Chapter 11.S, Problem 5P

a)

Summary Introduction

To determine: The probability that all three suppliers are disrupted using option 1.

Introduction: Supply chain management is one of the important elements of a business which impacts business product development. With expanding businesses in global conditions, supply chain activities can impact the cost effectiveness of these businesses.

Answer to Problem 5P

The probability that three suppliers are disrupted using option 1 is 0.025062.

Explanation of Solution

Given information:

Option 1

$\begin{array}{c}\text{Probability}\text{of}\text{super-event,}\text{S}=2.5\text{\%}\\ =\text{0}\text{.025}\\ \text{Probability}\text{of}\text{unique-event,}\text{U}=4\text{\%}\\ =\text{0}\text{.04}\\ \text{Purchasing \& Transportation cost}=\text{\$1,000,000}\\ \text{Disruption loss}=\$500,000\end{array}$

Option 2

$\begin{array}{c}\text{Probability}\text{of}\text{super-event,}\text{S}=0.4\text{\%}\\ =\text{0}\text{.004}\\ \text{Probability}\text{of}\text{unique-event,}\text{U}=20\text{\%}\\ =\text{0}\text{.20}\\ \text{Purchasing \& Transportation cost}=\text{\$1,010,000}\\ \text{Disruption loss}=\$500,000\end{array}$

Formula:

$\text{P}\left(\text{n}\right)=\text{S}+\left(\text{1}-\text{S}\right){\text{U}}^{\text{n}}$

Probability calculation when both suppliers are disrupted by option 1:

$\begin{array}{c}\text{P}\left(\text{n}\right)=\text{S}+\left(\text{1}-\text{S}\right){\text{U}}^{\text{n}}\\ \text{P}\left(3\right)=\text{0}\text{.025}+\left(\text{1}-\text{0}\text{.025}\right)\text{0}{\text{.04}}^3\\ =\text{0}\text{.025}+\left(\text{0}\text{.975}\times \text{0}\text{.000064}\right)\\ =\text{0}\text{.025062}\end{array}$

To find the probability of selection, one supplier substitutes the number of chosen suppliers, and the probability of super-event and unique-event in the above formula. Here, the number of suppliers chosen is ‘3’, S=0.025 and U=0.04, and is substituted in the above formula which gives probability as 0.025062.

Hence, the probability that two suppliers are disrupted using option 1 is 0.025062.

b)

Summary Introduction

To determine: The probability that all three suppliers are disrupted using option 2.

Answer to Problem 5P

The probability that three suppliers are disrupted using option 1 is 0.011968.

Explanation of Solution

Given information:

Option 1

$\begin{array}{c}\text{Probability}\text{of}\text{super-event,}\text{S}=2.5\text{\%}\\ =\text{0}\text{.025}\\ \text{Probability}\text{of}\text{unique-event,}\text{U}=4\text{\%}\\ =\text{0}\text{.04}\\ \text{Purchasing \& Transportation cost}=\text{\$1,000,000}\\ \text{Disruption loss}=\$500,000\end{array}$

Option 2

$\begin{array}{c}\text{Probability}\text{of}\text{super-event,}\text{S}=0.4\text{\%}\\ =\text{0}\text{.004}\\ \text{Probability}\text{of}\text{unique-event,}\text{U}=20\text{\%}\\ =\text{0}\text{.20}\\ \text{Purchasing \& Transportation cost}=\text{\$1,010,000}\\ \text{Disruption loss}=\$500,000\end{array}$

Formula:

$\text{P}\left(\text{n}\right)=\text{S}+\left(\text{1}-\text{S}\right){\text{U}}^{\text{n}}$

Probability calculation when both suppliers are disrupted by option 2:

$\begin{array}{c}\text{P}\left(\text{n}\right)=\text{S}+\left(\text{1}-\text{S}\right){\text{U}}^{\text{n}}\\ \text{P}\left(3\right)=\text{0}\text{.004}+\left(\text{1}-\text{0}\text{.004}\right)\text{0}{\text{.20}}^3\\ =\text{0}\text{.004}+\left(\text{0}\text{.996}\times \text{0}\text{.008}\right)\\ =\text{0}\text{.011968}\end{array}$

To find the probability of selection, one supplier substitutes the number of chosen suppliers, anf the probability of super-event and unique-event in the above formula. Here, the number of suppliers chosen is ‘3’, S=0.004 and U=0.20, and is substituted in the above formula which gives probability as 0.011968.

Hence, the probability that two suppliers are disrupted using option 1 is 0.011968.

c)

Summary Introduction

To determine: The total annual expected cost for option 1.

Answer to Problem 5P

The total expected cost for option 1 is $1,012,531.

