Chapter 11.S, Problem 3P

Summary Introduction

To determine: The number of suppliers to be chosen by Company W.

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

Answer to Problem 3P

TheCompany W can choose one supplier.

Explanation of Solution

Given information:

$\begin{array}{c}\text{Probability}\text{of}\text{super-event,}\text{S}=0.5\text{\%}\\ =\text{0}\text{.005}\\ \text{Shutdown}\text{cost, L}=\text{\$400,000}\\ \text{Probability}\text{of}\text{unique-event,}\text{U}=\text{5\%}\\ =\text{0}\text{.05}\\ \text{Marginal}\text{cost,}{\text{C}}_{\text{1(local)}}=\text{\$15,000}\\ {\text{C}}_{\text{2(distant)}}=\$25,000\end{array}$

Formula:

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

Calculation for supplier selection:

Probability of selecting one supplier:

$\begin{array}{c}\text{P}\left(\text{n}\right)\text{=S+}\left(\text{1-S}\right){\text{U}}^{\text{n}}\\ \text{P}\left(\text{1}\right)=\text{0}\text{.005}+\left(\text{1}-\text{0}\text{.005}\right)\text{0}{\text{.05}}^{\text{1}}\\ =\text{0}\text{.005}+\left(\text{0}\text{.995}\times \text{0}\text{.05}\right)\\ =\text{0}\text{.05475}\end{array}$

To find the probability of selecting one supplier, substitute the number of chosen suppliers, probability of super-event and unique-event in the above formula. Here, the number of suppliers chosen is ‘1’; S=0.005 and U=0.05 are substituted in the above formula, which gives the probability as 0.05475.

Probability of selecting two suppliers:

$\begin{array}{c}\text{P}\left(\text{n}\right)\text{=S+}\left(\text{1-S}\right){\text{U}}^{\text{n}}\\ \text{P}\left(2\right)=\text{0}\text{.005}+\left(\text{1}-\text{0}\text{.005}\right)\text{0}{\text{.05}}^2\\ =\text{0}\text{.005}+\left(\text{0}\text{.995}\times \text{0}\text{.0025}\right)\\ =\text{0}\text{.007488}\end{array}$

To find the probability of selecting two suppliers, substitute the number of chosen suppliers, probability of super-event and unique-event in the above formula. Here, the number of suppliers chosen is ‘2’; S=0.005 and U=0.05 are substituted in the above formula, which gives the probability as 0.007488.

Probability of selecting three suppliers:

$\begin{array}{c}\text{P}\left(\text{n}\right)\text{=S+}\left(\text{1-S}\right){\text{U}}^{\text{n}}\\ \text{P}\left(3\right)=\text{0}\text{.005}+\left(\text{1}-\text{0}\text{.005}\right)\text{0}{\text{.05}}^3\\ =\text{0}\text{.005}+\left(\text{0}\text{.995}\times \text{0}\text{.000125}\right)\\ =\text{0}\text{.00512}\end{array}$

To find the probability of selecting three suppliers, substitute the number of chosen suppliers, probability of super-event and unique-event in the above formula. Here, the number of suppliers chosen is ‘3’; S=0.005 and U=0.05 are substituted in the above formula, which gives the probability as 0.00512.

Calculation of cost:

Failure cost and no-failure cost is calculated for every number of suppliers selected.

• When one supplier is selected:
  • Calculation of no-failure cost:

When there is no failure of supplier, then there is no loss, when one supplier is selected. The only cost associated with the supplier selection is the marginal cost.

Probability of no-failure for one supplier,

$\begin{array}{c}\text{Probability of no-failure of supplier}=\text{1}-\text{P}\left(\text{1}\right)\\ =\text{1}-\text{0}\text{.05475}\\ =\text{0}\text{.9452}\end{array}$

$\begin{array}{c}\text{No-failure}\text{cost}=\text{1C}\\ =\text{1}\times \text{\$15,000}\\ =\text{\$15,000}\end{array}$

• Calculation of failure cost:

When the supplier selection fails, along with marginal cost, the supplier failure cost is also added to the failure cost.

