OPTIMIZATION OF ROCK STONE SELECTION IN SHORE PROTECTION PROJECTS - CASE STUDY: GAZA BEACH CAMP SHORE PROTECTION PROJECT

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1 OPTIMIZATION OF ROCK STONE SELECTION IN SHORE PROTECTION PROJECTS - CASE STUDY: GAZA BEACH CAMP SHORE PROTECTION PROJECT ABSTRACT Rifat N. Rustom 1 and Omar Al-Tabbaa 2 Rock stone of different sizes are used in stabilizing sea shores of varied slopes. Using large rock stone foundation (aprons) under wall gabions is considered as one of the effective methods used for stabilizing slopes. The availability of wide range of rock sizes with varying unit weight and cost dictates that the designers should optimize their usage of rock in the apron. Gaza Beach Camp Shore Protection Project is considered here as a case study to demonstrate the optimization of rock sizes in the apron used to carry the gabion walls to ensure the slope stability of the sea shore. The designer's specification required the usage of four types of rocks to ensure the stability, unit weight and voids ratio. Type 1 rock with size less than or equal to 5 kg, Type 2 of 40 kg or less, Type 3 of 200 kg or less, and Type 4 of 300 kg or less were used. Total bulk density of the mix, apron stability, percentage limits of each stone type as well as the total quantity of stones were the limiting factors for the optimization process. A linear programming model was developed and used in order to optimize the quantity of each type of stone (category) to produce a stone mix that satisfies the design requirements and at the same time the lowest cost criteria. Sensitivity analysis is used to verify the output and generate the different possible scenarios of the outcomes. KEWWORDS: Optimization, linear programming, sensitivity analysis, apron, gabions. 1. INTRODUCTION Gaza Beach Camp Shore Protection Project is aimed to protect the sea shore from erosion by the action of air and sea waves hence protecting the costal road and the houses of the inhabitants located near the coast. The total length of the road to be protected is about 1600 meters. A contract was made between the United Nations Relief and Works Agency (UNRWA) and a Palestinian local contractor to construct the project. The project was funded by the Dutch government [1]. The main raw material used in the project is imported rock stones with varied sizes. The construction process consists mainly of two stages: Stage I: Toe Protection - Aprons This stage consists of the construction of aprons as a rubble toe protection for the base of the gabion walls, see Figure 1. The aprons are intended to reduce wave reflection and scour of bottom sediments during storms [2]. The installation of the rubble toe protection included a layer of geotextile as a filter and to protect the rocks from being embedded in the soil [3]. A bedding layer of sand is used over the geotextile to protect it 1 Islamic University-Gaza, Palestine, Associate Professor of Civil Engineering 2 Al Khudari Contracting Company, Project Engineer C-102

2 from impact damage during installation or abrasion damage during its life time (due to wave action agitating the rock rubbles) [4]. Stage II: Gabions This stage consists of the construction of gabion (stone-filled wire baskets) baskets to provide slope stabilization along the coastal road. This mass gravity structure is intended to resist and balance the active soil pressure behind the wall [5]. The gabions are laid over the aprons. The gabions are filled with small boulder rocks (11 to 23) cm in size. A mild slope backfill layer is spread and compacted between the gabions and the road. The gabions are installed in rows ranging from one row at the bottom level to a maximum of three rows at the highest level as shown in Figure 1. Backfilling Geotextile Figure 1: Schematic cross section of the project [1] 2. PROBLEM IDENTIFICATION One of the main materials used in this project is rock stones with different sizes and weights to stabilize and protect the shore hence the coastal road. The stone sizes required (as per the specifications and drawings) are categorized in four groups; each one has a defined percentage by weight with certain allowance that the quantity might be plus or minus the percentage given. The difference in price of each type (category) of stone required dictates that the contractor to attempt to optimize the quantities selected based on the minimum cost criteria. The large stone size covers more volume in the gabions thus reducing the quantity of rock material used. On the other side the small stone size is easier to handle and placed in the gabions thus reducing the time and consequently the cost. A linear programming model is formulated in order to determine the optimum quantity of each type (category) that will satisfy the specifications required by the client (UNRWA) and at the same time achieve the lowest overall cost (i.e. max the contractor's profit). C-103

