A Robust Optimization Model for Strategic Production and Distribution Planning for a New Product

Renee J. Butler, Ph.D., P.E. University of Central Florida School of Industrial Engineering and Management Systems 4000 Central Florida Blvd. Orlando, FL 32816-2993 rjbutler@mail.ucf.edu

Jane C. Ammons, Ph.D., P.E. Georgia Institute of Technology School of Industrial and Systems Engineering 765 Ferst Drive Atlanta, GA 30332-0205 jammons@isye.gatech.edu

Joel Sokol, Ph.D. Georgia Institute of Technology School of Industrial and Systems Engineering 765 Ferst Drive Atlanta, GA 30332-0205 jsokol@isye.gatech.edu

July 22, 2003

A paper to be submitted to one of the following

Management Science Operations Research

Abstract

We study the strategic design of a supply chain for new products. Planning a supply chain for a

new product requires addressing both uncertainties in demand and cost and changes in market

conditions over time. We develop a mathematical programming model for new products that

includes both financial viability and robustness relative to uncertainty. An example based on a

Fortune 200 company’s efforts to develop a supply chain for a spin-off product illustrates the

model. We derive insights on the cost drivers for supply chain development decisions and the

tradeoff between investment and cash flow. We also demonstrate the effects of uncertainty and

delaying investment decisions in supply chain development.

2

1 Introduction

In this paper we present a robust optimization model that addresses uncertainty in

demand and supply chain costs for a new product launch. We begin with a discussion of the

uncertainties found in new product supply chains and an introduction to robust optimization. In

Section 2 we discuss previous robust supply chain models to provide a framework for the new

robust model we propose in Section 3. In Section 4 we relate our robust supply chain model to

others that have been proposed. We illustrate our robust supply chain model with an example in

Section 5.

Butler, Ammons, and Sokol (2003) develop and solve a deterministic model for strategic

supply chain design for new products using the expected values of the demand and costs.

However, new product supply chains experience uncertainties in demand, price, costs, and

capacity. We discuss sources of uncertainty in new product supply chains in more detail below.

Uncertainty in customer demand is very important in new product supply chain design.

Some information might be known about the present demand for the product if the new product

is entering an existing market. Alternatively, very little might be known about the demand if the

product creates a new market, as when cell phones were first introduced. Kurt Hellstrom,

president of Ericsson, states in PC Magazine, “Analysts predicted in 1980 that one million

mobile phones would be used worldwide by the year 2000. They were wrong by 599 million”

(Goetschalckx, 2003).

In the case where the company is making a product to enter an existing market, a current

selling price for the product is known. The price may differ for the entering company based on

relative quality/delivery compared to competitors. The market may also have an initial

3

adjustment when a new competitor enters the market. For products in an entirely new market,

the forecasted selling price may have a high degree of uncertainty.

In addition to cost uncertainties, capacity might be uncertain due to equipment failure and

quality issues. For spin-off products (new products launched by existing companies), capacity

uncertainty can also result from sharing resources with existing products.

2 Literature Review

In this section, we review several robust optimization approaches to supply chain

modeling. The important factors that distinguish our work from previous work are the types of

uncertainty as well our new modeling approach.

One of the traditional ways to address uncertainty is to solve a deterministic problem

using the mean values of the parameters. However, the optimal solution for the mean value

problem could be very sub-optimal if another scenario is realized (Kouvelis and Yu, 1997). Bai,

Carpenter, and Mulvey (1997) suggest performing sensitivity analysis for the uncertain

parameters and if the solution is found to be sensitive to the uncertain parameters, search for an

alternative feasible point. However, the single parameter sensitivity analysis still neglects the

effects of parameter interactions.

Stochastic optimization models using probability distributions can also be used to address

uncertainty. These models require a complete description of the uncertain parameters, grow

large quickly, and become difficult to solve. Stochastic production and distribution optimization

models are typically decomposed into two-stage stochastic programs to avoid excessive

computation times.

For new products, it is extremely difficult to identify probability distributions for

unknown parameters such as demand because there is no historical data. The uncertain

4

parameters could more easily be described as possible scenarios with the help of industry

experts. Defining scenarios is still a difficult task and requires the decision maker to first

identify the main factors that drive the uncertainty, and then describe the relationship between

them. Understanding these relationships helps to eliminate many unrealistic scenarios (Kouvelis

and Yu, 1997). It is desirable to limit the number of scenarios for tractability.

Once the scenarios are defined, they each need to be assigned a probability of occurrence.

From a modeling standpoint, the probabilities can vary or be equally likely. In reality, these

probabilities are difficult to determine, especially for new products. Industry experts find it

easier to predict the relative likelihood that one event is more likely than another rather than

prescribe exact probabilities. Therefore, sensitivity analysis is necessary to verify that the

solution is not highly dependent on the exact probabilities.

A common modeling approach that uses discrete scenarios is robust optimization; that is,

to find a near-optimal solution that is not overly sensitive to any specific realization of the

uncertainty (Bai, Carpenter and Mulvey, 1997). Alternatively, robust optimization can be

described as finding a solution whose objective value is suitably close to that of the optimal

solution for each scenario. The resulting overall solution might not be optimal for any of the

potential scenarios (Mulvey, Vanderbei, and Zenios, 1995).

There are several key differences between stochastic optimization and robust

optimization. Both try to utilize information about uncertainty in mixed-integer programs.

However, stochastic optimization considers only the minimization of expected costs or

maximization of expected profits and fails to consider the variance in the solution. Therefore,

the stochastic optimization solution could be very sub-optimal if the parameters change.

