W24 CIVE332 - Assignment 4

Page 1 of 4 CIVE 332 Civil Engineering Systems and Project Management Assignment No. 4 (Total marks: 100) Due Thursday, March 28 th 6:00pm on Crowdmark 1) What is the difference between deterministic and stochastic optimization models? What is the difference between descriptive and prescriptive optimization models? [4 marks] 2) Solve the following linear programs using the graphical method. [20 marks] Compute the value of the objective function and decision variables at optimality, and indicate which statement best describes the solution: a) Minimize : 𝒁 = ?𝒙 ? + 𝒙 ? Subject to: ?𝒙 ? + ?𝒙 ? ≥ ? ?𝒙 ? − ?𝒙 ? ≥ −? 𝒙 ? + 𝒙 ? ≤ ? ?𝒙 ? ≤ ? 𝒙 ? , 𝒙 ? ≥ ? a) This linear program has a unique optimal solution b) This linear program has alternate optima c) This linear program is infeasible d) This linear program is unbounded b) Maximize: 𝐙 = ?𝐱 ? + ?𝐱 ? Subject to: ?𝐱 ? + ?𝐱 ? ≤ ? ?𝐱 ? + ?𝐱 ? ≤ ?? ?𝐱 ? ≤ ? 𝒙 ? , 𝒙 ? ≥ ? a) This linear program has a unique optimal solution b) This linear program has alternate optima c) This linear program is infeasible d) This linear program is unbounded 3) A manager of a shipping company that delivers packages to customers in different cities. The manager has four different transportation options available (A, B, C, and D) with varying fixed costs per delivery and different delivery capacities. The fixed cost per delivery of the four options are $140/unit, $130/unit, $170/unit, and $135/unit, respectively. His company needs to deliver packages to four different cities each week, with varying costs per delivery and different delivery demands. The delivery capacities and demands for each city, along with the shipping costs per unit are presented in the table below. Formulate a linear program that will identify the optimal shipping strategy for this company to minimize costs while meeting the delivery demands of all four cities. [10 marks]

Page 2 of 4 Transportation Option City 1 City 2 City 3 City 4 Delivery Capacity A $30/unit $19/unit $42/unit $38/unit 700 units B $45/unit $38/unit $27/unit $36/unit 1310 units C $24/unit $17/unit $29/unit $29/unit 570 units D $20/unit $20/unit $31/unit $41/unit 1100 units Demand 420 units 580 units 700 units 620 units 4) Solve the following linear program using analytical method. Compute the value of the objective function and decision variables at optimality, and indicate which statement best describes the solution: [15 marks] Minimize: 𝒁 = 𝒙 ? + ?𝒙 ? Subject to: • ?𝒙 ? − ?𝒙 ? ≥ −? • ?𝒙 ? − ?𝒙 ? ≤ ? • 𝒙 ? + 𝒙 ? ≥ ? • 2𝒙 ? + 𝒙 ? ≤ ? • 2𝒙 ? ≤ ? • 𝒙 ? , 𝒙 ? ≥ ? a) This linear program has a unique optimal solution b) This linear program has alternate optima c) This linear program is infeasible d) This linear program is unbounded 5) A company which produces two different types of reinforcement: carbon fibre-reinforced polymer bars and glass fibre-reinforced polymer bars. The production of these reinforcements is limited by the availability of moulding production time, assembling time, and labour availability, as shown in the table below. The company has a total of 24 hours of moulding production time, 8 hours of assembling time, and 12 hours of labour time available each week. Each bundle of glass fibre- reinforced polymer bars sells for $550, and each carbon fibre-reinforced polymer bars sells for $700. [20 marks] Resources Reinforcement carbon fibre- reinforced polymer bars glass fibre- reinforced polymer bars Moulding production hours 2 1 Assembling time 1 2 Labour hours 6 4 a. Formulate a linear program that will suggest a production policy for maximizing sells. b. Graph the feasible region in decision space for this problem and Identify the feasible extreme points.

Page 4 of 4 We need to calculate the total profit of using a particular dragline at a specific site. To do this, we multiply the daily profit when the specific dragline that at old site ? is shifted from that site and used at new mining site ? by the number of days to exhaust the coal seam at site ? . This gives total profit 𝑃 ?? of assigning the dragline from old mine ? to the new mine ? , but this profit value does not include the significant cost of moving that dragline to site ? . The costs of shifting the draglines, numbered 1,2,3 and indexed by ? , between the old mining sites and the new mine sites, numbered 1,2,3, and indexed by ? , are denoted by ? ?? . Structure, and write without summation notation, a model to determine which dragline will be shifted to which new mine site in order to achieve the maximum total profit. 8) A maximum flow problem. [10 marks] The diagram shown below shows a road network in a city where goods are transported. The cost of transportation per unit of good on each link (??) is ? ?? . Nodes 1,2, and 3 are inflow nodes, and nodes 7 and 8 are outflow nodes. At each inflow node the allowable amount of good entry is ? ? . At each outflow node the amount required to leave is ? ? . Set up a linear program model that can be used to find the minimum cost for transporting goods through the system .