Optimization Objective Functions

The objective of the research is to optimize the dynamic layout problem under multi objective optimization functions (MOO) to mimic the objective of real sites. Three congruent objective functions are employed in this research to generate optimal dynamic layouts for construction project. The first objective function (f1) is minimizing the total handling cost and time of interaction flows between the site facilities in order to reduce cost and time of the construction projects. The second objective function (f2) is minimizing the likelihood of accidents happened in order to improve the safety level. The third objective function (f3) is minimizing the harmful effect of construction activity on the surrounding neighboring in order to grab attention

168 to the environmental concern. The objective functions (f1), (f2) and (f3) are mathematically defined as follows:

f1 = ∑P_P=1∑n_i=1∑n_j=1C_ij RD_ijT_ij equation (5. 1)

This objective function (f1) is to minimize the total handling cost and time of interaction flows between the site facilities based on the proximity weight (Cij) defined by the user that reflects the cost and time closeness relationship between any pair of facilities i and j and the real travel distance (RDij) between them during their interaction duration (Tij) in the same stage duration over the entire project duration in order to reduce cost and time of the construction projects.

f2 = ∑PP=1∑ni=1∑nj=1Sij eijTij equation (5. 2)

This objective function (f2) is to minimize the likelihood of accidents happened based on the proximity weight (Sij) defined by the user that reflects the safety concern closeness relationship between any pair of facilities i and j and the Euclidean distance (eij) between them during their interaction duration (Tij) in the same stage duration over the entire project duration in order to improve the safety level.

f3 =∑PP=1∑ni=1∑nz=1Viz eizTiz equation (5. 3)

This objective function (f3) is to minimize the harmful effect of construction activity on the surrounding neighboring based on the proximity weight (Viz) defined by the user that reflects the environmental concern closeness spatial relationship for the sensitivity of facility i is to be located in the environmental protection zone z and the Euclidean distance (eiz) between them during their interaction duration (Tiz) in the

169 same stage duration over the entire project duration in order to grab attention to the environmental concern.

Where:

P: number of project stages (the proposed system will consider the project duration for a project with no stage number as one stage).

n: number of all type of construction facilities. i,j: facility i and facility j.

Cij: a relative proximity weight that reflects the cost and time closeness relationship between facilities i and j (will explain in section 5.3.6).

RDij: real distance between facilities i and j (will explain in section5.3.7).

T_ij: the interaction duration between facilities i and j durations during the stage duration over the entire project duration.

S_ij: a relative proximity weight that reflects the safety concern closeness relationship between facilities i and j (will explain in section 5.3.6).

e_ij: Euclidean distance between facilities i and j (will explain in section 5.3.7). z: protection zone.

V_iz: a relative proximity weight that reflects the environmental concern closeness spatial relationship for the sensitivity of facility i is to be located in the environmental protection zone z (will explain in section 5.3.6).

e_iz: Euclidean distance between facility i and protection zone z (will explain in section 5.3.7).

170 Tiz: the interaction duration between facility i and protection zone z durations during the stage duration over the entire project duration.

Since the multiple objective functions defined in this research are congruent, they can be transformed into single objective function using the following equation:

Objective function = min (w1 •f1 + w2 • f2 + w3 • f3) equation (5. 4) Where w1, w2 and w3 are weights defined by the user for the three congruent objective functions respectively. This method is called weighted sum method, which is a method combining a set of objectives into a single objective by pre-multiplying each objective with a user defined weight (Deb, 2001) to solve and minimize multi objectives optimization problems.

Furthermore, to calculate the layout weight for each project’s stage a weight for a relocation cost (RC) will be added to each layout stage if any dynamic facility has been relocated except in the first stage there is no relocation cost added. The relocation cost (RC) can be mathematically defined as follows:

RC =∑ Fi+ Ril−k RDil−k m

i=1 equation (5. 5) This equation aims at calculating the dynamic facility relocation cost (RC) and comprises two cost components. First component (Fi) which is a proximity weight defined by the user to reflect the cost of dismantling and erecting of dynamic facility i. Second component to calculate the cost of moving dynamic facility I to new location based on the proximity weight (Rl−k_i ) defined by the user that reflects the relocation cost for transport dynamic facility i one meter from its old location l to the new

171 location k and the real travel distance (RD_il−k) from the old location l to the new location k.

Where:

m: number of dynamic facilities. i: dynamic facility i.

Fi: a relative proximity weight from 0 to 1000 with increment of 100 that reflects the cost of dismantling and erecting of dynamic facility i.

Rl−k_i : a relative proximity weight from 0 to 9 that reflects the relocation cost for transport dynamic facility i one meter from its old location l to the new location k. RD_il−k: real distance between locations l and k for dynamic facility i (will explain in section 5.3.7).

l: old location for dynamic facility i. k: new location for dynamic facility i.

This relocation cost will be calculated to express the cost of relocating the dynamic facilities. The relocation cost is calculated for each dynamic facility at the beginning of the stage if any of the following two conditions is occurred: (1) the orientation of a facility is changed. (2) the location of a facility is changed.