Stochastic optimization requires complete probability distributions, whereas robust optimization

5

utilizes discrete scenarios to describe the uncertain parameters. Additionally, robust

optimization can be used to model risk averse decision makers. According to Bai, Carpenter,

and Mulvey (1997), robust optimization also requires less management attention than stochastic

optimization because it is not as critical to adjust the recourse variables as uncertainty is

resolved.

We divide the robust supply chain models into two basic categories: regret models and

variability models. The “regret” of a scenario is measured as the difference between the cost of

the chosen solution for that scenario and the cost of the optimal solution for that scenario. The

difference can be expressed in absolute terms or as a percentage. Variability models control the

spread of the costs by adding standard deviation, variance, or other measures to the objective

function. Other robust models use a variety of approaches, including minimizing the maximum

costs and building excess capacity to handle uncertainty. Below we discuss select examples for

various types of robust models in more detail.

2.1 Regret Models

Kouvelis and Yu (1997) define two regret criteria for robustness. The robust deviation

decision is “the [decision] that exhibits the best worst-case deviation from optimality, among all

feasible decisions over all realizable input data scenarios.” They also define the relative robust

decision as “the one that exhibits the best worst-case percentage deviation from optimality.”

These criteria serve to control the absolute worst that might happen. The worst that may happen

for new product launches is that the new product fails, which happens to many new products. In

fact, only one in one hundred new products find long-term success (Stevens and Burley, 1997).

With this conservative approach, a new product would never be launched.

6

Gutierrez, Kouvelis, and Kurawarwala (1996) design a different robustness approach.

Instead of addressing the worst case, they require a robust network design to be within p% of the

optimal solution for any realizable scenario. Therefore, they in effect add a constraint to their

model to ensure robustness.

Our model combines the features of these regret models. Whereas regret models that use

a min-max problem structure, like that of Kouvelis and Yu (1997), tend to be more difficult to

solve than other robust models (Snyder and Teo, 2002), our model utilizes a weighted average of

the regret measure and solves more quickly.

2.2 Variability Models

An alternative definition of robustness is to find a near-optimal solution that is not overly

sensitive to any specific realization of the uncertainty (Bai, Carpenter and Mulvey, 1997). These

robust optimization models include a measure of variability rather than regret. The goal is to

minimize expected costs (maximize expected profit) and to reduce the variability over the

scenarios. Variability can be measured by variance (Hodder and Dincer, 1986; Mulvey,

Vanderbei and Zenios, 1995; Bok, Lee, and Park, 1998; Yu and Li, 2000) or by standard

deviation (Goetschalckx, et al., 2001), both of which make the objective function non-linear.

Both methods also assume symmetric risk, so that it is equally bad for costs to be below the

mean or above. Several other measures of variability have been used, including the von

Neumann-Morganstern expected utility function (Bai, Carpenter and Mulvey, 1997) and the

upper partial mean (Ahmed and Sahinidis, 1998), to allow asymmetry, but these functions are

often hard to compute. Additionally, when coefficients in a model are uncertain, the constraints

may not necessarily be satisfied for all scenarios. In such a situation, it is convenient to

introduce additional variables that represent the slack or surplus in the constraints. These

7

variables, called recourse variables, are included in the objective function as an infeasibility

penalty (Mulvey, Vanderbei and Zenios, 1995; Yu and Li, 2000). We discuss the variability

models in more detail below.

2.3 Other Robust Models

In addition to the regret and variability models we have discussed, there are several other

approaches to robust a supply chain design. Kouvelis and Yu (1997) minimize the maximum

costs of the supply chain, Voudouris (1996) and Sabri and Beamon (2000) address uncertainty

by building excess capacity in the supply chain, Applequist, Penky, and Rekalaitis (2000)

propose a new metric called risk premium for evaluating supply chains, and Vidal and

Goetschalckx (2000) use extensive sensitivity analysis.

2.4 Literature Summary

Table 1 defines the stochastic features included in the papers discussed in this paper.

Almost all of the papers consider stochastic demand, while a few consider stochastic demands

and costs together. Goetschalckx, et al. (2001) address stochastic demand combined with

stochastic costs in a simulation-based approach. We address stochastic demands and costs using

a mixed-integer programming model. We do not include stochastic lead times or supplier

reliability because these factors are important only when determining inventory levels. Our

model is strategic, not operational, and inventory decisions are not included.

We also present an overview of the literature for robust optimization definitions to show

the similarities and differences between the different robust definitions. We use a regret model

rather than a variability model to avoid a symmetric penalty that drives all solutions toward the

average costs. We do not want the company to miss the opportunity to make high profits if the

new product is very successful. Further, we combine the robust criteria developed by Kouvelis

8

and Yu (1997) with the modeling approach of Gutierrez, Kouvelis, and Kurawarwala (1996) to

create a weighted average regret model. This ensures that one worst-case scenario is not the

driver of the solution. We discuss the model development in the next section.

Table 1. Stochastic Features in Selected Robust Supply Chain Models

Stochastic Feature Dem

and

Pric

es

Exch

ange

rate

Fixe

d fa

cilit

y co

sts

Prod

uctio

n co

sts

Tran

spor

tatio

n co

sts

Cap

acity

Lead

tim

es

Supp

lier

relia

bilit

y

Hodder and Dincer (1986) X X X Malcolm and Zenios (1994) X Gutierrez, Kouvelis and Kurawarwala (1996) X

Bok, Lee and Park (1998) X

Ahmed and Sahinidis (1998) X X

Applequist, Penky, and Rekalaitis (2000) X X X

Vidal and Goetschalckx (2000) X X X X

Yu and Li (2000) X X

Goetschalckx, et al. (2001) X X X X

Butler (2003) X X X X X X

3 Robust Lagrangian Model

We extend the viable supply chain model of Butler, Ammons, and Sokol (2003) to

develop a robust scenario-based formulation to encompass multiple scenarios. The viable supply

chain model maximizes the net profit for a production and distribution system with a constraint

to require the cumulative net cash position (revenues less costs) for each period to remain above

a predetermined (possibly negative) threshold. This model limits the potential losses associated

with a failed new product venture (14) and (15), allows facilities to open and close over the

planning horizon (10) - (13), and ties future demand to past delivery performance (6) and (7).