Explanation of Solution

Given information:

Option 1

$\begin{array}{c}\text{Probability}\text{of}\text{super-event,}\text{S}=2.5\text{\%}\\ =\text{0}\text{.025}\\ \text{Probability}\text{of}\text{unique-event,}\text{U}=4\text{\%}\\ =\text{0}\text{.04}\\ \text{Purchasing \& Transportation cost}=\text{\$1,000,000}\\ \text{Disruption loss}=\$500,000\end{array}$

Option 2

$\begin{array}{c}\text{Probability}\text{of}\text{super-event,}\text{S}=0.4\text{\%}\\ =\text{0}\text{.004}\\ \text{Probability}\text{of}\text{unique-event,}\text{U}=20\text{\%}\\ =\text{0}\text{.20}\\ \text{Purchasing \& Transportation cost}=\text{\$1,010,000}\\ \text{Disruption loss}=\$500,000\end{array}$

Formula:

$\text{Total expected cost}=\text{Annual}\text{purchasing}\text{cost}+\text{Expected}\text{disruption}\text{cost}$

Calculation of total expected cost:

$\begin{array}{c}\text{Total expected cost}=\text{Annual}\text{purchasing}\text{cost}+\text{Expected}\text{disruption}\text{cost}\\ \text{=\$1,000,000+}\left(\$500,000\times 0.025062\right)\\ =\$1,012,531\end{array}$ (1)

The total expected cost is calculated by taking the summation of the annual purchasing cost and expected disruption cost. The product of $500,000 and 0.025062 is added with $1,000,000, which gives $1,012,531 as the total expected cost for option 1.

Hence, the total expected cost using option 1 is $1,012,531.

d)

Summary Introduction

To determine: The total annual expected cost for option 2.

Answer to Problem 5P

The total expected cost for option 2 is $1,015,984.

Explanation of Solution

Given information:

Option 1

$\begin{array}{c}\text{Probability}\text{of}\text{super-event,}\text{S}=2.5\text{\%}\\ =\text{0}\text{.025}\\ \text{Probability}\text{of}\text{unique-event,}\text{U}=4\text{\%}\\ =\text{0}\text{.04}\\ \text{Purchasing \& Transportation cost}=\text{\$1,000,000}\\ \text{Disruption loss}=\$500,000\end{array}$

Option 2

$\begin{array}{c}\text{Probability}\text{of}\text{super-event,}\text{S}=0.4\text{\%}\\ =\text{0}\text{.004}\\ \text{Probability}\text{of}\text{unique-event,}\text{U}=20\text{\%}\\ =\text{0}\text{.20}\\ \text{Purchasing \& Transportation cost}=\text{\$1,010,000}\\ \text{Disruption loss}=\$500,000\end{array}$

Formula:

$\text{Total expected cost}=\text{Annual}\text{purchasing}\text{cost}+\text{Expected}\text{disruption}\text{cost}$

Calculation of total expected cost:

$\begin{array}{c}\text{Total expected cost}=\text{Annual}\text{purchasing}\text{cost}+\text{Expected}\text{disruption}\text{cost}\\ \text{=\$1,010,000+}\left(\$500,000\times 0.11968\right)\\ =\$1,015,984\end{array}$ (2)

The total expected cost is calculated by taking the summation of the annual purchasing cost and expected disruption cost. The product of $500,000 and 0.11968 is added with $1,010,000 which gives $1,015,984 as the total expected cost for option 2.

Hence, the total expected cost using option 2 is $1,015,984.

e)

Summary Introduction

To determine: The best options.

Answer to Problem 5P

Option 1 is best.

Explanation of Solution

Given information:

Option 1

$\begin{array}{c}\text{Probability}\text{of}\text{super-event,}\text{S}=2.5\text{\%}\\ =\text{0}\text{.025}\\ \text{Probability}\text{of}\text{unique-event,}\text{U}=4\text{\%}\\ =\text{0}\text{.04}\\ \text{Purchasing \& Transportation cost}=\text{\$1,000,000}\\ \text{Disruption loss}=\$500,000\end{array}$

Option 2

$\begin{array}{c}\text{Probability}\text{of}\text{super-event,}\text{S}=0.4\text{\%}\\ =\text{0}\text{.004}\\ \text{Probability}\text{of}\text{unique-event,}\text{U}=20\text{\%}\\ =\text{0}\text{.20}\\ \text{Purchasing \& Transportation cost}=\text{\$1,010,000}\\ \text{Disruption loss}=\$500,000\end{array}$

On comparing equations (1) and (2), it can be inferred that option 1 has the least expected cost. Therefore, it is advisable for Company BJ to choose option 1.

Hence, Option 1 is best.