Chapter 13, Problem 2PA

Flextrola, Inc., an electronics systems integrator, is planning to design a key component for its next-generation product with Solectrics. Flextrola will integrate the component with some software and then sell it to consumers. Given the short life cycles of such products and the long lead times quoted by Solectrics, Flextrola only has one opportunity to place an order with Solectrics prior to the beginning of its selling season. Flextrola’s demand during the season is normally distributed with a mean of 1000 and a standard deviation of 600.

Solectrics’ production cost for the component is $52 per unit, and it plans to sell the component for $72 per unit to Flextrola. Flextrola incurs essentially no cost associated with the software integration and handling of each unit. Flextrola sells these units to consumers for $121 each. Flextrola can sell unsold inventory at the end of the season in a secondary electronics market for $50 each. The existing contract specifies that once Flextrola places the order, no changes are allowed to it. Also, Solectrics does not accept any returns of unsold inventory, so Flextrola must dispose of excess inventory in the secondary market.

1. a. What is the probability that Flextrola’s demand will be within 25 percent of its forecast? [LO13-1]
2. b. What is the probability that Flextrola’s demand will be more than 40 percent greater than Flextrola’s forecast? [LO13-1]
3. c. Under this contract, how many units should Flextrola order to maximize its expected profit? [LO13-1]
4. d. If Flextrola orders 1200 units, how many units of inventory can Flextrola expect to sell in the secondary electronics market? [LO13-2]
5. e. If Flextrola orders 1200 units, what are expected sales? [LO13-2]
6. f. If Flextrola orders 1200 units, what is expected profit? [LO13-2]
7. g. A sharp manager at Flextrola noticed the demand forecast and became wary of assuming that demand is normally distributed. She plotted a histogram of demands from previous seasons for similar products and concluded that demand is better represented by the log normal distribution. Figure 13.15 plots the density function for both the log normal and the normal distributions, each with mean = 1000 and standard deviation = 600; Figure 13.16 plots the corresponding distribution functions. Using the more accurate forecast (i.e., the log normal distribution), approximately how many units should Flextrola order to maximize its expected profit? [LO13-1]

Summary Introduction

To determine: The probability that the Company F demand will be within 25 percent of the forecast.

Explanation of Solution

Given information:

Production cost (P)         = $52 per unit

Selling price (S)         = $72 per unit

Customer price (C)         = $121 per unit

Unsold inventory cost (U)     = $50 per unit

Mean (M)             = 1,000

Standard deviation (SD)     = 600

Calculation of Z – value:

The 25% mean forecast is a value which it is 25% less and 25%. The value for 25% less is 750 units. The value for 25% more will 1,250 units.

When forecast is 1,250 units:

$\begin{array}{c}\text{Z}=\frac{1,250-M}{SD}\\ =\frac{1,250-1,000}{600}\\ =\frac{250}{600}\\ =0.4167\end{array}$

When forecast is 750 units:

$\begin{array}{c}\text{Z}=\frac{750-M}{SD}\\ =\frac{750-1,000}{600}\\ =\frac{-250}{600}\\ =-0.4167\end{array}$

The probability values for the Z-values of 0.4167 and -0.4167 are subtracted to find the probability of Company F demand being within 25% of the forecast. Using the Excel =NORMSDIST (0.4167) function, the probability value is 0.6615 and =NORMSDIST (-0.4167) function, the probability value is 0.33845.

Calculation of probability:

$\begin{array}{c}\text{Probability}=0.6615-0.33845\\ =0.32305\\ =0.3230\end{array}$

The probability that the Company F demand will be within 25 percent of the forecast is 0.3230.

Summary Introduction

To determine: The probability that the Company F demand will be 40% more than Company F forecast.

Explanation of Solution

Given information:

Production cost (P)         = $52 per unit

Selling price (S)         = $72 per unit

Customer price (C)         = $121 per unit

Unsold inventory cost (U)     = $50 per unit

Mean (M)             = 1,000

Standard deviation (SD)     = 600

Calculation of Z – value:

The 40% more value is 1,400 units.

When forecast is 1,400 units:

$\begin{array}{c}\text{Z}=\frac{1,400-M}{SD}\\ =\frac{1,400-1,000}{600}\\ =\frac{400}{600}\\ =0.6667\end{array}$

Using the Excel =NORMSDIST (0.6667) function, the probability value is 0.4167.

Calculation of probability:

$\begin{array}{c}\text{Probability}=1-0.7475\\ =0.2525\end{array}$

The probability that the Company F demand will be 40% more than Company F forecast is 0.2525.