For this formulation, we define Ω to be the set of all scenarios. Each scenario ω ∈Ω

occurs with probability ωρ . The input data for each scenario ω consist of a set of prices, costs,

capacities, and demands. Each scenario has a corresponding set of continuous variables, but

there is only one set of binary facility variables for all scenarios in the robust model. The

optimal net profit for each scenario *Oω is obtained by solving the viable supply chain model for

each scenario and is an input parameter for the robust model.

Below, we define the notation for the input parameters and the decision variables.

Indices for Decision Variables and Parameters I set of products 1, 2, …, niIj set of products that can be made on machine j J set of machines 1, 2, …, nj K set of distribution centers 1, 2, …, nkL set of customers 1, 2, …, nl M set of transportation modes 1, 2, …, nm T set of time periods 1, 2, …, nt

Ω set of scenarios 1, 2, …, nω Parameters

Based on these sets, the model parameters are defined as follows. The superscripts

represent the echelon of the supply chain: 1 = factory level, 2 = distribution center level, and 3 =

customer level. The cost and price parameters are discounted for each time period to represent

the time value of money.

itSω selling price per unit of product i in period t and scenario ω 1

ijtCω production costs per unit of product i made on machine j in period t and scenario ω 2

iktCω unit handling costs of product i at distribution center (DC) k in period t and scenario ω

2ijktmTω transportation cost per unit of product i from machine j to DC k by mode m in period

t and scenario ω 3ikltmTω transportation cost per unit of product i from DC k to customer l by mode m in period

t and scenario ω 1

jtFω fixed cost for operating machine j in period t and scenario ω

9

2ktFω fixed cost for operating DC k in period t and scenario ω

1jtFω one-time fixed cost to purchase/convert machine j in period t and scenario ω

1jtFω one-time cost to decommission machine j in period t and scenario ω

2ktFω one-time fixed cost to open DC k in period t and scenario ω

2ktFω one-time cost to close DC k or break lease in period t and scenario ω

ωρ the probability of occurrence of scenario ω λ robustness penalty parameter

*Oω the optimal net profit for scenario ω

iltDω forecasted demand of product i for customer l during period t in scenario ω

tH discounted cash flow threshold for period t 1

jPω production capacity for machine j for scenario ω 2kPω handling capacity for DC k for scenario ω

1ijA factor to adjust product i into standard units of production for machine j 2ikA factor to adjust product i into standard units for DC k

jη lead-time for equipment purchase, refurbishment, or certification of machine j

kγ lead-time for distribution center construction at location k

lβ percentage decrease of future demand of customer l due to past shortages Decision Variables The decision variables are described below. Variables of type D, v, x, and z are

continuous, whereas y variables are binary. The superscripts represent the echelon level.

iltDω demand for product i that the company sees from customer l in period t and scenario ω based on past delivery performance

1ijtxω units of product i made by machine j in period t and scenario ω

2ijktmxω units of product i shipped from machine j to DC k in period t by mode m in scenario

ω 3

ikltmxω units of product i shipped from DC k to customer l in period t by mode m in scenario ω

3iltxω units of unsatisfied demand for customer l for product i in period t and scenario ω

1ijtzω units of product i in inventory at machine j at the end of period t for scenario ω

10ijzω initial machine level inventory for new products = 0 for all scenarios

2iktzω units of product i in inventory at DC k at the end of period t for scenario ω

2 initial DC inventory level for new products = 0 for all scenarios ikzω 0

tvω cumulative net profit after period t for scenario ω 10

1jty 1 if machine j is used in period t, 0 otherwise 2kty 1 if DC k is open in period t, 0 otherwise 1jty 1 if machine j is purchased in period t, 0 otherwise 1jty 1 if machine j is decommissioned in period t, 0 otherwise 2kty 1 if DC k is opened in period t, 0 otherwise 2kty 1 if DC k is closed in period t, 0 otherwise

ωα robust deviation for scenario ω

The robust profit for each scenario ω is defined as

( , ) R x yω ω =

( ) ( )

( ) ( )

3 3 1 1 2 2 2

, , , , , , , , , ,

1 1 1 1 1 1 2 2 2 2 2 2

, ,

it ikltm ikltm ijt ijt ikt ijktm ijktmi k l t m i j t i j k t m

jt jt jt jt jt jt kt kt kt kt kt ktj t k t

S T x C x C T x

F y F y F y F y F y F y

ω ω ω ω ω ω ω ω

ω ω ω ω ω ω

⎡ ⎤− − − +⎢ ⎥⎢ ⎥− + + − + +⎢ ⎥⎣ ⎦

∑ ∑ ∑

∑ ∑(1)

We use Lagrangian relaxation to incorporate the relative robustness for each scenario (the

percentage deviation of the robust profit from the optimal profit) as a penalty term in the

objective function. The objective function (2) shows the expected net profit minus a nonnegative

robustness penalty (λ ) times the relative robust deviation ( ωα ). Here, ωα is a decision variable.

The robust Lagrangian model with the relative robustness measure is shown below.