$\begin{array}{c}\text{Failure}\text{cost}=\text{L}+\text{1C}\\ =\text{\$400,000}+\left(\text{1}\times \text{\$15,000}\right)\\ =\text{\$415,000}\end{array}$

• Calculation of total cost:

The total cost is calculated by summing up the values obtained by multiplying the cost with respective probabilities.

$\begin{array}{l}\text{Total}\text{cost}=\left(\text{\$15,000}\times \text{0}\text{.9452}\right)+\left(\text{\$415,000}\times \text{0}\text{.54750}\right)\\ =\$36,900\end{array}$ (1)

The total cost of selecting one supplier is $36,900.

• When two suppliers are selected:
  • Calculation of no-failure cost:

When there is no failure of suppliers, then there is no loss, when two suppliers are selected. The only cost associated with the supplier selection is the marginal cost.

Probability of no-failure for two suppliers,

$\begin{array}{c}\text{Probability of no-failure of supplier}=\text{1}-\text{P}\left(2\right)\\ =\text{1}-\text{0}\text{.007488}\\ =\text{0}\text{.992512}\end{array}$

$\begin{array}{c}\text{No-failure}\text{cost}={\text{C}}_1+C_2\\ \text{=\$15,000+\$25,000}\\ =\text{\$40,000}\end{array}$

• Calculation of failure cost:

When the supplier selection fails, along with marginal cost, the supplier failure cost is also added to the failure cost.

$\begin{array}{c}\text{Failure}\text{cost}=\text{L}+{\text{C}}_1+C_2\\ =\text{\$400,000}+\$40,000\\ =\text{\$440,000}\end{array}$

• Calculation of total cost:

The total cost is calculated by summing up the values obtained by multiplying the cost with respective probabilities.

$\begin{array}{l}\text{Total}\text{cost}=\left(\text{\$40,000}\times \text{0}\text{.992512}\right)+\left(\text{\$440,000}\times \text{0}\text{.007488}\right)\\ =\$\text{42,995}\end{array}$ (2)

The total cost of selecting two suppliers is $42,995.

• When three suppliers are selected:
  • Calculation of no-failure cost:

When there is no failure of suppliers, then there is no loss, when three suppliers are selected. The only cost associated with the supplier selection is the marginal cost.

Probability of no-failure for three suppliers,

$\begin{array}{c}\text{Probability of no-failure of supplier}=\text{1}-\text{P}\left(3\right)\\ =\text{1}-\text{0}\text{.005124}\\ =\text{0}\text{.9948}\end{array}$

$\begin{array}{c}\text{No-failure}\text{cost}=C_1+C_2+C_3\\ =\text{\$15,000+2}\left(\$25,000\right)\\ =\text{\$65,000}\end{array}$

• Calculation of failure cost:

When the supplier selection fails, along with marginal cost, the supplier failure cost is also added to the failure cost.

$\begin{array}{c}\text{Failure}\text{cost}=\text{L}+C_1+C_2+C_3\\ =\text{\$400,000}+\$65,000\\ =\text{\$465,000}\end{array}$

• Calculation of total cost:

The total cost is calculated by summing up the values obtained by multiplying the cost with respective probabilities.

$\begin{array}{l}\text{Total}\text{cost}=\left(\text{\$65,000}\times \text{0}\text{.994876}\right)+\left(\text{\$465,000}\times \text{0}\text{.005124}\right)\\ =\$67,050\end{array}$ (3)

The total cost of selecting three suppliers is $67,050.

From Equations ((1), (2) and (3)), it can be inferred that the total cost is low, when one supplier is selected. Therefore, it is advisable for Company W to select one supplier.

Hence, Company W can choose one supplier.