3 3. PROBLEM PARAMETERS A. Decision Variables: The decision variables identified in this case are the quantities required from each rock stone category. Table 1 lists the required properties as well as the cost of each category. Table 1: Rock Stones Properties [1] Category Variable Size Cost (NIS/ton)* Bulk Density (Kg/m3) Type 1 X 1 rock size less than or equal to 5 kg Type 2 X 2 rock size less than or equal to 40 kg Type 3 X 3 rock size less than or equal to 200 kg Type 4 X 4 rock size less than or equal to 300 kg * 1$ = 4.75 NIS B. Percentage Allowances by Weight of the Stone Category: The allowable ranges of the stones as required by the specifications are listed below: A. 0 % X 1 2 %, by weight of the total sample B. 5 % X 2 10%, by weight of the total sample C. 10% X 3 50%, by weight of total the sample D. 30% X 4 70%, by weight of the total sample C. Apron Stability: The percentage of the combination of stones Type 2 and Type 4 must be equal to or larger than 40 % of the total quantity. D. Total Bulk Density of the Mix: The overall bulk density of the combined stones shall range between 2500 to 2750 kg/m3; the bulk density is calculated by dividing the weight of stones by the unit volume (apron volume). E. Total weight The total weight of stones required must be equal to or larger than 45,000 tons for the entire project. This quantity is calculated based on the geometric properties of the apron cross sections. F. Total cost The total cost of the stones used (including transportation) must be equal to or less than 3,510,000 NIS (bid price), or 78.0 NIS/ton. 5. DEVELOPMENT OF THE MATHEMATICAL MODEL Linear programming method was used to find out the optimum solution that satisfies the mandated specifications and at the same time achieve the minimum cost for the contractor (i.e. maximize the profit). The linear programming method requires that the specifications and conditions to be converted into a number of inequality equations and an objective function to be formulated based on minimizing the cost of the stone quantities. C-104

4 LINDO [6] computer software was used to solve these inequality equations and to determine the optimum solution. 5.1 Model Formulation In linear programming problems, the objective of the decision maker is to maximize (usually revenues or profit) or minimize (usually cost) some function in terms of the decision variables. The function to be maximized or minimize is called the objective function. On the other hand, the value of the decision variables must satisfy a set of constants (i.e. constrains) and each constant must be a linear equation or linear inequality [7]. Let, X 1 = proportion by weight of rock stone Type 1, X 2 = proportion by weight of rock stone Type 2, X 3 = proportion by weight of rock stone Type 3, and X 4 = proportion by weight of rock stone Type 4, Then the model can be formulated as follows: The objective function here is to minimize the cost per ton of the stone quantity subject to the given constraints (specifications). Objective Function: Minimize C = 70X X X X 4. (1) Where C is the cost of one ton of the rock stone mix Subjected to the following constraints: 1) The maximum percent of the quantity of each stone type to be used in the project according to the percentage allowance of weight as stated in the Percentage Allowances by Weight of the Stone Category are: X (2) X (3) X (4) X (5) X (6) X (7) X (8) X 1 + X 2 + X 3 + X 4 = 1... (9) 2) Given the bulk density for the stone types as shown in Table 1, the overall mix bulk density should be greater than 2.55 ton/m 3 and less than 2.62 ton/m 3. This condition can be represented by the following equations as: C-105