Maximize ( ( , ) )R x yω ω ω ωω

ρ λ∈Ω

−∑ α (2)

Subject to

( )*

*

- ,

O R x yO

ω ω ϖω

ω

α≤ ω∀ (3)

1 1 1 1

j

ij ijt j jti I

A x P yω ω∈

≤∑ , , j t ω∀ (4)

2 2 2 2

, ,ik ijk tm k k t

i j m

A x Pω ω≤∑ y , , k t ω∀ (5)

3 3

,ik ltm ilt ilt

k m

x x Dω ω+ =∑ ω , , , i l t ω∀ (6)

3( 1)ilt l il t iltD x Dω ωβ −− = ω , , , i l t ω∀ (7)

1 1 2( 1)

,ij t ijt ijktm ijt

k m

z x x zω ω ω− + − =∑ 1ω , , , i j t ω∀ (8)

11

12

2zω2 2 3

( 1), ,

ik t ijktm ikltm iktj m l m

z x xω ω ω− + − =∑ ∑ , , , i k t ω∀ (9)

( )1 1 1

( 1)j

jt j t j ty y y

η− −− ≤ , j t∀ (10)

1 1( 1 )

1j t jty y− − ≤ jty , j t∀ (11)

( )2 2 2

( 1) kk t k t k ty y y γ− −− ≤ , k t∀ (12) 2 2

( 1 )k t kt k ty y− − ≤ 2y , k t∀ (13)

t tv Hω ≥ , t ω∀ (14)

( )3 3 1 11

, , , ,t t it ikltm ikltm ijt ijt

i k l m i j

v v S T x C xω ω ω ω ω ω ω−= + − −∑ ∑

( )2 2 2

, , ,ikt ijktm ijktm

i j k m

C T xω ω ω− +∑ ( )1 1 1 1 1 1jt jt jt jt jt jt

j

F y F y F yω ω ω− + +∑

( )2 2 2 2 2 2kt kt kt kt kt kt

j

F y F y F yω ω ω− + +∑ , t ω∀ (15)

0ωα ≥ ω∀ (16) 1 2 3 3 1 2, , , , , , 0ilt ijt ijktm ikltm ilt ijt iktD x x x x z zω ω ω ω ω ω ω ≥ , , , , , , i j k l m t ω∀ (17)

1 0,1jty ∈ , 1 0,1jty ∈ , 1 0,1jty ∈ ,

2 0,1kty ∈ , 2 0,1kty ∈ , 2 0,1kty ∈ , , j k t∀ (18)

Because the objective function (2) includes the expected net profits, we use the relative

robustness measure in the model to insure that the robust configuration performs reasonably well

in scenarios with lower profit. For new product applications, this ensures that the model

addresses the scenarios where the product does not enter the market successfully. The absolute

deviation requirement places more emphasis on scenarios with higher net profits, whereas the

relative robustness requirement places more emphasis on scenarios with lower net profits. Also,

the penalty term ωλα is multiplied by the scenario probability so that one scenario with a large

regret but a small probability does not drive the model.

We use one λ value so that there is only one parameter to adjust and we can create an

efficient frontier of the solution with different penalty values. The penalty parameter is a goal

programming weight measured in dollars which represents the value of being 100% away from

the optimal net profit. Thus, the value of will be on the order of the total net profit for the time

λ

λ

horizon. By changing , the relative importance of the expected profit and the closeness to

optimal profit for scenario ω can be controlled.

λ

13

0For all scenarios, because ( )* - ,O R x yω ω ω ≥ *Oω is an upper bound on Rω . For a

scenario ω in which the new product fails and the optimal profit * 0Oω ≤ , ( )*

*

- ,0

O R x yO

ω ω ω

ω

≤ .

Therefore, we use *Oω in the denominator of the relative robustness constraint to obtain a

positive deviation even when the optimal net profit for the scenario is negative. Otherwise, the

objective function value would increase as the scenario profit Rω becomes more negative.

To better understand the effect of λ , we make a substitution for ωα in the objective

function. Since 0λ ≥ and 0ωα ≥ , the negative coefficient in front of ωλα in the maximize

objective function will force ωα to be as small as possible. The robustness constraints (3) are

the only requirement for ωα , so the robustness constraints will be tight at equality at optimality.

Therefore, we substitute *

*

( , )O R x yO

ω ω ω

ω

− into the objective function for ωα . After simplifying,

we obtain the objective function

*Max 1 ( , )R x yO ω ω ω

ω ω

λ ρ∈Ω

⎧ ⎫⎛ ⎞⎪ +⎜ ⎟⎨⎜ ⎟⎪ ⎪⎝ ⎠⎩ ⎭∑ ⎪

⎬ . (19)

For a given λ , *Oω

λ is larger for less profitable scenarios because of the smaller *Oω value in

the denominator. For new product launches, this correctly puts more weight on scenarios with

lower positive net profit to increase the chances that the product survives in the long-term.

In summary, we use the relative robustness definition of Kouvelis and Yu (1997) to

measure closeness to the optimal net profit. However, the measure could be replaced with any of

the robustness measures proposed in the literature and described in Section 2. In the next

section, we illustrate other robust measures with this model.

4 General Robust Model

The robust Lagrangian model we develop in this paper generalizes many of the previous

robust optimization models in the literature. We show that all of the regret models of Kouvelis

and Yu (1997) and Gutierrez, Kouvelis and Kurawarwala (1996) can be derived from our general

model. Additionally, many of the variability models, like those of Hodder and Dincer (1986),

Goetschalckx, et al. (2001), and Ahmed and Sahinidis (1998), can be obtained by using a

different robustness measure in our model.

Below, we explicitly discuss the connections between the models. To illustrate the

relationship between our model and previous models, we identify the changes required for the

robust objective function (20) and the general robustness constraint (21) of the robust Lagrangian

model and then rewrite the previous models’ objective function and robustness constraint in

terms of our model.

