Lecture 13 - Decision Tree Example

Welcome friends. In our previous sessions, we discussed that how we have to take different types of decisions in uncertainty. Uncertainty is a very common issue in various operations management cases, right from the forecasting, where we are dealing with lot of uncertainties. We discussed different types of possibilities of uncertainties and we also discussed that how we can best use analytics for helping us in dealing with those uncertainties. Then we are also talking of inventory. In last three four sessions, we are seeing that how different types of uncertainties from the demand side, from the supply side are possible and you have to take those uncertainties in light of the customer service you want to give to your customers. So, uncertainties are inevitable and you have to handle your case in these uncertain environment. In last session, we introduced the concept of decision tree that when there are multiple possibilities, you can use those multiple possibilities for having a good prediction and for that purpose, we introduced the concept of expected value also and we saw that how in a very systematic manner, we can actually calculate the impact of our decisions. You have decisions and then there are different possibilities which are possible from that decision. Now, in this particular session, we are going to discuss with the help of a detailed example that how decision tree is used in decision making. The name itself is decision tree. So, you can understand that it is coming for the decision making purpose. So, we are going to discuss with the help of this simple example because the theory of decision tree, we have already covered in our previous session. Now, here in this case, a company is going to open a particular factory. So, you can say that the case is related to facility decision and these decisions, facility decisions are very common in our supply chain management, in network management because generally you have to take 2, 3 different types of decisions whenever you are deciding about the facility. One is location, second is size and nowadays, one more decision is related to facility decision that is ownership. So, these types of decisions are very common when we are talking of facility decisions. Where to locate your facility? The facility should be of big size, medium size, small size, whether I should take the facility from the spot market or I should go for a lease agreement or I should construct my own facility. So, all these are the issues related to ownership also. So, these are the three very critical issues and then there may be few also, few more also, but these are beyond our scope of decision tree. For example, whether you want to have a general purpose facility or a special purpose facility. General purpose facility gives you more flexibility while special purpose facility gives you minimum flexibility, but the economies of scale, operating cost, these are more in case of special purpose facility because you have one type of equipment, one type of stacking arrangement that is very efficiently used for all the items which you are stocking in that facility or all the items which you are producing from that facility. When you have general purpose facility, you may go for frequent change overs and frequent change overs may give you flexibility, but the operating cost may be relatively higher. So, efficiency will be low in case of general purpose. So, depending upon your business interest objective, the kind of SKUs you have in your offering, you may decide whether to have general purpose or special purpose facility. Location size ownership, these are the issues which we are going to discuss with the help of decision tree. Now, in this particular example which I am discussing, the current demand for a particular product in the market is 500 units which can go up and down by 20 percent. Now, if you remember, we discussed the case of binomial distribution while preparing the decision tree. So, 500 is the current demand and it can go up and down as per 20 percent plus and 20 percent down with equal probabilities. That means, 0.5 percent, 0.5 and 0.5, it can go plus 50 percent, plus 20 percent, it can reduce by 20 percent also. Then it says the retail price of the product is 4 dollars and that they can set up the factory that will cost them 2 dollar per product. So, current price is 4 dollar, cost is 2 dollar. They can also subcontract at the rate of 3 which can also go up and down by 20 percent. Now, it is possible that either the cost is own, 3 is subcontract. This is also liable to change with 20 percent plus, 20 percent minus for the and with equal probabilities. For the coming 2 years, they have to decide which option will be best suited for them and the discount rates over the period is 10 percent. So, let us see what are the different options available to them which are there for making this choice. So, 3 options are suggested, get all the demand from the spot market as needed. So, you can get all the items, you have 500 units requirement and buy all 500 from the spot market and that is one option. Option number 2, you set up your own factory and get additional because your factory will have a limited size. Right now, factory is let us say of 500 units, factory size. In next period, if demand increases by 20 percent that means next period demand becomes 600. So, you have 500 from factory plus 100 from spot market. So, this is option number 2. Option number 1 from spot all 500 and third is you are going in a contract with some party where you have a lease arrangement with a minimum change that allows variable uses of the subcontracting with additional requirement from the spot market. So, you have the lease agreement where we have, you are going to use some minimum value of the