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In this section, the nonlinear objective function model is developed. The objective function – excluding last two terms – is adopted from Ben Arieh et al. (2009). New limiting constraints are developed and discussed. The following section describes the methods for the model linearization for performance improvement.

The objective function has four main terms representing the cost of i) platforms components mass assembly, ii) platform customization by components assembly, iii) platform customization by components disassembly, and iv) labor training cost for each platform:

Minimize Z (Total Cost) = ∑ ∑ ∑ ( ) ∑ ∑ ∑ (

) ∑ ∑ ∑ ∑ (‎3.1) Subject to: ∑ (‎3.2) (‎3.3) (‎3.4) (‎3.5) (‎3.6) (‎3.7) ∑ (‎3.8) ∑ (‎3.9) ∑ (‎3.10) ∑ (‎3.11) (‎3.12) (‎3.13) (‎3.14) (‎3.15)

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(‎3.16)

The term describes the cost of mass production of assembling component j to a platform. The term represents the cost of any component j. The second objective function term explains the costs of assembling extra components into a certain platform to form a product variant. The third term represents the cost of disassembling components from a platform to form another product variant. The fourth term describes the costs of producing multiple platforms (labor training for each platform assembly). This platform term controls the formation of new platforms by the introduction of binary variable , therefore, the target is to minimize the total cost of platform mass production (minimize number of product platforms), components to be added to a platform to form product variants, components to be disassembled from a platform to form a new variant and the costs associated with switching between platforms.

Constraint set (2) assigns each product to at least a single platform. Constraint set (3) determines that if variant k in platform i need component j to be added to the platform i then . Constraint set (4) determines that if variant k is not in platform i, then no components should be added to that platform to form that product. Constraint set (5) confirms whether product k is in platform i and does not have component j already installed in platform i, then that component should be removed (i.e. ). Constraint set (6) is preventing the removal of a component from a platform to form a product, if that component is not in the platform. Constraint set (7) forces the model if component l must precede component j to have component l in case platform i contains a certain product k that contains component j in its structure.

Constraint set (8) ensures that if no product is assigned to a certain platform, then the platform should not contain any components. Constraint set (9) forces the model to not include any platform costs if no product uses that platform. Constraint set (10) allows the presence of any platform which is used to construct at least one product. The set (11) is to ensure the binary nature of all decision variables in the model. Constraint sets (11-16) ensures that products should not be assigned to a platform, if their demand rate is zero.

3.4.1 Linearized changeable modular product platform assembly model

The next step is model linearization. Although only the objective function has nonlinear component, it still leads to a computationally expensive model. A linearization scheme is adopted from Peterson (1971). This linearization is as follows:

If z * x are two multiplied linear variables, with z being a non-negative variable that has an upper bound M (equals 1 in this model, because z is binary), then the product of z and x can be replaced by the linear variable y such that:

( ) (‎3.17) (‎3.18)

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In the proposed model, every two multiplied variables are replaced by one new variable and two sets of constraints. To adapt the linearization scheme to the model, suppose that variable x has indices i and j ( ), and variable has and indices ( ). Accordingly, variable y must have all of their indices ( ). Hence, the valid relation between the value of and the other variables

is:

( ) (‎3.19)

The decision variables used in the MPMP model are binary. As a result, all the upper bound M values are one. The above transformation is done on every two multiplied nonlinear variables that exist in the objective function:

(‎3.20)

(‎3.21)

(‎3.22)

Minimize Z (Total Cost) = ∑ ∑ ∑ ( ) ∑ ∑ ∑ (

) ∑ ∑ ∑ ∑ (‎3.23) Subject to: (‎3.24) (‎3.25) (‎3.26) (‎3.27) (‎3.28) (‎3.29) (‎3.30) (‎3.31) (‎3.32) * + (‎3.33)

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3.5 Initial‎Comparison‎between‎MPMP‎Model‎and‎Ben‎Arieh’s‎Model

Table 3.1 lists some of the differences between the two models. The proposed MPMP model has numerous advantages over Ben Arieh‘s. MPMP has a linear objective function that enables it to solve large number of products and components. MPMP has a new variable which determines the optimal number of the platforms used to derive new products. The platform component in Ben Arieh‘s model forces the model to use specific number of platforms, whether they are needed or not. The families‘ variable in Ben Arieh‘s is not tightly constrained with other model variables. That weak formulation enables variables to take erroneous values and producing inaccurate results. MPMP model can handle periods with no product demand, while Ben Arieh‘s does not. In Ben Arieh‘s model, the subtraction of components‘ costs in the term shown in table 3.1 obtains negative costs and produce platforms with components that can be assembled in different places (Figure 3.4)

Table ‎3.1 Comparison between MPMP model and Ben Arieh‘s model

Ben‎Arieh’s‎Model MPMP Model Complexity of objective

function Non-linear Linear

Number of Platform Determination

Trial and error ( In Ben Arieh‘s model, the platform determination component is ∑ , A is constant equal to platform cost, I: Maximum

number of platforms)

New variable introduced to determine optimal number of

platforms ( ) Assignment of product to

family formulation

Weak formulation ( is not linked tightly to other variables , ,

)

Tight formulation Ability to handle zero demand

products

Cannot handle and will produce erroneous results

Can handle (Constraints 3.12 – 3.16)

Model stability with higher platforms

Not stable and produce negative costs with high number of platforms

(∑ ∑ ∑ ( ) )

Stable model ∑ ∑ ∑