General Robust Lagrangian Model

Below we restate the robust objective and the robustness constraint of the robust

Lagrangian model in general terms for use in the model comparisons.

Robust objective Max ( )( )( , )R x yω ω ω ωω

ρ λ− α∑ (20)

General robustness constraint Robustness measure ωα≤ ω∀ (21)

0, 0, 0,1x yω ω α ≥ ≥ ∈ ω∀ (22)

14

We show that this is a general form for several models.

Expected Value Net Profit Models

We can easily equate our model to a net profit model with and without a regret-type

robustness constraint. We can obtain an expected profit model by setting λ in our

Lagrangian model. This model maximizes the net profit without a robustness constraint:

= 0

Max ( , )R x yω ω ωω

ρ∈Ω∑ .

If we impose ωα α= in our constrained model, where α is a parameter for the allowable

percentage scenario deviation from optimality, our model is the same as that of Gutierrez,

Kouvelis and Kurawarwala (1996):

Min ( , )R x yω ω ωω

ρ∈Ω

−∑

Subject to *

*

( , )O R x yO

ω ω ω

ω

α−≤ ω∀

Worst-case Regret Models

The models presented in Kouvelis and Yu (1997) are conservative models that try to

minimize the effect of the worst-case scenario by minimizing the largest observed deviation from

the optimal for all scenarios. We can obtain worst-case models from our Lagrangian model by

setting ωα α= and the penalty parameter λ sufficiently large so that the robustness penalty of

the worst scenario dominates the expected value. When ωα α= , the robustness constraint sets

α equal to the largest deviation, ( )*

*

,max

O R x yO

ω ω ω

ω

α⎛ ⎞−⎜=⎜ ⎟⎝ ⎠

⎟ . By substitution, the objective

function becomes max ( , )R x yω ω ωω

ρ λα−∑ . When 1 ( , )R x yω ω ωω

λ ρα⎛ ⎞

⎟> ⎜⎝ ⎠∑ , the λα term

15

will dominate the objective function and we obtain the min-max relative robustness model of

Kouvelis and Yu (1997):

Max α−

Subject to *

*

( , )O R x yO

ω ω ω

ω

α−≤ ω∀

0, 0, 0,1x yωα ≥ ≥ ∈ ω∀ .

Similarly, we can obtain the robust deviation criterion of Kouvelis and Yu (1997_. For

δ such that *Oω ωα δ≤ for all ω , and for λ sufficiently large, we get the following model:

Max δ− Subject to * ( , )O R x yω ω ω δ− ≤ ω∀ 0, 0, 0,1x yωδ ≥ ≥ ∈ ω∀ .

Variability Models

We can also extend our model to generalize the variability models. To do this, we

replace the relative robustness measure with a measure of deviation from the average profit

R Rω ωω

ρ=∑ .

To obtain the Hodder and Dincer (1986) model, we replace the relative robustness

measure with the variance. However, this makes the model non-linear:

Max ( ( , ) )R x yω ω ω ωω

ρ λ∈Ω

−∑ α

Subject to ( ) 2R Rω ω ω

ω

ρ α− ≤∑ ω∀

0, 0, 0,1x yω ωα ≥ ≥ ∈ ω∀ .

The model of Goetschalckx, et al. (2001) is obtained by using the standard deviation

instead of the variance:

Max ( ( , ) )R x yω ω ω ωω

ρ λ∈Ω

−∑ α

16

Subject to ( ) 2R Rω ω ω

ω

ρ α− ≤∑ ω∀

0, 0, 0,1x yω ωα ≥ ≥ ∈ ω∀ .

Similarly, the model of Ahmed and Sahinidis (1998) is obtained by using the Upper

Partial Mean as follows. When the robust profit is larger than the average profit, 0ωα = and

there is no penalty. When the robust profit is less than the average, then R Rω ωα = − . The

model is the following:

Max ( ( , ) )R x yω ω ω ωω

ρ λ∈Ω

−∑ α

Subject to R Rω ωα− ≤ ω∀

0, 0, 0,1x yω ωα ≥ ≥ ∈ ω∀ .

Scenario Probability

In our model, we use a general form of the probabilities for each scenario ωρ . Kouvelis

and Yu (1997) and others assume that the scenarios are all equally likely. This could easily be

obtained by setting 1ωρ = Ω for all ω .

In summary, this section illustrates how our model generalizes many previous robust

models. The connection between our model and the regret models of Gutierrez, Kouvelis and

Kurawarwala (1996) and Kouvelis and Yu (1997) is straightforward. Our model can also be

used to frame the variability models of Hodder and Dincer (1986), Goetschalckx, et al. (2001),

and Ahmed and Sahinidis (1998) by using a term that measures the deviation from the average

profit for the robust measure in the robustness constraint.

5 Application and Results

In this section, we illustrate the robust Lagrangian model on a sample instance based on

representative industry data; however, we manipulate some of the data to illustrate the features of

17

the robust model. In the example, the company is introducing two new products: a low-grade

product and a high-grade product, both of which are manufactured at a single facility and sold to

four customers. Five machines, each with different processing capabilities and costs, are

available for purchase. Seven company-owned warehouses and third-party warehouses are

available as potential distribution sites. Each machine and distribution center experiences a

delay between purchase and the period in which they become available for use.

In this section, we discuss the uncertain parameters and how the uncertainties are used to

generate scenarios. We solve the viable model for each scenario to obtain the optimal net profit

for each scenario. All of the scenarios, probabilities, and optimal net profits are then used as

inputs for the robust model. We solve the robust model with varying λ values from zero to the

maximum net profit for all scenarios. (For 0λ = , we obtain the expected value solution.) We

also report on the computation time needed to solve the model and present and interpret the

results.