facility and if you are going for extra requirement, you are going to take that from spot market. So, these are the 3 options. One is have your own factory and then additional requirements coming from the spot market. Another is 100 percent requirement from the spot market and third is have a facility which is a contract facility, but if you require more items, you can go to the spot market. So, 2 or 3 are more or less similar. The only difference is in 2, you have the ownership, in 3, you are taking a lease agreement. Now, with these inputs, let us see the data which are given to us. Current demand is 500 units per year. As I already mentioned, this is binomial uncertainty. Demand is going up and down by 20 percent plus minus with for plus also 20 percent, for minus also 20 percent equal probabilities. Lease price is 2 dollars per unit, spot market price is 3 dollars per unit, spot prices are also liable to change with plus minus 20 percent. If it is going up, again the equal probability and down also. You are getting a revenue of 4 dollars per unit and the discount rate is 10 percent. Discount rate 10 percent means, I hope most of you are aware of time value of money. So, that means, whatever is 100 rupee today, the value of that 100 rupee tomorrow will be 90 rupees. What is 90 rupee tomorrow, day after tomorrow, their value will become 81 rupees. So, discount factor means, the value is depleting over a period of time, the time value. We all know that the value, the purchasing power of the money decreases with time. So, that factor at which the purchasing power of the money is reducing is denoted by discount rate. So, as we are now able to see that, there are two important things. One is, two changes are happening. One changes in the demand and another changes in the spot prices, one factor, two factor. And these two factors can be represented as a decision tree. Starting point 500, demand can increase, demand can decrease. And then again, from here also, demand can increase, demand can decrease, like that. And every time, things are happening with equal probability. Now, and same thing is happening in the subcontracting price also. Right now, the subcontract price is 3 rupees, prices may increase by 20 percent, prices may decrease by 20 percent with equal probability and so on. Now, we need to put all these information in a single decision tree diagram. And let us see, what is the shape of that decision tree diagram that comes here in front of you. We have two variables, one is this 500, another is this 3. So, we are starting with this 500 and 3 in the beginning. 500 is demand and 3 is price, price of the spot market. Then there are possibilities that, demand is also increasing and price is also increasing. So, now you see, there are four possibilities, demand increase, price increase, demand decrease, price decrease, demand increase, price decrease and demand decrease, price increase. Now, probability of each phenomena is 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, that both these things will happen. Therefore, combined probability will be 0.5 into 0.5, 0.25 and same 0.25, 0.25, 0.25. So, there are four possible scenarios, two where demand is increasing and two where demand is decreasing and that is where you have four scenarios, 0.25, 0.25, 0.25, these are the four probabilities. You please remember that, you can see or you can check your decision tree, that the sum of all these branches, the probability associated with all these branches should be equal to 1, sigma probability should be equal to 1. Now, when demand is increasing, price is also increasing, it becomes 600, 3.6, when demand is increasing, but price is decreasing, 600, 2.4, then demand is decreasing, price increasing, 400, 3.6 and demand is also decreasing, price is also decreasing, that is 400, 2.4. So, this is, let us say our t0, this becomes t1 and this becomes t2 and similarly, I request all of you, that you can make t3 also. Now, here you see that, because four different branches are coming from t0 to t1, so you have four alternatives. Now, every node in t1 is giving four, you can say branches. So, ideally 4 into 4, 16 branches are there, but when you see this t2 situation, only 9 exclusive alternatives are there, you can count 1, 2, 3, 4, 5, 6, 7, 8, 9. So, actually 9 alternatives are there, which are 9 unique alternatives are there. So, on the same lines, both can increase, both can decrease, one can increase, another can decrease, one will decrease, another will increase, these are the alternatives, which are available. So, in this way, more branches of this decision tree are possible, people can expand it for t10 or even more periods. Now, after understanding this initial construction of decision tree, let us see, how are we calculating profit related to decision tree, using this decision tree activity. Like, for the period number 2, if we calculate the profit at each node, now period number 2 here, this is the period 2. In this case, demand and price are all these different combinations, demand is 720, price is 4.32. So, total cost associated with this 720 and 4.32 is, 720 is this and the price of not assigning a lease and obtaining all the warehouses space from the spot market. So, the spot market price is 4.32, you are using option 1, where you are taking all the requirement from the spot market and therefore, 720 multiplied by 4.32, the total cost is this much. The total revenue will be, the selling price is 4 rupees, yes, selling price is 4 rupees, 4 dollars. So, our revenue will be 720 into 4, 2880. So, in fact, it is a case of loss, it is a case of loss, that you are getting a loss of 0.230. So, if impact I want to calculate for is this, if this decision is coming for all spot market. So, here the impact is minus 230.4 and similarly, you have to