True 2. All design problems have only linear inequality constraints. False 3. All design variables should be independent of each other as far as possible. True 4. If there is an equality constraint in the design problem, the optimum solution must satisfy it.
True 5. Each optimization problem must have certain parameters called the design variables. True 6. A feasible design may violate equality constraints. False 7. False 8. True 9. The constraint set for a design problem consists of all the feasible points. True The number of independent equality constraints can be larger than the number of design variables for the problem.
False The feasible region for an equality constraint is a subset of that for the same constraint expressed as an inequality. A lower minimum value for the cost function is obtained if more constraints are added to the problem formulation. Data for the three available truck models are given in Table E2.
There are some limitations on the operations that need to be considered. The labor market is such that the company can hire at most truck drivers. Garage and maintenance facilities can handle at the most 25 trucks. How many trucks of each type should the company purchase? Truck Truck load Average truck Crew required No.
Given: The maximum amount of money the company can spend, the data given in Table E2. Each plant processes iron ore into two different ingot stocks. They are shipped to any of the three fabricating plants where they are made into either of the two finished products. In total, there are two reduction plants, two ingot stocks, three fabricating plants, and two finished products.
For the coming season, the company wants to minimize total tonnage of iron ore processed in its reduction plants, subject to production and demand constraints.
Formulate the design optimization problem and transcribe it into the standard model. The total tonnage of iron ore processed by both reduction plants must equal the total tonnage processed into ingot stocks for shipment to the fabricating plants.
The total tonnage of iron ore processed by each reduction plant cannot exceed its capacity. The total tonnage of ingot stock manufactured into products at each fabricating plant must equal the tonnage of ingot stock shipped to it by the reduction plants.
The total tonnage of ingot stock manufactured into products at each fabrication plant cannot exceed its available capacity. The total tonnage of each product must equal its demand. Given: The maximum number of reduction plants, ingot stocks, fabricating plants, and finished products available, the constraints shown above, and the constants shown in the table above. Required: It is desired to minimize the total tonnage of iron ore that is processed in reduction plants.
Several formulations for the design problem are possible. For each formulation proper design variables are identified. Expressions for the cost and constraint functions are derived. Step 3: Definition of Design Variables For this formulation, twenty-four design variables are chosen which designate the twenty-four different paths of processing the iron ore, i. For a particular set of i, j, k, and l, R i, j, k, l means that the tonnage of iron ore processed at reduction plant i, yielding ingot stock j, shipped to the fabricating plant k and manufactured into product l.
Step 4: Optimization Criterion Optimization criterion is to minimize total tonnage of iron ore processed at the reduction plants, and the cost function is defined as.
In terms of design variables and the given data, these two constraints are:. In terms of design variables and the given data, these constraints can be written as:. These coefficients are given as:. These constraints have been satisfied automatically since the twenty-four design variables paths are chosen which satisfy these conditions. Step 3: Definition of Design Variables The design variables are chosen as follows:. Design a water canal having a cross-sectional area of m2.
Least construction costs occur when the volume of the excavated material equals the amount of material required for the dykes, i. Formulate the problem to minimize the dug- out material A 1. Transcribe the problem into the standard design optimization model created by V. Given: The specific, required cross-sectional area of the canal, least construction costs occur when the volume of the excavated material is equivalent to the amount of material required for the two dykes, and the dimensions as shown in Figure E2.
Required: It is desired to minimize the dug-out material A 1. Step 3: Definition of Design Variables w, w 1 , w 2 , w 3 , H 1 and H 2 m are chosen as design variables which are defined as shown in Figure E2. Step 3: Definition of Design Variables w 1 , H 1 , H 2 m , and s unitless are chosen as design variables which are defined below in relation to Figure E2.
Step 4: Optimization Criterion Optimization criterion is to minimize the volume of excavation, and the cost function is defined as:. Step 3: Definition of Design Variables A 1 , A 2 , w, w 1 , w 2 , w 3 , H 1 , H 2 m , and s unitless are chosen as design variables which are defined above in Figure E2.