5.1 Uncertainty

In order to define scenarios for the sample problem, we first identify the significant

sources of uncertainty in the model. We discuss the potential uncertainty in new product supply

chain models in the introduction of this paper. For illustrative purposes, we select demand and

transportation costs as the uncertain parameters for the example in this section.

We vary the growth rates for the products in the scenarios to model how well the

company develops market share with customers and how well the product is received. The

growth rates vary by customer. This incorporates both the success of the company with different

customers as well as the customers’ potential growth. Market experts also believe that

18

19

technological and market changes may shift demand from the low-grade product to the high-

grade product in the future, so this possibility is captured in the scenarios.

Transportation cost uncertainty due to fluctuating oil prices illustrates uncertainty in a

cost parameter. Production cost, distribution center handling cost, and facility cost uncertainty

can be handled in a similar manner.

We identify four levels of demand: slow growth, medium growth, fast growth with

different demand allocation to customers and product failure. We also categorize two levels of

transportation costs: regular and high. We combine these uncertainties to generate 12 scenarios.

We discuss the scenarios in more detail below.

5.2 Scenario Generation

For the uncertain parameters in this example, we generate 12 scenarios. For each of the

three demand levels, we vary the percentages of demand allocated to the two main customers

(Customers 2 and 3) at 20%, 40%, and 60% of the total demand volume in Scenarios 1 - 7. In

this way, we ensure correlated customer demand.

We also evaluate the medium growth demand scenarios with higher transportation costs

in Scenarios 8, 9, and 10. In Scenario 11, there is only demand for the high-grade product. In

addition the product demand for Scenario 12 declines to zero. For this example, the

probabilities of the scenarios are assumed equally likely. Thus, the probability of success is very

high, around 92%, so that the model reflects the company’s desire to have capacity in place if the

product is successful. We list the details of the 12 scenarios in Table 2 below.

Table 2. Scenario Descriptions

Scenario Probability Demand Growth Percent Demand Allocated to

Customers 1, 2, 3, 4 Transportation Cost 1 8.3% Slow 5%, 60%, 20%, 15% Regular 2 8.3% Medium 5%, 60%, 20%, 15% Regular 3 8.3% High 5%, 60%, 20%, 15% Regular 4 8.3% Slow 5%, 20%, 60%, 15% Regular 5 8.3% Medium 5%, 20%, 60%, 15% Regular 6 8.3% High 5%, 20%, 60%, 15% Regular 7 8.3% Medium 5%, 40%, 40%, 15% Regular 8 8.3% Medium 5%, 40%, 40%, 15% High to DC2, DC5, DC6,

C2 (Japan) 9 8.3% Medium 5%, 40%, 40%, 15% High to DC2, DC4, DC5,

DC6, C2, C4 (Japan and Taiwan)

10 8.3% Medium 5%, 40%, 40%, 15% All high 11 8.3% Medium, only high-

grade product 5%, 40%, 40%, 15% Regular

12 8.3% Failure 5%, 40%, 40%, 15% Regular 5.3 Computational Results

For our example problem, we solve the robust Lagrangian model using the MIP solver of

CPLEX 7.5 (ILOG, Inc., 2001) on a Pentium 4 1.8 GHz PC with 1.8 GB of RAM. The problem

size is shown in Table 3 below. The robust models require 302 to 388 seconds to solve. The

computation time for the robust model with different λ values are shown at the bottom of Table

4. The 12 scenarios solved with the viable model to obtain the optimal net profit for each

scenario each require an additional 11 to 69 seconds of processing time. The analysis of this

representative case shows that a commercial software package provides solutions to a reasonably

sized problem in less than one hour.

20

Table 3. Test Case Size

Characteristic Single Scenario

Robust Model

Products 2 2 Machines 5 5 Distribution centers 7 7 Customers 4 4 Transportation modes 1 1 Periods 16 16 Scenarios 1 12 Integer variables 576 576 Continuous variables 1,762 23,839 Constraints 954 7,897 Computation time (seconds) 11-69 302-388

We solve the robust model with different λ penalty values to illustrate how the penalty

affects the weighting of the scenarios. Table 4 shows the value of the weight, *1 O ωω

λ ρ⎛ ⎞+⎜ ⎟⎝ ⎠

, for

each scenario as λ varies. The scenarios are arranged in ascending order of optimal net profit.

The weight for Scenario 12, the lowest performing scenario, increases as λ increases.

21

Table 4. Scenario Weighting for Different Penalty Values (Percent)

Percent Weight for Differentλ Values (millions)

Sce io nar Prob lity abi

Optimal Net Profit 0 6.5 13 17 27 67 92 134

12 8.3 (13,025,700) 8.3 12.5 16.6 19.4 25.5 51.2 67.2 94.1 4 8.3 98,588,900 8.3 8.8 9.4 9.8 10.6 14.0 16.1 19.7 1 8.3 98,693,900 8.3 8.8 9.4 9.8 10.6 14.0 16.1 19.7

10 8.3 104,006,000 8.3 8.8 9.3 9.7 10.4 13.7 15.7 19.1 9 8.3 108,484,000 8.3 8.8 9.3 9.6 10.3 13.4 15.4 18.6 5 8.3 108,672,000 8.3 8.8 9.3 9.6 10.3 13.4 15.4 18.6 8 8.3 108,692,000 8.3 8.8 9.3 9.6 10.3 13.4 15.4 18.6 7 8.3 108,764,000 8.3 8.8 9.3 9.6 10.3 13.4 15.4 18.6 2 8.3 108,830,000 8.3 8.8 9.3 9.6 10.3 13.4 15.4 18.6

11 8.3 130,596,000 8.3 8.7 9.1 9.4 10.0 12.6 14.2 16.9 6 8.3 132,589,000 8.3 8.7 9.1 9.4 10.0 12.5 14.1 16.8 3 8.3 134,127,000 8.3 8.7 9.1 9.4 10.0 12.5 14.0 16.7

Computation Time (seconds) 388 384 369 304 323 348 306 302

We solve the model with 0λ = to obtain the expected value solution. Then we solve the

model with increasing λ values. For 0 $17λ< ≤ million, the solution is the same as the

expected value solution. For $17λ > million, the model provides a more conservative solution,

which we call the robust solution. Because the value of λ represents the value of being 100%

away from optional, λ should be near the average of the optimal net profits ($92 million).

