EcoWrap Solutions has developed a new biodegradable packaging material with potential applications in shipping packaging and food containers. There is great interest in sustainable packaging materials, but quality standards are high. Launching in either industry is a risky, all$-$or$-$nothing bet. They would have to invest in expensive, specialized equipment to mass produce the material, plus extra equipment needed for food$-$grade packaging. If consumers like the new material, the company would be guaranteed a large profit. But if it doesn$$t hold up under real$-$world use, it will be quickly abandoned, leaving the company at a loss.

Shipping packaging needs less investment and has a $50\%$ chance of success, but doesn$$t have as much upside. The food container product line has more profit potential, but requires more investment and only has a $40\%$ chance of success.

The similarity of these products means that if one of the applications works, the other is more likely to work also. The probability of both applications being successful is $0\mathrm{.}30.$ More information on the probabilities is available in the hint below.

EcoWrap is preparing to launch products in these industries, and needs to choose from two possible launch strategies. The following tables provide the expected profit of each product, in millions of dollars:

Concurrent Launch: Launch both products simultaneously to capitalize on being the first entrant in both respective markets. This strategy holds the potential for the highest profits but requires a significant upfront investment, exposing the company to a higher level of financial risk.

Profits

Success

Failure

Shipping packaging

$$60$

$-$$$20$

Food container

$$100$

$-$$$40$

Sequential Launch: Introduce one product first and assess its market performance before launching the second product. This approach mitigates some of the financial risks, as the company could decide not to launch the second product if the first one is unsuccessful, thereby avoiding additional investment costs. If they choose this strategy, they would launch Shipping Packaging first because the equipment investment is lower. However, their profit for food containers would be lower with the delayed launch.

Profits

Success

Failure

Shipping packaging

$$60$

$-$$$20$

Food container

$$90$

$-$$$40$

Build a decision tree to answer the following questions.

Hint: Conditional Probabilities

From the text we know that the probabilities of success are:

P$($shipping success$)=$

$0\mathrm{.}50$

P$($food success$)=$

$0\mathrm{.}40$

P$($food and shipping success$)=$

$0\mathrm{.}30$

You can create two uncertainty nodes in the tree; one for shipping and one for food. If the first node is shipping, then its probabilities will be:

P$($shipping success$)=0\mathrm{.}5$ and P$($shipping failure$)=0\mathrm{.}5$

But then the second node for food needs the conditional probabilities of success and failure conditioned on $($given that$)$ the outcome for shipping was success or failure.

That is$,$ we need:

P$($food success$|$shipping success$)$ and P$($food success$|$shipping failure$).$

We can find the conditional probabilities for food using the Multiplication Rule of probability, which says $($see page $189$ in the textbook$)$:

P$($A and B$)=$ P$($A$|$B$)$P$($B$)$

Setting A $=$ food success and B $=$ shipping success,

P$($food and shipping success$)=$ P$($food success$|$shipping success$)$ P$($shipping success$)$

and rearrange to

P$($food success$|$shipping success$)=$ P$($food and shipping success$)/$ P$($shipping success$)$

then plugging in the probabilities given above,

P$($food success$|$shipping success$)=0\mathrm{.}30/0\mathrm{.}50=0\mathrm{.}60$

This means that if shipping is successful, the probability of success for food increases from $0\mathrm{.}40$ to $0\mathrm{.}60.$

Applying this rule again in the case of shipping failure gives:

P$($food success$|$shipping failure$)=$ P$($food success and shipping failure$)/$ P$($shipping failure$)$

but we need to first find P$($food success and shipping failure$).$

Since:

P$($food success$)=$ P$($food success and shipping failure$)+$ P$($food success and shipping success$)$

P$($food success and shipping failure$)=$ P$($food success$)-$ P$($food success and shipping success$)$

P$($food success and shipping failure$)=0\mathrm{.}4-0\mathrm{.}3=0\mathrm{.}1$

Then

P$($food success$|$shipping failure$)=0\mathrm{.}1/0\mathrm{.}5=0\mathrm{.}2$

$1.$ What is the expected profit of a concurrent launch? Express you answer in millions of dollars, with no decimal points. Ex: $$50$ million $=>50.$

$2.$ If they launch concurrently, what is the expected profit given that Shipping Packaging is successfull?

$3.$ If they launch concurrently, what is the expected profit given that Shipping Packaging fails?

$4.$ What is the expected profit of a sequential launch that launches shipping packaging first?

$5.$ If they launch sequentially, what is the expected profit given that Shipping Packaging is successfull?

$6.$ If they launch sequentially, what is the expected profit given that Shipping Packaging fails?

$7.$ The sequential launch strategy has less risk, but at the cost of reduced profit for food containers of $$90$

$8.$ With the sequential launch strategy, does the decision of whether to launch food containers change depending on whether shipping is successful? yes or no

$9.$ Value of control: There may be another opportunity to reduce risk. A major ecommerce shipping company is interested in partnering with EcoWrap. The company's resources and experience guaranteed development and launch of a succesful shipping packaging product. However the two companies would share in the profits. Reducing EcoWraps profit and overall value versus if they had developed it alone. What is the greatest potential value this partnership could have for EcoWrap? i$.$e$.$ what is the value of controlling the outcome of shipping packaging launch success?