As we saw in the previous video, the sensitivity report is a very valuable tool for what-if analysis. We also saw that it has two main limitations. First, the report can only tell you what will happen to the optimal solution, when you make one change at a time and everything else remains the same. For example, we saw that the shadow price of a capacity constraint can tell you the reduction in total cost for every unit of additional capacity. That's great but it assumes that all other capacities are going to stay at the same current values. So, if we would like to know the effect on the total costs, when we change two capacity limits at the same time, we'll have to make the change in the model and resolve. The other limitation is that the prices and costs in the sensitivity report are only valid in a specified range. For example, this sensitivity report is the one that we generated in our last video for our transportation problem. Cell $G$16 refers to the total amount shipped from Supplier A to all customers. You can see that the total amount is 60, and that the capacity of Supplier A is also 60. The capacity is given by the right hand side of the constraint. The shadow price is -4, and indicates that for every unit of additional capacity, the total cost decr eases by $4. If this is all you are interested in evaluating in your what-if analysis, then the only thing that you need to remember is that this price is only valid within 20 units of increase. This is given in the Allowable Increase column of the table. What you're going to learn in this video is how to evaluate the scenarios that involve more than a single change, or where the change is outside the valid range of the sensitivity report. You will also learn how to use Excel charts to visualize sensitivity results. We will continue to use a transportation problem to illustrate these concepts. Let's assume that there is uncertainty in the demand of customer 4, and that we would like to see how the optimal solution changes with changes in this demand. Locate and open the Excel file Transportation Problem- Parameter Analysis. This file contains the transportation model that we have been discussing in this module. Note that the demand for customer 4 is now 40 instead of 65. This is because we have added the ASP function PsiOptParam. This function tells the software that we will like to solve the problem for values of this demand that vary from 40 to 90. If we go to the Analysis group of the ASP tab and click on Reports, we'll find Parameter Analysis on their Optimization reports. These dialog allows us to identify the variables and the objective function that we want in our report, and also the parameters that we want to vary. Since we want to analyze the changes in all the variables and the total cost, click on the double arrow. Also click on the double arrow for the Parameters, leave the rest unchanged, including the 11 major access points. By clicking OK, ASP generates the analysis report one time. This is the same table that is in the analysis report worksheet, included in the file. Which I have already formatted, and I have included for those of you without access to ASP. The table shows 11 solutions to the problem, 1 solution per demand value. The demand varies from 40 to 90, and as expected the cost increases with the demand. The most important insight from this analysis is that, some suppliers ship the same amount to the same customers regardless of the changes in the demand of customer 4. For example supplier D, ships four units to customer 1 in all the scenarios. Also supplier E only ships to customer 4. Shipments from supplier B are also pretty stable. Suppliers A and C experience the most changes, when the demand for customer 4 varies. It is possible to create more complicated scenarios using the parameter analysis tool, by adding more parameters that can change values. However, we're not going to cover that in this course. We have seen that the interpretation of the solver reports requires some technical knowledge of optimization terminology. You have learned how to interpret shadow prices and reduce costs, but you can't expect clients or other people in your organization to know these concepts. Data visualization can help present results in forms that are more understandable to those not familiar with optimization. Let's start with shipments. Instead of showing the results in a table, you can create a chart with one bar per supplier. The length of the bar is the total shipments sent by the supplier, and the colors represent the customers. From the supplier side, you can see that suppliers B, D and E supply to only one customer. From the customer's side, you can see that customer 3 receives units from three different suppliers. The chart gives you insights on the level of coordination that the solution requires. Clearly, it is easier to coordinate shipments from a single supplier to a single customer. Now, let's take a look at the capacity constraints. Instead of showing the values in the sensitivity report, you can create a bar chart that compares the total shipments and the capacity for each supplier. In this chart, orange bars indicate capacity and the blue bars indicate total shipments. The chart reveals that three suppliers, A, D, and E, utilize all of their capacity. On the other hand, suppliers B and C, have extra capacity. As you know, the sensitivity report includes reduced cost information. Reduced cost are a little more difficult to visualize. The key concept as we discussed in our previous video, is that at reduced cost tells you the amount by which a current cost needs to decrease, in order to make it attractive. For example, these tables shows the reduced costs for supplier A. Take a look at the first row which corresponds to A1. This is the variable that represents the units sent from supplier A to customer 1. In the optimal solution, A1 has a value of 15 units. The cost of sending these 15 units is $3 per unit as shown by the objective coefficient. The reduced cost in this case is 0. Because $3 per unit is already attracted to send units from supplier A to customer 1. Now take a look at A2. It's final value is 0. This means that at the current cost of 7, it is not attractive to send units from supplier A to customer 2. The reduced cost is also 7, so the cost has to drop to 0 for this option to be attractive. Well, this is probably not going to happen. So if everything else stays the same, sending units from supplier A to customer 2 is not a good idea. The best way to visualize this information is to convert all reduced costs to negative values. Then you can create a bar chart that shows the current costs as positive values, and the reduced costs as negative values. Do this only for those variables with a final value of 0. The chart shows that there are variables, such as A2, D2 and E3, that require very large cost reductions to make those shipment options attractive. It also shows that it would require a modest cost reduction for B4 and C4 to become viable options. In this video, you have learned how to do a parameter analysis to review changes in the optimal solution when key input data changes. Parameter analysis provides information about the robustness of the solution in the face of uncertainty of input values, such as customer demand. You have also learned how to present results from an optimization model to those who might not be familiar with prescriptive analytics concepts. This is pre-advanced analytics, and I'm glad that you're mastering all these material. By the way, if you want to see how the charts were created, you can review the Excel file Transportation Problem- Visualization.