OR6205: Deterministic Operations Research 1 Solutions to HW #3 •
4.1-­‐4 a)
b) CP Solution Feasibility Objective A (0, 3/2) Infeasible 6750 B (0, 6/5) Infeasible 5400 C (0, 1) Feasible 4500 D (1/4, 1) Feasible 5625 E (2/5, 1) Infeasible 6300 F (1, 1) Infeasible 9000 G (2/3, 2/3) Feasible 6000* H (1, 2/5) Infeasible 6300 I (1, 1/4) Feasible 5625 J (1, 0) Feasible 4500 K (6/5, 0) Infeasible 5400 L (3/2, 0) Infeasible 6750 M (0, 0) Feasible 0 The point G is optimal. c) Start at the origin M = (0, 0). Both adjacent points C = (1, 0) and J = (0, 1) are feasible and have better objective values, so one can choose to move either one of them. Suppose we choose C, which is not optimal since its adjacent CPF solution D is better. The other corner point that is adjacent to C is B, but it is infeasible, so move to D. Its adjacent G is both feasible and better. The CPF solutions that are adjacent to G, namely D and I, both have lower objective values. Hence, G is optimal. If one chooses to proceed to J instead OR6205: Deterministic Operations Research 2 Solutions to HW #3 of C after the starting point, then the simplex path follows the points M, J, I, G, and using similar arguments, one obtains the optimality of G. •
4.2-­‐1 a) Augmented form: maximize 4500x1 + 4500x2 subject to x1 + x3 = 1 x2 + x4 = 1 5000x1 + 4000x2 + x5 = 6000 400x1 + 500x2 + x6 = 600 x1, x2, x3, x4, x5, x6 ≥ 0 b) CPF Solution BF Solution Nonbasic Variables Basic Variables A (0, 1) (0, 1, 1, 0, 2000, 100) x1, x4 x2, x3, x5, x6 B (1/4, 1) (1/4, 1, 3/4, 0, 750, 0) x4, x6 x1, x2, x3, x5 C (2/3, 2/3) (2/3, 2/3, 1/3, 1/3, 0, 0) x5, x6 x1, x2, x3, x4 D (1, 1/4) (1, 1/4, 0, 3/4, 0, 75) x3, x5 x1, x2, x4, x6 E (1, 0) (1, 0, 0, 1, 1000, 200) x2, x3 x1, x4, x5, x6 F (0, 0) (0, 0, 1, 6000, 600) x1, x2 x3, x4, x5, x6 c) o
BF solution A: Set x1 = x4 = 0 and solve x3 = 1 x2 = 1 4000x2 + x5 = 6000 à x5 = 2000 5000x2 + x6 = 600 à x6 = 100 o
BF solution B: Set x4 = x6 = 0 and solve x1 + x3 = 1 à x3 = 3/4 x2 = 1 5000x1 + 4000x2 + x5 = 6000 à x5 = 750 400x1 + 500x2 = 600 à x1 = 1/4 o
BF solution C: Set x5 = x6 = 0 and solve x1 + x3 = 1 x2 + x4 = 1 5000x1 + 4000x2 = 6000 400x1 + 500x2 = 600 OR6205: Deterministic Operations Research 3 Solutions to HW #3 From the last two equations, x1 = x2 = 2/3, and from the first two, x3 = x4 = 1/3. o
BF solution D: Set x3 = x6 = 0 and solve x1 = 1 x2 + x4 = 1 à x4 = 3/4 5000x1 + 4000x2 = 6000 à x2 = 1/4 400x1 + 500x2 + x6= 600 à x6 = 75 o
BF solution E: Set x2 = x3 = 0 and solve x1 = 1 x2 = 1 5000x1 + x5 = 6000 à x5 = 1000 400x1 + x6 = 600 à x6 = 200 o
BF solution F: Set x1 = x2 = 0 and solve x3 = 1 x4 = 1 x5 = 6000 x6 = 600 •
4.4-­‐3 a) Optimal solution: (x1*, x2*) = (20, 20) and Z* = 60 OR6205: Deterministic Operations Research 4 Solutions to HW #3 b)
e)
f) Last two steps: OR6205: Deterministic Operations Research 5 Solutions to HW #3 g) LINGO: •
Forest Pest Control Spray Program Case Study a) What is the problem addressed in this paper? Spruce budworm destroys large areas of forest in the state of Maine every year. To control budworm damage, the Maine Forest Service conducts aerial spraying of the forests with insecticides. The OR6205: Deterministic Operations Research 6 Solutions to HW #3 investigators developed a mathematical model for improving planning and operating decisions of the aerial control spray program. b) What is the objective, the decision variables, and the types of constraints in the model. Briefly describe. The objective of the problem is to minimize the total aircraft cost for spraying all blocks. The decision variables tijk are the aircraft times spent for every feasible combination of aircraft team type (i), airfield base (k), and spray block (j), where aircraft times include spraying time, ferrying time, orientation time upon arrival at the block, turning time to align at the beginning of each swath, and ground time. The constraints are: (1) the total time for spraying each block by all aircraft team types, flying from all airfields, must be sufficient to allow for complete spraying of that block which is dependent on its the location and geometry, along with the aircraft and airfield characteristics; (2) the time window for each aircraft and airfield combination is limited to a certain amount of time that depends on weather conditions at the airfield and the time needed for aircraft to be loaded by insecticide and take off, thus it depends on airfield (location) and aircraft type; (3) non-­‐negativity constraints on all variables. c) If you were to use a modeling language, which would have been the sets and the associated attributes (model parameters and variables)? The sets to be used in this problem would have been the following: o