Formulate the minimum mass design problem using a hollow circular cross section. The material should not fail under bending stress or shear stress. Solution Given: The equations to calculate maximum bending and shearing stress in the beam, the force applied to the beam, the length of the beam, the density of the beam, the maximum values of R o and R i , and the allowable bending and shear stress for the beam. Required: It is desired to create a beam design, as shown in Figure E2.
The beam should not fail due to bending or shear at any point. Step 5: Formulation of Constraints g 1 : bending stress should be smaller than the allowable bending stress g 2 : shear stress smaller than allowable shear stress.
Formulate the minimum-weight design problem and transcribe it into the standard form. Use Newton N and millimeters mm as units in the formulation. Open navigation menu. Close suggestions Search Search. User Settings. Skip carousel. Carousel Previous. Carousel Next. What is Scribd? Uploaded by Afshar Arabi. Also, this file include Matlab codes.
There is one pdf file as a solution manual for each of chapters. In addition, there are some Excel files for chapters 6, 14 and Also, product include Matlab codes for appendix B.
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Smallman, A. Introduction to Optimum Design, 3rd Ed. Jump to Page. Search inside document. Solution Given: The lot size, building floor space and parking area requirements, and the data given in the problem statement. Solution Given: The maximum and minimum radius of the mug, the maximum height of the mug, and the maximum surface area of the mug.
Solution Given: The minimum radius of each tube, the similarity between each tube, and the maximum surface area of all tubes combined. Solution Given: The cost of land in the downtown urban renewal section, the maximum width and depth available, and the minimum area available in the lot. Solution Given: The profits from selling products A and B, the amount of raw material available of products C and D, the amount of products C and D required to produce products A and B, and the market for products A and B.
Levy Solution Given: The amount of bottles of pure alcohol which can be produced each week, the two types of alcohol which are produced, the amount of empty bottles available per week, the amount of each alcohol which can be produced based on the weekly sugar supply, and the profits for each alcohol type. Solution Given: The desired interior can volume, the minimum and maximum ratio of height to diameter, and the maximum height.
Solution Given: The construction costs for the sides, ends, and the bottom of the container and the minimum volume requirement. Solution Given: The required volume of the tank, the fabrication cost of the sheet metal per unit area, and the limiting relation between the height and the diameter. Solution Given: The cost of a horizontal member in two, separate directions, the cost of a vertical member, and the minimum volume which must be enclosed. Solution Given: The cost function of each generator, shown in Figure E2.
Solution Given: The number of manufacturing facilities the company owns, the capacity of the ith facility to produce b i units of an item, the number of distribution centers the product should be shipped too, the minimum number of items, a j , required by the jth distribution center, and the cost to ship an item from the ith plant to the jth distribution center. The design should satisfy the following constraints: 1. Use N and m as the units, and the corresponding values for various parameters.
Solution Given: The equations to calculate bending and average shear stress in a beam, the constraint that the depth of the beam will not exceed twice its width, the applied moment, the applied shear force, and the maximum allowable bending and shear stresses in the beam.
Formulation 1: Step 1: Problem Statement Shown above Step 2: Data and Information Collection Shown above Step 3: Definition of Design Variables For this formulation, twenty-four design variables are chosen which designate the twenty-four different paths of processing the iron ore, i.
Solution Given: The specific, required cross-sectional area of the canal, least construction costs occur when the volume of the excavated material is equivalent to the amount of material required for the two dykes, and the dimensions as shown in Figure E2. Afshar Arabi. Alberto Vasquez Martinez. Surya Prangga. Bakkiya Raj. Muhammad Iskandar. Anastasia Ayu Pratiwi.