The optimal facility decisions for each scenario are shown in Table 5. The entries in the

table represent the period when the facility/equipment is purchased. No facilities are closed if

once opened. The table shows the optimal facility decisions for each scenario’s optimal solution,

the expected value solution, and the robust solution. The decisions to purchase Machine 1 and

build Distribution Center 7 are consistent among all scenarios. All scenarios except Scenario 11

(only high-grade product) and Scenario 12 (product failure) also purchase Machine 2. The

scenarios with higher demand differ in the additional facilities used (Machine 4 or 5, Distribution

22

23

Center 4 or 6) and in the period of expansion. The expected value solution calls for the purchase

of Machines 1 and 2 and Distribution Center 7. The robust solution provides a more

conservative capacity development plan by not adding Machine 2.

Table 5. Facility Decisions

Equipment/Facility Scen

ario

1

Scen

ario

2

Scen

ario

3

Scen

ario

4

Scen

ario

5

Scen

ario

6

Scen

ario

7

Scen

ario

8

Scen

ario

9

Scen

ario

10

Scen

ario

11

Scen

ario

12

Expe

cted

Val

ue

Rob

ust

Machine 1 P1 P1 P1 P1 P1 P1 P1 P1 P1 P1 P1 P1 P1 P1 Machine 2 P5 P2 P1 P5 P2 P1 P2 P2 P2 P2 P3 Machine 3 Machine 4 P4 P4 P4 P4 P4 P4 P4 P6 Machine 5 P5 P4 P3 Distribution Center 1 Distribution Center 2 Distribution Center 3 Distribution Center 4 P7 P7 P7 P7 P7 P7 P7 Distribution Center 5 Distribution Center 6 P7 P7 Distribution Center 7 P1 P1 P1 P1 P1 P1 P1 P1 P1 P1 P1 P1 P1 P1

Key: PX = Purchase facility/equipment in Period X

A comparison of the scenario optima, expected value, and robust solutions are shown by

net profit and by percent of optimal net profit in Table 6. The robust solution places more

emphasis on the scenarios with lower profits; therefore, it performs better than the expected

value solution for Scenarios 11 and 12. The robust solution also loses less money if the product

were to fail, losing approximately $16 million if the product does not survive, whereas the

expected value solution loses $31 million.

24

Table 6. Net Profit and Percent of Optimal Net Profit by Scenario

Net Profit ($000) Percent of Optimal

Scenario Optimal Net

Profit

Expected Value

Solution Robust

Solution

Expected Value

Solution Robust

Solution Scenario 1 98,694 98,192 95,847 1% 3% Scenario 2 108,830 06,285 101,150 2% 7% Scenario 3 134,127 112,871 106,427 16% 21% Scenario 4 98,589 98,087 95,764 1% 3% Scenario 5 108,672 106,170 101,063 2% 7% Scenario 6 132,589 112,796 106,369 15% 20% Scenario 7 108,764 106,250 101,130 2% 7% Scenario 8 108,692 106,194 101,087 2% 7% Scenario 9 108,484 106,166 101,065 2% 7% Scenario 10 104,006 102,246 98,077 2% 6% Scenario 11 130,596 89,580 106,080 31% 19% Scenario 12 -13,026 -31,202 -15,957 140% 23% Weighted Average 102,418 92,802 91,509

Table 7 shows the minimum net cash position for each scenario and each solution. A

cash flow threshold is not enforced in any of the solutions. The minimum net cash position for

the expected value model ranges from -$17 to -$22 million for Scenarios 1 – 11 and is -$31

million for Scenario 12. The robust solution exhibits less severity and variability in the

minimum cash position, with Scenarios 1 – 11 at approximately -$12 million and Scenario 12 at

-$16 million.

25

Table 7. Minimum Net Cash Position by Scenario ($000)

($000)

Expected Value

Solution Robust

Solution Scenario 1 -21,699 -12,172 Scenario 2 -19,587 -12,100 Scenario 3 -17,467 -12,100 Scenario 4 -21,700 -12,172 Scenario 5 -19,590 -12,100 Scenario 6 -17,472 -12,100 Scenario 7 -19,581 -12,100 Scenario 8 -19,582 -12,100 Scenario 9 -19,582 -12,100 Scenario 10 -19,678 -12,100 Scenario 11 -18,263 -12,100 Scenario 12 -31,202 -15,957 Minimum -31,202 -15,957

6 Conclusions

In this paper, we develop a robust optimization model for the strategic integrated design

of a multi-period production/distribution system for new product launches. This model extends

the methodology for new product supply chains developed by Butler, Ammons, and Sokol

(2003) by incorporating uncertainty into the simultaneous determination of production

equipment types and sizes, distribution facility locations, customer allocation, and production

and transportation flows. The model limits the potential losses involved in new product launches

by placing a threshold on the net cash position (revenues less costs) for each period in each

possible scenario.