calculate, you have to calculate for all the spot market related issues, it is possible that you have at the market demand increases, but the price decreases, it remains 2.88 for some reason, I do not know, but it will be 720 into 2.88, but our revenue is this much 2880. So, in this case, it is easy to understand that there will be a positive, profit will be positive in this case. So, we have to calculate the impact that if we are going to buy all the space in the spot market, what will be the impact of this. Now, that impact is coming with a particular probability, this it is not that you are directly getting this impact, because 25 percent probability that demand will increase, price will also increase and then again 25 percent probability of getting a impact of minus 230.4. So, these probabilities will multiply and then you will get this expected impact. So, expected profit is therefore, calculated for all such you can say possibilities, then we are going to get the expected impact that what is the total expected impact for our this particular decision. Now, another situation you can consider that I am going to take a limited amount of space from the lease market and then I will see whether that is profitable or not, whether the expected impact of that particular situation is better or not. So, in this way we are calculating profit or loss for each of these nodes and we can continue this calculation for all these nodes. Now, these are happening for period 2 and these are in reference of this decision of period 1. The decision of or the situation of 603.6, this is giving you the decision of period 2, 4 options and then you have calculated the calculations for option 1, 2, 3 and 4. We will discuss how to summarize these calculations in few minutes. Now, we are doing this discussion when we are having all the requirement fulfilled from our spot market, but as I discussed in the beginning that we have 3 options and if let us say I want to compare between these options that whether, so our decision tree will come like this, you have a present situation, you take spot market, then you take lease and then you take ownership. Whether you are doing spot, lease or ownership, demand may increase, demand may go this way, this way, there are 4 options which are possible. Same 4 options are possible here also and same 4 options are possible here also. This is happening in T1 and same T1 will result into T2, 4 options are possible here also. So, 4 options like this and so on. So, now the calculation which we were showing you are only for this part, that is what we are showing and when you have to compare, you need to do entire calculation of spot versus lease that if I am doing the spot calculation, my expected profit or expected impact I can I will say because profit will be possible and loss is also possible. So, in that case what is your expected impact in case option 1, expected impact for option 2 and option 3 and all these impacts are going to happen in future after second period, after third period, after fourth period, as many periods you want to discuss depends on you. So, we have to do a backward calculation also that what is the current impact of these future cash flows which are going to come to us because decision I am going to make today. So, therefore, I consider the concept of time value of money also and in time value of money we are considering that how a future cash flow which is going to come to us will have some value at the present time because you have to do some investment today whether you are building your own factory or you are doing some lease agreement. So, you need to pay today some amount and you need to see that whether it is profitable or it is less profitable accordingly you will be making a decision whether to go for a spot market, whether to go for lease market, whether to take for your own ownership. Now, we discussed one type of case which is all the requirement going to be fulfilled from the spot market. Take another consideration for the comparison purpose like we are going to make a plan of capacity 500 and rest would have been bought at the revised prices. Now, the same kind of evaluation process we are going to follow what we did for the spot market case. Now, in this case take a note of 720 432. Now, when the demand is 720 you are having a capacity of 500 on your own. So, you have to buy only this additional capacity of 220 at the rate of 4.32 and remaining you have on your own, but for developing this capacity there will be some initial capital investment you have to do and therefore, the concept of time value of money comes into the picture. For example, if I am going to buy everything from the spot market there will be no need of time value of money, but you may consider one option in respect of other option therefore, time value of money is coming because in one case you are investing something in another case there is no investment upfront. So, therefore, because some of you may be thinking why we are talking of time value of money we can do all these comparison in future only, but there are some investments happening today therefore, we need to make the decision in today's context only. And if for example, because if you see our decision tree these are the possible scenarios 320 480 etcetera where you have less than 500 demand and because of less than 500 demand these additional capacity you have will remain unused. So, that is a risk also you have built extra capacity and that will not be used. So, now after so you have to do both these calculations as part of your homework also that develop a model like this itself and here you have to calculate the expected value like in this case if you see we are in case 2 it will be let us say 720 4.32. So, here you will be calculating 220 at the rate of 4.32 while