slides by john loucks st. edward’s university

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© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Slides by

JohnLoucks

St. Edward’sUniversity

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Chapter 8Nonlinear Optimization Models

A Production Application Blending: The Pooling Problem Forecasting Adoption of a New

Product

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Introduction

Many business processes behave in a nonlinear manner.

For instance, The quantity demanded for a product is often a nonlinear function of the price.

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Introduction

A nonlinear optimization problem is any optimization problem in which at least one term in the objective function or a constraint is nonlinear.

Nonlinear terms include The nonlinear optimization problems

presented on the upcoming slides can be solved using computer software such as LINGO and Excel Solver.

31 2 1 2 3, 1/ , 5 , and log .x x x x x

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Par, Inc. manufactures golf bags• Two Models: Standard (S) and Deluxe (D)

Four Operations Required for each Bag• Cutting and Dyeing• Sewing• Finishing• Inspection and Packaging

Example: Production Application Par, Inc.

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Example: Production Application Assume that the demand for Standard (S) and

Deluxe (D) golf bags are (projected sales – cost differential):

S = 2250 – 15Ps

D = 1500 – 5Pd

where Ps = the price of a Standard bag Pd = the price of a Deluxe bag.

We will need to isolate Ps and Pd:- 15Ps = 2250 – S 5Pd = 1500 – D- Ps = 2250/15 –S/15 Pd = 1500/5 –

1/5D- Ps = 150 – 1/15S Pd = 300 – 1/5D

Example: Production Application Profit Contribution as a Function of Demand The profit contributions (revenue – cost) are:

PsS – 70S (Standard bags) PdD – 150D (Deluxe bags)

• Solving for Ps we get:• PsS – 70S• (150 – 1/15S)S – 70S• 80S – 1/15S^2

• Solving for Pd we get• PdD – 150D• (300 – 1/5D)D – 150D• 150D – 1/5D^2

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Example: Production Application Total Profit Contribution

Total Profit Contrib. = 80S – 1/15S^2 + 150D – 1/5D^2

This function is an example of a quadratic function

because the nonlinear terms have a power of 2.

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Par Inc, Unconstrained Solution

If we were to just solve the optimization equation, then we would find that it is:

S = 600, D = 375, Ps = 110, Pd = 225 BUT WE HAVENT INCLUDED THE CONSTRAINTS!

• 7/10s + 1d <= 630 (Cutting and Dyeing)• 1/2s + 5/6d <= 600 (Sewing)• 1s + 2/3d <= 708 (Finishing)• 1/10s + 1/4d <= 135 (Inspecting and

Packing)• s, d >= 0

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Par Inc

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Par Inc

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Par Inc

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Dual Values

Recall that the dual value is the change in the value of the optimal solution per unit increase in the right-hand side of the constraint.

The interpretation of the dual value for nonlinear models is exactly the same as it is for LPs.

However, for nonlinear problems the allowable increase and decrease are not usually reported.

This is because for typical nonlinear problems the allowable increase and decrease are zero.

That is, if you change the right-hand side by even a small amount, the dual value changes.

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Blending: The Pooling Problem

Blending problems arise when a manager must decide how to blend two or more components (resources) to produce one or more products.

It is often the case that while transporting or storing the blending components, the components must share a pipeline or storage tank.

In this case, the components are called pooled components.

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Blending: The Pooling Problem

Two types of decisions arise: What should the proportions be for the

components that are to be pooled? How much of the pooled and non-pooled

components will be used to make each of the final products?

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Example: Blending - The Pooling Problem

Grand Strand refinery wants to refine three petroleum components into regular and premium gasoline in order to maximize total profit contribution. Components 1 and 2 are pooled in a single storage tank. Component 3 has its own storage tank.

The maximum number of gallons available for the three components are 5000, 10,000, and 10,000, respectively. The three components cost $2.50, $2.60, and $2.84, respectively. Regular gasoline sells for $2.90 and premium sells for $3.00. At least 10,000 gallons of regular gasoline must be produced.

The product specifications for regular and premium gasoline are shown on the next slide.

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Grand Strand

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Product Specifications• Regular gasoline At most 30% component 1

At least 40% component 2At most 20% component 3

• Premium gasolineAt least 25% component 1At most 45% component 2At least 30% component 3

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Define the 6 Decision Variables y1 = gallons of component 1 in the pooling tank y2 = gallons of component 2 in the pooling tank

xpr = gallons of pooled components 1 and 2 in regular gas

xpp = gallons of pooled components 1 and 2 in premium gas

x3r = gallons of component 3 in regular gasolinex3p = gallons of component 3 in premium gasoline

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Define the Objective Function Maximize the total contribution to profit (which isrevenue from selling regular and premium gasolinesminus cost of buying components 1, 2, and 3):

Max 2.90(xpr + x3r) + 3.00(xpp + x3p) – 2.50y1 – 2.60y2 – 2.84(x3r + x3p)

(Note: xpr + x3r = gallons of regular gasoline sold,

xpp + x3p = gallons of premium gasoline sold,

x3r + x3p = gallons of component 3 consumed)

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Define the 11 ConstraintsComponents 1 and 2 consumedSimilar to a “flow-in-flow-out” constraint:

1) y1 + y2 = xpr + xpp

Component availability:2) y1 < 5,0003) y2 < 10,000

4) x3r + x3p < 10,000Minimum production of regular gasoline:

5) xpr + x3r > 10,000

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Define the 11 Constraints (continued)Regular gasoline specifications:

We have to measure the proportion of a component in the pool when calculating the specification constraints

6)

7)

8)

13

1 2.3( )pr pr r

y x x xy y

23

1 2.4( )pr pr r

y x x xy y

3 3.2( )r pr rx x x

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Define the 11 Constraints (continued)Premium gasoline specifications:

9)

10)

11)

Non-negativity: xpr , xpp , x3r , x3p , y1 , y2 > 0

13

1 2.25( )pp pp p

y x x xy y

23

1 2.45( )pp pp p

y x x xy y

3 3.3( )p pp px x x

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Solution (with pooling optimal sol. = $5831.43

Without Pooling optimal sol. = $7100

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Forecasting Adoption of a New Product

Forecasting new adoptions (purchases) after a product introduction is an important marketing problem.

