In this application brief we will look at a manufacturing decision to make a key sub-assembly of a product in-house or to outsource. If it's made in-house, we also will look at whether it should be assembled with highly automated process or a more labor intensive process.

Therefore, we have three options in our manufacturing decision:

- Manufacture in-house with a high degree of automation.
- Manufacturing in-house with manual assembly.
- Outsource the sub-assembly to a vendor.

We have estimated that the initial investment required for each of our three options is as follows:

- Automated assembly: $900,000
- Manual assembly: $200,000
- Outsource: $0

The production of the sub-assembly is expected to run for five years.

A decision tree is a good tool to explore all of the possibilities of our manufacturing decision. Let's look at the tree below.

There are two possible outcomes for the demand of this sub-assembly, high and low. If demand for this sub-assembly is high, 20,000 per year will be produced. If demand is low, 5,000 per year will be produced. We could add more outcomes for demand to better address the range of demand, but we will limit it to two outcomes to keep the discussion simple.

Another benefit of a decision tree is showing any timing effects. In our example, there are three times that are important. Production start-up, one year from start-up, and the end of year five. If we manufacture in-house, there will be an initial investment. We will assume this investment is made at start-up. The nodes in our tree are color coded to reflect the timing effects.

If we decided to use manual assembly or outsourcing initially, at the end of year 1 we want to see what demand was for the first year and change strategy if it makes sense. Note that if we initially chose automation, we are already producing at the lowest cost, so even if demand is low, it doesn't make sense to switch.

## Calculating Payoffs

So what are the payoffs that we are analyzing? In our example, we may or may not make an investment in order to save cost in the production of the sub-assembly. Using this convention, an investment is a negative cash flow, and any cost saving (versus the highest cost option of outsourcing) is a positive cash flow.

We need to determine the net payoff at each node. For this example, we will discount all cash flows to present value using a discount rate of 10%. The following spreadsheet was set up to make the payoff calculations. The yellow cells are changed according to the situation, and we choose the appropriate present value as the value for a node.

The decision tree is shown again with notations showing what payoffs are used at the nodes.

#### Investments at Start-Up

For the three red chance nodes, we enter the amount of investment required at start-up as a negative number.

#### Year 1-5 Present Value

For end nodes that will not involve an intermediate investment decision, we enter the present value of the cost savings for all five years.

#### Year 1 Present Value

There are two cases where we would make a decision to switch methods after year 1:

- Manual assembly and high demand.
- Outsource and high demand.

The green decision nodes denote these decision points. Up to this point (end of year 1), we enter the cost savings present value for year 1.

#### Year 1-5 Net Present Value

As we exit the green decision nodes at year 1, in some case we may decide to make an investment in automated or manual assembly. In these end nodes we need to calculate the net present value of year 2-5 cost savings and the investment made at the end of year 1.

#### Year 2-5 Present Value

For the remaining cases emanating from the green decision nodes, we are not changing anything, so we need to calculate the present value of year 2-5 cost savings.

## Rolling Back the Tree

Now that the decision tree is constructed and the payoff values at each node calculated, we are ready to roll back the tree. Starting at the end nodes, we calculate the expected value of each node. We repeat this procedure, and work backward to the root node.

During roll back, when we reach a decision node, a decision rule must be invoked to make a decision that meets our goal. The decision rule can be either choose the minimum value of it child nodes, or choose the maximum value. Using our convention that cost savings is positive, we will choose the maximum value to maximize cost savings.

DTace will do rollback calculations, and the user specifies the decision rule to use. The decision tree has already been rolled back, and the expected value at the root node is 0.75 ($750,000).

## Optimal Path Through the Tree

When DTace rolls back a tree, it also determines the optimal path through the tree. The optimal path shows which decisions result in the highest expected value (or lowest depending on decision rule).

For our example, the optimal path is initially select outsourcing. If after 1 year the demand is high then we should invest in automation and produce in-house. If demand is low after 1 year, we should not invest, and continue to outsource.

## Conclusion

Decision trees are a good way of boiling a complicated set of scenarios down to an understandable graphic. For our manufacturing decision, we found that initially outsourcing the sub-assembly was the best course of action. After year 1, if demand is high we can invest in automation and bring production in-house to minimize costs. If demand is low, we can continue to outsource and avoid any additional investment.

Excel is a registered trademark of Microsoft Corporation. Used with permission from Microsoft.