Ch. 7: Forecasting

1
Planning modeling

• Chapter outline
• The basic planning problem
• The basic pricing problem
• 3. Nonlinear cost and demand functions
• XYFunction
• Mathematical model of an XY Function
• Spreadsheet Model o
• f XY Function
• Approximating the cost with a Cubic Function
• Preparing a Five year Plan
• The Impact of Pricing

2
The Basic Planning Problem
Down is a skeleton model for a five year
projection of profit for a corporation.

• Assumption
• Selling price is 60 (not change)
• Fixed cost 1500 (grow at constant rate)
• Number of units 80 (grow at a constant rate)
• Variable cost 45 per unit

3
The Basic Planning Problem
4
The Basic Planning Problem
Data table is created to view What-if Analysis on
the growth rates for fixed costs and units sold
5
The Basic Pricing Problem

• The central issue in pricing is deterring how
  quantity sold depends on the price.
• Demand in most cases is elastic, that is when the
  price increases, the demand decreases.
• The simplest assumption is that demand is a
  linearly decreasing function of price.
• Organization assumptions or judgments
• At price 70, sales will be 2,400
• One dollar increase in price 37 units decrease
  in the sale
• Used the above information to express the number
  of units sold as a function of price.
• PRICE UNIT
• (PRICE 70) 0 70
  2400
• (PRICE 70) gt 0
  INCREASE DECREASE
• (PRICE 70) lt 0
  DECREASE INCREASE
• Changed in number of units -37 (PRICE -
  70)
• Using the second assumption, then
• Number of unit 2400 Changed in number of
  units
• Number of units 2400 - 37 (PRICE - 70)
  4990 ( 37 PRICE )

6
The Basic Pricing Problem
4990-37B3
50000 35 B4
B3 B4
B6-B5
7
Maximum profit occurs around 42,750 at the price
85. More accurate value can be generated by
entering more numbers in the column providing the
input to the table. However exactness might be
misleading because of the uncertainty in the
demand function.
8
Nonlinear Cost and Demand Function

• Most applications in real life have nonlinear
  relationships. They follow a curved, nonlinear XY
  functions.
• EX.
• Torrington Corporation, deciding whether they
  should introduce a new product.
• Production prepares a cost estimate for making up
  to 150 units. (variable cost will not be a linear
  function of quantity)
• Graph is prepared to show the curverepresenting
  variable cost
• The problem is to develop formulas to give the
  cost values for any value of the quantity.

9
Nonlinear Cost and Demand Function Mathematical
Models of an XY Functions
Suppose Torrington wishes to determine the cost
associated with the quantity 70. Find the slope
of any line segment. Consider the cost at 50
which 3,000 then slope is calculated as Slope
(Y2-Y1)/(X2-X1) (4,500-3,000)/(100-
50) 30 Cost Y1 Slope ( X-X1) Quantity
(X) 70, X150, and Cost Y1 3,000 Then Cost
for X 3,000 30 (70 50) 3,600
10
Nonlinear Cost and Demand Function Spreadsheet
Models of an XY Functions

• Cells A3 to C7 (Number of units, variable cost,
  slop) contain a lookup table that Excel uses to
  find the necessary parameters for a given number
  of units.
• Cells B9 to B11 are user-entered data.(Input
  factors)
• Cells B12 to B14 use the lookup table to find
  required items of data.
• Cell B15 computes the variable cost for the
  number of units entered in cell B10,
• Variable cost UP ( NUMBER OF UNIT LEFT )
  SLOPE using the formula B13 ( B10-B12)B14
• Cell B16 computes the total cost by adding the
  fixed and variable costs using B11 B15
• Cell B17 compute revenue using B9B10
• Cell B18 profits using B17-B16

11
Nonlinear Cost and Demand Function Spreadsheet
Models of an XY Functions
The table in the left shows the data table
comparing number of units with profits Traini
ng Exercise Calculate profits against number of
units being sold, where number of units start
from 0 up to 150 with increment of 10 units. Find
the break-even point using Goal seeker?
12
Nonlinear Cost and Demand Function Approximating
the Cost with a Cubic Function

• Curves are good facility for representing
  nonlinear functions.
• Polynomial is a class of functions that are often
  satisfactory.
• The linear (first-order) function has the
  following form
• 2 (10 X)
• Quadratic function assumes the form
• -24 (56 X2)
• and an example of a cubic function is
• (5.3 X2) ( 21.6 X3)

13
Nonlinear Cost and Demand Function Approximating
the Cost with a Cubic Function
The general approach 1- Try polynomials,
quadratic, cubic, and so on, on the spreadsheet
representing the situation. 2- Calculate the
values for the given curve 3- Calculate the
square of the deviations (differences), add
them 4- Minimize the sum with Excel Solver by
allowing the coefficient of function to be
changed.
14
Nonlinear Cost and Demand Function Approximating
the Cost with a Cubic Function

• Cell A3 uses the formula AVERAGE(A2, A4).
• Column C calculates the cubic function based on
  the current coefficients in cells F3 to I3. For
  example Cell C2 uses the formula
• F3 (G3A2)(H3A22) (I3A23).
• Column D calculates the squared difference
  between the variable cost and the cube function.
  For example, cell D2 uses the formula
  (B2-C2)2.
•  Cell D10 shows the original sum of deviations.

15
Nonlinear Cost and Demand Function Approximating
the Cost with a Cubic Function

• The target cell D10 needs to be minimized. There
  are no constraints. The cells to vary are the
  cubic coefficient in cells F3 to I3.
• The better way to judge how good the
  approximation is to compare the given curve with
  the calculated curve.
• If the management feels that the approximation is
  not good enough, a fourth, fifth order polynomial
  or other type of function can be tried.

16
Preparing a Five-Year plan

• The lookup table is no longer required as cubic
  equation replaces it
• Cell B7 in the cubic model now computes the
  variable cost using the cubic function
  D6(E6B5)(F6B52)(G6B53)
• This Modified Model can be used to perform
  scenario analysis

17
Preparing a Five-Year plan

• The management is particularly interested in
  three scenarios
• They are also interested in growth over the nest
  five years

18
Preparing a Five-Year plan
19
The Impact of Pricing

• Apply the cubic function approach to the analysis
  of pricing
• The Marketing suggests the following pegs to
  approximate the price versus quantity
• The first step is to develop a cubic function for
  quantity based on price.

Price Quantity 20 250 40 150
60 100 80 60
20
The Impact of Pricing