objective function and that lines always cross in the positive quadrant.
• Transportation models: Tolerance for total costs should be about 1-5.
• Waiting-line models: Specific tolerances vary a great deal depending on the values in your question. Your tolerances need to allow for using a formula from scratch and for taking an answer from a previous calculation and multiplying it by a given number.
• Learning curves: Tolerances for learning rates should be about 0.5. Tolerances for time required for the units and cost of the units vary based on the size of the values in the problem.