HighTec service. Hadi Partovi. Ravish Yadav. Mahmoud AbdAllah. Sunny Chosa. Chloe Jazmines. Eva Gregory. More From Afshar Arabi. Hasan Mohammadi. Elger, Barbara a. LeBret, Clayton T. Crowe, John a. Popular in Science. Rochana Ramanayaka. Anuj Kumar Sharma. Shibu Kumar S. Formulate the minimum cost design problem. Solution Given: The lot size, building floor space and parking area requirements, and the data given in the problem statement.
Required: It is desired to find the building cross-sectional area and its height to meet all the requirements and minimize cost of the building. Procedure: We follow the five step process to formulate the problem as an optimization problem. Note that for a meaningful design, h must be a multiple of 3. The company manufactures gasoline and lube oil from the crudes. Yield and sale price barrel of the product and markets are shown in Table E2.
How much crude oils should the company use to maximize its profit? Formulate the optimum design problem. Table E2. Required: It is desired to find the amount of each crude oil which should be used, subject to the above constraints, to maximize profit.
The height and radius of the mug should be not more than 20 cm. The mug must be at least 5 cm in radius. The surface area of the sides must not be greater than cm2 ignore the area of the bottom of the mug and ignore the mug handle — see figure. Solution Given: The maximum and minimum radius of the mug, the maximum height of the mug, and the maximum surface area of the mug. The area of the bottom of the mug is ignored. Required: It is desired to find the dimensions of the beer mug which will maximize the amount of beer it can hold.
An end view of the units is shown in Fig. There are certain limitations on the design problem. The smallest available conducting tube has a radius of 0. Further, the total cross sectional area of all the tubes cannot exceed cm2 to ensure adequate space inside the outer shell. Formulate the problem to determine the number of tubes and the radius of each tube to maximize the surface area of the tubes in the exchanger.
Solution Given: The minimum radius of each tube, the similarity between each tube, and the maximum surface area of all tubes combined. Required: It is desired to find the number of tubes and the radius of each tube which will maximize the surface area of the tubes in the heat exchanger.
The available width along the street is m, while the maximum depth available is m. We want to have at least 10, m2 in the lot. To avoid unsightliness, the city requires that the longer dimension of any lot be no more than twice the shorter dimension. Formulate the minimum-cost design problem. Solution Given: The cost of land in the downtown urban renewal section, the maximum width and depth available, and the minimum area available in the lot.
In addition, the longer dimension can be no more than twice the shorter dimension. Required: Minimize the cost required to build such a parking lot, subject to the given constraints. Available raw materials for the products are kg of C and 80 kg of D. To produce 1 kg of A, we need 0.
To produce 1 kg of B, we need 0. The markets for the products are 70 kg for A and kg for B. How much A and B should be produced to maximize profit? Formulate the design optimization problem. Solution Given: The profits from selling products A and B, the amount of raw material available of products C and D, the amount of products C and D required to produce products A and B, and the market for products A and B.
Required: It is desired to find the amount of A and B which should be produced to maximize profit. The amount of vitamins A and B in 1 kg of each food and the cost per kilogram of food are given in Table E2.
Formulate the design optimization problem so that we get at least the basic requirements of vitamins at the minimum cost. Required: It is desired to find the amount of each food which should be consumed to provide the basic vitamin requirements at the minimum cost. They can produce bottles of pure alcohol each week. They bottle two products from alcohol: i wine, 20 proof, and ii whiskey, 80 proof.
Recall that pure alcohol is proof. They have an unlimited supply of water but can only obtain empty bottles per week because of stiff competition. The weekly supply of sugar is enough for either bottles of wine or bottles of whiskey. They can sell whatever they produce. How many bottles of wine and whisky should they produce each week to maximize profit?
Levy Solution Given: The amount of bottles of pure alcohol which can be produced each week, the two types of alcohol which are produced, the amount of empty bottles available per week, the amount of each alcohol which can be produced based on the weekly sugar supply, and the profits for each alcohol type.