there is a case of 380 4.32 there will be no cost 0 cost because you are not going to use any capacity from the spot market. There are cases like 480 2.88 again it is 0 whenever it is more than 500 only in that case there will be a cost applicable in option number 2. So, now once we have calculated the impacts or all the branches for period number 2 let us go back to the time machine and see what is the net present value of all these future cash flows. So, let us see how are we going to do that for this node 603.6 there are 4 branches coming. So, first important thing is we need to calculate the expected profit in period 2 over all these 4 possible states which are coming from this particular node of the period number 1 and that is expected profit for this particular node for period number 1 this comma 1 is stands for period. So, because each of them are having equal probability. So, I have taken this 0.25 that is the probability outside it, but for example, if they are coming with different probabilities. So, you have to multiply these impacts with their respective probability incidentally these are equal in my case. So, I can take them as a common factor outside the bracket. So, this expected profit for this particular node after this calculation coming to be 240 and then we have to see because this profit will come after 1 year after 1 year I am expecting this profit of 240 dollars. So, we have to go for present value calculation of this. So, expected profit at each node in period 1 is the profit during period 1 plus the present value of the expected profit in period 2. So, now at node 1 if I go back to the this particular node. So, during period 1 whatever profit I am getting plus whatever future profits I am going to get their present value. So, that is what I am going to calculate now. So, expected profit of a particular period when D is sum P is sum at node 1 is the expected profit all 4 nodes in period 2 that may result from this particular node and present value of expected profit P V E P stands for present value of expected profit is the present value of this expected profit and P D P 1 and the total expected profit is the sum of the profit in period 1 and the present value of the expected profit in period 2 which I have just explained. So, this calculation is the present value of this expected value in period 1 is P V E P for this particular scenario expected profit for this divided by discounted rate factor. So, the net present value we know that whatever cash flow I am getting cash flow in future divided by 1 plus k into P into the power of n. So, if I am getting in next 1 period n will become 1 what I am getting 2 period ahead n will become square if I am getting 3 period ahead the value of n will change to 3. So, since I am talking only 1 period ahead this is 1 plus k where k is the discount factor and in this case we have consider k is value as 10 percent. So, the present value of expected profit is 280 the profit which you are going in the year 2 that is 240, but its present value is equivalent to 280 and therefore, the total expected profit at this particular node where D is this P is this in is the sum of the profit in period 1 plus this 218.2 and that is 600 into 4 minus 600 minus 3.6 because this is the revenue this is the cost and this. So, 240 plus 218.2 458.2 that is the total expected profit at this particular node and similarly you need to go backward to time t 0 also. Now, if you see whatever calculation I did for t 1 the same calculation will go for t naught also and taking this calculation forward after this you can go for period t naught also where you can see that the initial demand was 500 initial price was 3 and then expected profit at this particular level is whatever the expected profits at all the nodes of year 1 plus the profit you are getting at that particular level. So, all these are the 4 different terms 25 percent of this 25 percent of this situation 25 percent of this situation and 25 percent of this situation because 4 possible situations are emerging. So, the expected profit in year 1 is 954 and then you have to calculate the present value of this at the rate of 10 percent that becomes the present value 867. So, you your present value of profit is 867 and the profit which is coming because of this particular requirement in this period only that is 500 is the demand 3 is the price 4 is the revenue. So, 500 rupees directly you can consider. So, 500 plus 867 1367 is the total expected profit using the net present value concept for this spot market. We have option 1 giving us 1367.78 as the expected profits present value this is what I write for this. Now, another option which we have considered in this problem where 500 own and rest of spot market. Now, let us say for building this 500 on your own there is a cost of let us say 1000 dollar and option 3 500 lease rest spot market and this 500 lease is coming to you because you are doing agreement in the beginning itself for next 2 years. So, this is available to you at the rate of 2 per unit. So, you have a fixed rate which is applicable for next 2 years. So, the question is whether this option is better, this option is better or this option is better. We have explained you the process of calculation of how we are going to calculate the expected profit with the net present value. I request all of you to do the same calculations for option 2 and option 3 also and then you can compare which option is the best one and accordingly you will take a decision. So, this is also a very interesting use case of our predictive analytics that you are able to predict the expected profit and accordingly you are going to take a decision that which is a more profitable situation option for me. With this we come to end of this particular session. Thank you very much.