Our robust model provides a framework that generalizes many previous robust models,

including worst-case and expected value models, regret and variability models. Our model also

provides a general form of scenario probabilities.

We select relative robustness as an appropriate robustness measure for new products

because it ensures that the robust configuration performs well in scenarios with lower profit. The

robustness measure is a penalty term in the objective function weighted by the probability of

occurrence for each scenario. Using the weighted average ensures that the model solution is not

driven by one remote case. However, determining the exact probabilities for the scenarios is a

potential vulnerability of this approach. The scenario probability vulnerability could be

addressed with sensitivity analysis.

Our computational studies show that the robust model can be solved in a reasonable

amount of time using the CPLEX 7.5 MIP solver (ILOG, Inc.). The results show that altering

the penalty parameter to change the weighting of the scenarios in the objective function can

produce alternative solutions by creating more risk aversion. Our test results show how the

importance of the lower performing solutions increases as λ increases. The robust solution

performs better in scenarios with lower net profits and limits the company’s losses in the event of

product failure. However, the facility decisions for the robust model and the expected value

model are the same for Periods 1 and 2. Our model allows decisions to be implemented in a

stage-wise manner, by implementing the Period 1 decisions and re-solving the model before

purchasing additional equipment. Solving the model sequentially allows decisions to be made

after some of the uncertainty is diminished.

There are several promising areas for future work based on this research. Applying the

model in different new product launch situations could provide more insight as to how cost

structures and constraints affect solutions. The models could be extended to include a more

detailed after-tax analysis and incorporate taxes, tariffs, and duties for an international supply

chain. The model could also be modified to incorporate inventory costs and decisions and

26

27

customer service level requirements extending the model to the operational and strategic levels.

Extending the model to evaluate several new product ideas simultaneously could determine

which combination of products to launch with limited resources in order to pool risk or improve

the probability that at least on product succeeds. We could also try to exploit the scenario

structure assuming alternative demand elasticity curves for different scenarios to address the

non-linearities that arise when the demand for a product is dependant on the price. Another

extension suggested in Clarke (2003) is the use of a weighted robustness constraint with a

maximum net profit objective function to capture the overall robustness factor for the instance.

7 References

Ahmed, S. and N. Sahinidis, “Robust process planning under uncertainty,” Industrial Engineering and Chemical Research 37 (1998) 1883 – 1892.

Applequist, G., J. Penky and G. Rekalaitis, “Risk and uncertainty in managing chemical manufacturing supply chain,” Computers and Chemical Engineering 24 (2000) 2211 – 2222.

Bai, D., T. Carpenter and J. Mulvey, “Making a case for robust optimization models,” Management Science 43/7 (1997) 895 – 907.

Bok, J. H. Lee, and S. Park, “Robust investment model for long-range capacity expansion of chemical processing networks under uncertain demand forecast scenarios,” Computers and Chemical Engineering 22/7 (1998) 1037 –1049.

Butler, R., J. Ammons, and J. Sokol, “A strategic production and distribution model for financial viability in new product supply chains,” working paper, Georgia Institute of Technology, Atlanta, Georgia (2003).

Clarke, L., Personal communication from the author, June 30, 2003.

Goetschalckx, M., S. Ahmed, A. Shapiro and T. Santoso, “Designing flexible and robust supply chains,” IEPM Quebec, August 2001.

Gutierrez, G., P. Kouvelis, and A. Kurawarwala, “A robustness approach to uncapacitated network design problems,” European Journal of Operations Research 94 (1996) 362 – 376.

28

Goetschalckx, M., Logistics System Design, Course Materials, http://www.isye.gatech.edu/people/faculty/Marc_Goetschalckx/course_materials.html, viewed 6/19/2002.

Hodder, J. and M.C. Dincer, “ A multifactor model for international plant location and financing under uncertainty,” Computers and Operations Research 13/5 (1986) 601 – 609.

ILOG, Incorporated, ILOG CPLEX, Release 7.5, 2001.

Kouvelis, P and G. Yu, Robust Discrete Optimization and Its Applications, Dordecht, The Netherlands: Kluwer Academic Publishers (1997).

Malcolm, S.A. and S. A. Zenios, “Robust optimization for power systems capacity expansion under uncertainty,” Journal of Operations Research Society 45/9 (1994) 1040 – 1049.

Mulvey, J., R. Vanderbei and S. Zenios, “Robust optimization of large-scale systems,” Operations Research 43 (1995) 264 – 281.

Sabri, E. and B. Beamon, “A multi-objective approach to simultaneous strategic and operational planning in supply chain design,” The International Journal of Management Science 28 (2000) 581 – 598.

Snyder, L., M. Daskin and C. Teo, “The stochastic location model with risk pooling,” working paper, Northwestern University, Evanston, Illinois (2002).

Vidal, C. and M. Goetschalckx, “Modeling the effect of uncertainties on global logistics systems,” Journal of Business Logistics 21/1 (2000) 95 – 120.

Voudouris, V., “Mathematical programming techniques to debottleneck the supply chain of fine chemical industries,” Computers and Chemical Engineering 20 (1996) S1269 – S1274.

Yu, C. and H. Li, “A robust optimization model for stochastic logistics problems,” International Journal of Production Economics 64 (2000) 385 – 397.

8 Acknowledgements

This research has been supported in part by Kodak Corporation and a grant from the National Science Foundation under Grant Number SBE-0123532. Any opinion, findings, and conclusions expressed in this material are those of the authors and do not necessarily reflect the views of Kodak or the National Science Foundation. The authors would like to thank Charles Barrentine, Chris Johnson, Kevin Farrelly, Roxana Ahmed, Lloyd Clarke, Marc Goetschalckx, and Paul Griffin.