5 (X 1 /2.75+X 2 /2.63+X 3 /2.51+X 4 /2.45) (X1+X 2 +X 3 +X 4 )/ (10) (X 1 /2.75+X 2 /2.63+X 3 /2.51+X 4 /2.45) (X1+X 2 +X 3 +X 4 )/ (11) 3) Total Cost of the stone material per ton should be less than budgeted (bid price): (70X X 2 +77X 3 +80X 4 ) 78.(12) 4) For Apron Stability: The percentage of the combination of Type 2 and Type 4 should be equal to or larger than 40 % of the total quantity: (X 2 +X 4 ) 0. 4(X 1 +X 2 +X 3 +X 4 ).(13) 5) All stone types should be used in the mix X 1, X 2, X 3, X 4 0.(14) 6. RESULTS 6.1 Decision Variables After defining the mathematical model representing the objective function and the constraints (equations 2 to 14), LINDO computer package was used to determine the decision variables (X 1, X 2, X 3, X 4 ). The Optimum cost per ton that can satisfy all required specifications and conditions stated is found to be 77.85NIS (minimum C value). The percentage of each stone type that makes one cubic meter of the mix is also determined and shown in Table 2. Table 2: Percentage of stone types used in the mix Stone Type Percent X1 2% X2 10% X3 50% X4 38% Total 100% 6.2 Sensitivity Analysis Sensitivity analysis is concerned with how changes in the original linear programming model parameters (objective function parameters and the right hand side values) affect the optimal solution [7]. The output of the sensitivity analysis provides a good tool for the contractor to foresee the impact of changes and allowable margins in the parameters of the objective function and the right hand side (RHS) coefficients that influenced by the market changes. Tables 3 and 4 show the sensitivity analysis output for the objective function coefficients and the constraints' right hand side values as obtained by LINDO. C-106

6 It is apparent from the slack/surplus values that only four out of twelve constraints (equations 2, 4, 6, and 9) are binding. This is could be of value to the contactor as these slacks provide more flexibility in responding to the limitations imposed by the large number of constraints and the market requirements. Table 3: Objective Function Coefficients Ranges: Stone Type Cost per ton Allowable Increase Allowable Decrease Reduced Cost X infinity X infinity X infinity X infinity Table 4: Constraints (RHS) Ranges, Slacks or Surpluses and Dual Prices Eq. Requirement No. 2 Max. quantity of stone X1 3 Min. quantity of stone X2 4 Max. quantity of stone X2 5 Min. quantity of stone X3 6 Max. quantity of stone X3 7 Min. quantity of stone X4 8 Max. quantity of stone X4 9 One ton unit condition Current Allowable Allowable Slack or Dual Prices RHS Increase Decrease Surplus infinity infinity infinity infinity Max bulk density infinity Min Bulk density 0.40 infinity Cost per ton 78.0 infinity Stability condition infinity CONCLUSION Linear Programming method is proven to be an effective mathematical tool to optimize construction involved processes. The model formulated in this research study to determine the optimum percentage of each stone type in the mix given a large number C-107

7 of constraints helped the contractor to identify the minimum cost of material required for the project. Should the linear programming model was not used, the contractor would have made several trial batches to attempt to satisfy the clients conditions. This indeed would require extensive time, cost and resources in addition to not necessarily reaching the optimum mix. The sensitivity analysis provided a good tool for the contractor to foresee the impact of changes and allowable margins in the parameters of the objective function and the RHS coefficients to coop with the market changes. 8. REFERENCES [1] UNRWA Tender Documents, Beach Camp Shore Protection Project, Phase I, Volume I, [2] U.S. Environmental Protection Agency (EPA), Guidance Specifying Management Measures for Sources of Non-point Pollution in Coastal Waters, Ch. 6, Part IV, [3] Das, Braja M. Principles of Foundation Engineering, PWA Publishing Company, Boston, [4] Koerner, M. R., Designing with Geosynthetics, 2 nd Ed., New Jersey: Prentice Hall, [5] Maccaferri. Gabion Retaining Walls, Maccaferri Publications, [6] LINDO. Linear Interactive and Discrete Optimizer, User instructions Hand Book, [7] Winston, Wayne L. Introduction to Mathematical Programming. Duxbury Press, California, C-108