We introduce here a forecasting model developed by Frank Bass.

Nonlinear programming is used to estimate the parameters of the Bass forecasting model.

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The Bass model has three parameters that must be estimated.• m is the number of people estimated to

eventually adopt a new product• q is the coefficient of imitation which

measures the likelihood of adoption due to a potential adopter influenced by someone who has already adopted the product

• p is the coefficient of imitation which measures the likelihood of adoption assuming no influence from someone who has already adopted the product.

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Developing the Forecasting Model• Ft , the forecast of the number of new adopters

during time period t , is Ft = (likelihood of a new adoption in time

period t) x (number of potential adopters

remaining at the end of time period t – 1)

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Developing the Forecasting Model• Essentially, the likelihood of a new adoption is

the likelihood of adoption due to innovation plus the likelihood of adoption due to imitation.

• Let Ct - 1 denote the number of people who have adopted the product up to time t - 1.

• Hence, Ct - 1 /m is the fraction of the number of people estimated to adopt the product by time t – 1.

• The likelihood of adoption due to imitation is q(Ct - 1 /m).

• The likelihood of adoption due to innovation and imitation is p + q(Ct - 1 /m).


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Developing the Forecasting Model• The number of potential adopters remaining at

the end of time period t – 1 is m - Ct - 1 .• Hence, the complete forecast model is given

by

Ft = (p + q(Ct - 1 /m)) (m - Ct - 1)


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Nonlinear Optimization Problem Formulation

Ft = (p + q(Ct - 1 /m)) (m - Ct - 1), t = 1,

…., N Et = Ft - St , t = 1, …., N

where N = number of time periods of data available

Et = forecast error for time period t St = actual number of adopters (or a

multiple of that number such as sales) in time period t


2

1Min

N

tt

E

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Example: Forecasting New-Product Adoption

Maid For YouMaid For You is a residential cleaning service

firm that has been quite successful developing a client base inthe Chicago area. The firm plans to expand to other majormetropolitan areas during the next few years.

Maid For You would like to use its Chicagosubscription data (on the next slide) to develop a modelfor forecasting service subscriptions in regions where itmight expand. The first step is to estimate values for p(coefficient of innovation) and q (coefficient of imitation).

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Subscribers and Cumulative Subscribers (1000s)

Month Subscribers St Cum. Subscribers Ct

1 0.53 0.53 2 2.94 3.47 3 3.60 7.07 4 4.8511.92

5 3.4415.36 6 2.7618.12 7 1.8219.94 8 0.9320.87 9 0.6121.48

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Forecasting (General Form)

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Define the Objective Function Minimize the sum of the squared forecast errors:

2 2 2 2 2 2 2 2 21 2 3 4 5 6 7 8 9Min E E E E E E E E E


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Define the ConstraintsDefine the forecast for each time period:

1) F1 = pm2) F2 = (p + q( 0.53/m)) (m – 0.53)3) F3 = (p + q( 3.47/m)) (m – 3.47)

4) F4 = (p + q( 7.07/m)) (m – 7.07)5) F5 = (p + q(11.92/m)) (m – 11.92)6) F6 = (p + q(15.36/m)) (m – 15.36)7) F7 = (p + q(18.12/m)) (m – 18.12)8) F8 = (p + q(19.94/m)) (m – 19.94)9) F9 = (p + q(20.87/m)) (m – 20.87)

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Define the Constraints (continued)Define the forecast error for each time period:

10) E1 = F1 – 0.5311) E2 = F2 – 2.9412) E3 = F3 – 3.60

13) E4 = F4 – 4.8514) E5 = F5 – 3.4415) E6 = F6 – 2.7616) E7 = F7 – 1.8217) E8 = F8 – 0.9318) E9 = F9 – 0.61

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Optimal Forecast Parameter Values Parameter

Value p 0.08 q 0.62

m 21.26

The value of the imitation parameter q = .62 issubstantially larger than the value of the innovationparameter p = .08. Subscriptions gain momentumover time due mainly to very favorable word-of-mouth.

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Optimal Solution


Month Forecast Subscribers Error 1 1.77 0.53 1.24 2 2.05 2.94 -0.89 3 3.29 3.60 -0.31 4 4.12 4.85 -0.73 5 4.03 3.44 0.59 6 3.14 2.76 0.38 7 1.93 1.82 0.11 8 0.88 0.93 -0.05 9 0.27 0.61 -0.34

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Subscribers versus Forecasts


Subscribers

Month

Subs

crib

ers (

1000

s) 5

4

3

2

1

1 2 3 4 5 6 7 8 9

Forecast

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End of Chapter 8

slides by john loucks st. edward’s university

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