Summary Introduction

To determine: The number of units to be ordered to maximize the expected profit.

Explanation of Solution

Given information:

Production cost (P)         = $52 per unit

Selling price (S)         = $72 per unit

Customer price (C)         = $121 per unit

Unsold inventory cost (U)     = $50 per unit

Mean (M)             = 1,000

Standard deviation (SD)     = 600

Calculation of critical ratio:

$\begin{array}{c}\text{Critical ratio}=\frac{C-S}{\left(S-U\right)+\left(C-S\right)}\\ =\frac{121-72}{\left(72-50\right)+\left(121-72\right)}\\ =\frac{49}{22+49}\\ =\frac{49}{71}\\ =0.6901\end{array}$

Using the standard normal distribution table and the roundup rule, the value of Z for a critical ratio of 0.6901 is 0.50.

Calculation of optimal order quantity:

$\begin{array}{c}\text{Optimal order quantity}=M+(Z\times SD)\\ =1,000+(0.5\times 600)\\ =1,000+300\\ =1,300\text{ units}\end{array}$

The number of units to be ordered to maximize the expected profit is 1,300 units.

Summary Introduction

To determine: The number of units of inventory Company F can expect to sell in the secondary electronics market.

Explanation of Solution

Given information:

Production cost (P)         = $52 per unit

Selling price (S)         = $72 per unit

Customer price (C)         = $121 per unit

Unsold inventory cost (U)     = $50 per unit

Mean (M)             = 1,000

Standard deviation (SD)     = 600

Order quantity (Q)         = 1,200 units

Calculation of z-value:

$\begin{array}{c}\text{Z}=\frac{Q-M}{SD}\\ =\frac{1,200-1,000}{600}\\ =\frac{200}{600}\\ =0.3333\end{array}$

Using the standard normal distribution table and the roundup rule, the value of ‘I’ for a Z-value of 0.4 is 0.6304

Calculation of expected leftover inventory:

$\begin{array}{c}\text{Inventory}=I\times SD\\ =0.6304\times 600\\ =378.24\text{ units}\end{array}$

The number of units of inventory Company F can expect to sell in the secondary electronics market is 378.24 units.

Summary Introduction

To determine: The expected sales.

Explanation of Solution

Given information:

Production cost (P)         = $52 per unit

Selling price (S)         = $72 per unit

Customer price (C)         = $121 per unit

Unsold inventory cost (U)     = $50 per unit

Mean (M)             = 1,000

Standard deviation (SD)     = 600

Order quantity (Q)         = 1,200 units

Calculation of expected sales:

$\begin{array}{c}\text{Expected sales}=Q-\text{Expected leftover inventory}\\ =1,200-378.24\\ =821.76\text{ units}\end{array}$

The expected sales is 821.76 units.

Summary Introduction

To determine: The expected profit.

Explanation of Solution

Given information:

Production cost (P)         = $52 per unit

Selling price (S)         = $72 per unit

Customer price (C)         = $121 per unit

Unsold inventory cost (U)     = $50 per unit

Mean (M)             = 1,000

Standard deviation (SD)     = 600

Order quantity (Q)         = 1,200 units

Calculation of expected profit:

$\begin{array}{c}\text{Expected profit}=\left(C\times \text{Expected sales}\right)+\left(U\times \text{Leftover inventory}\right)-\left(S\times Q\right)\\ =\left(121\times 821.76\right)+\left(50\times 378.24\right)-\left(72\times 1,200\right)\\ =99,432.96+18,912-86,400\\ =31,944.96\end{array}$

The expected profit is $31,944.96.

Summary Introduction

To determine: The number of units to be ordered to maximize the expected profit.

Explanation of Solution

Given information:

Production cost (P)         = $52 per unit

Selling price (S)         = $72 per unit

Customer price (C)         = $121 per unit

Unsold inventory cost (U)     = $50 per unit

Mean (M)             = 1,000

Standard deviation (SD)     = 600

Calculation of critical ratio:

$\begin{array}{c}\text{Critical ratio}=\frac{C-S}{\left(S-U\right)+\left(C-S\right)}\\ =\frac{121-72}{\left(72-50\right)+\left(121-72\right)}\\ =\frac{49}{22+49}\\ =\frac{49}{71}\\ =0.6901\end{array}$

The critical ratio is 0.6901. Based on the graph of the distribution function, the demand is approximately 1,125 units.

The number of units to be ordered to maximize the expected profit is 1,125 units.