Required: It is desired to find the amount of bottles of wine and whisky which should be produced, each week, to maximize profit. The can is a right circular cylinder with interior height h and radius r. The ratio of height to diameter must not be less than 1. The height cannot be more than 20 cm. Solution Given: The desired interior can volume, the minimum and maximum ratio of height to diameter, and the maximum height.
Required: It is desired to find the design which minimizes the area of sheet metal for the can. Use the data in the following table. Required: It is desired to find the design of the shipping container which minimizes the ratio given. Solution Given: The construction costs for the sides, ends, and the bottom of the container and the minimum volume requirement.
Required: It is desired to find the dimensions of the material container which minimize cost. The tank is to be housed in a shed with a sloping roof. Solution Given: The required volume of the tank, the fabrication cost of the sheet metal per unit area, and the limiting relation between the height and the diameter.
Required: It is desired to find a design of the tank which minimizes cost. The frame must enclose a total volume of at least m3.
Solution Given: The cost of a horizontal member in two, separate directions, the cost of a vertical member, and the minimum volume which must be enclosed. Required: It is desired to find a design which minimizes the cost of the steel framework. All costs and power are expressed on a per unit basis. The total power needed is at least 60 units.
Formulate a minimum-cost design problem to determine the power outputs P1 and P2. Solution Given: The cost function of each generator, shown in Figure E2. Required: It is desired to find the power outputs, P1 and P2, which minimizes cost.
A company has m manufacturing facilities. The facility at the ith location has capacity to produce bi units of an item. The product should be shipped to n distribution centers. The distribution center at the jth location requires at least aj units of the item to satisfy demand. The cost of shipping an item from the ith plant to the jth distribution center is cij.
Solution Given: The number of manufacturing facilities the company owns, the capacity of the ith facility to produce bi units of an item, the number of distribution centers the product should be shipped too, the minimum number of items, aj, required by the jth distribution center, and the cost to ship an item from the ith plant to the jth distribution center.
Required: It is desired to design a transportation system which minimizes costs and meets the constraints set by the two types of facilities.
Design a symmetric two-bar truss both members have the same cross section , as shown in Fig. The truss consists of two steel tubes pinned together at one end and supported on the ground at the other.
The span of the truss is fixed at s. Formulate the minimum mass truss design problem using height and the cross-sectional dimensions as design variable. The design should satisfy the following constraints: 1. Because of space limitations, the height of the truss must not exceed b1, and must not be less than b2. The ratio of the mean diameter to thickness of the tube must not exceed b3. The height, diameter, and thickness must be chosen to safeguard against member buckling.
Solution Given: Constraints listed above and the factor of safety against buckling in the data section above. Required: It is desired to design a truss which minimizes mass using height and the cross sectional dimensions as design variables. Use N and m as the units, and the corresponding values for various parameters. It is also desired that the depth of the beam shall not exceed twice its width.
Solution Given: The equations to calculate bending and average shear stress in a beam, the constraint that the depth of the beam will not exceed twice its width, the applied moment, the applied shear force, and the maximum allowable bending and shear stresses in the beam. Required: It is desired to design a beam which minimizes cross-sectional area without yielding due to shear or bending stresses.
At the present time, he has , kg of soybean oil, , kg of cottonseed oil, and kg of milk-base substances. The milk-base substances are required only in the production of margarine.
There are certain processing losses associated with each product: 10 percent for shortening, 5 percent for salad oil, and no loss for margarine. In addition, sales forecasts indicate a strong demand for all produces in the near future. The profit per kilogram and the base stock required per kilogram of each product are given in Table E2. Formulate the problem to maximize profit over the next production scheduling period.
Required: It is desired to create a production schedule which will maximize profit. Design of a system implies specification for the design variable values. True 2. All design problems have only linear inequality constraints. False 3. All design variables should be independent of each other as far as possible. True 4. If there is an equality constraint in the design problem, the optimum solution must satisfy it.
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