Primitive sets: (Aircraft Types), (Airfields), (Spray Blocks), and (Insecticides) o
Derived sets: (Aircraft Types)x(Airfields), (Spray Blocks)x(Airfields), and o
(Aircraft Types)x(Spray Blocks)x(Airfields). d) Using as guide the “Operations Research Modeling Approach” of Chapter 2 of the textbook, describe in a few sentences the phases or steps of the modeling approach performed in this particular OR project. Elaborate on the validation of the model and its use in conducting “what if” analyses. The approach for this project follows the Operations Research modeling approach referenced in Chapter 2 of the textbook. (1) The first step is to define the problem and to gather relevant information. The investigators interviewed the management of Maine Forest Service and reviewed the spraying practice of the previous years to come up with the objective to minimize spray costs – the dominant cost of the operation -­‐ that can be controlled by better decisions on allocating aircraft types to airfields and blocks. (2) The second step is to formulate the mathematical model to represent the problem. Different modeling approaches were considered and LP was adopted for its computational efficiency. Appropriate decision variables were defined that reflect operating management decisions. The objective function (minimum aircraft cost) and the restriction on the variables (constraints) were formulated. (3) A LP code based on the simplex algorithm was developed to derive solutions to the problem. (4) In order to test the model, data from the previous year (1983) project were used to generate solutions from the model. The model-­‐derived solutions were compared to the actual 1983 operation results. The management confirmed the validity of the results. (5) The model was prepared for application. All necessary raw data were collected, other model parameters were derived from raw data by formulas and a complete software package was developed with a matrix and report generator attached to the LP code. (6) The model was implemented to analyze next year’s (1984) project. The model facilitated what-­‐if analyses such as impact if an airfield closes or if a new type of aircraft is available for spraying. OR6205: Deterministic Operations Research 7 Solutions to HW #3 e) Were there any other modeling approaches considered for this project before the linear programming approach was adopted? A network-­‐flow approach and an integer-­‐programming approach were considered as well. The network-­‐
flow approach was excluded because of the “lack of realism” and the integer programming approach was too cumbersome. f) To estimate the model parameters, data were collected or derived. Which data were collected by the investigators, which were derived and which were supplied by which sources? The Maine Forest Service provided a map of the state with the blocks and the airfields and the investigators calculated distances from the airfields to blocks. 4 The investigators developed a probabilistic model of the available aircraft time for each aircraft type on each airfield by studying weather conditions for the time of the year and aircraft characteristics. Aircraft characteristics such as speeds were taken from aircraft manufacturer’s specifications. The Edson report and hand-­‐collected data were used to calculate the mean time per period for the C-­‐54 and Thrush aircraft. The Maine Forest Service also provided the historical data that allowed the parameters for time actually spent spraying and flying to be derived. The total hours available for each aircraft/airfield combination was derived from time windows and availability of aircraft crews. The investigators flew on small Cessna aircraft following the spray teams on the actual spray operation and measured the various aircraft times, such as orientation times, turning times, etc. The costs of each aircraft were determined by calculating the fixed and variable costs based on data obtained from manufacturer’s specifications. g) If the # of aircraft teams were 5, the # of spray blocks was 500 and the # of airfields were 10 – what would have been the size of the model (# of variables and # of constraints). Assume aircraft teams can be stationed at all airfields, can spray all types of insecticides and their operating range is large enough to reach any block from any airport. o
Variables: |S| = 25,000 (all combinations of aircraft teams, blocks and airfields) o
Constraints: Constraints (2.2) are 500, one for each spray block. Constraints (2.3) are 50, one for each aircraft type/airfield combination. The total is 550 functional constraints and 25,000 non-­‐
negativity ones.