MAT1200 Operations Research 1, S2-2016 Assignment 3 Due date: Friday 9 September 2016 Weight 12% Total Marks: 100 Assignment 3 is based on the material from modules 6 and 7. Full working must be shown, and reasons provided to justify your answers. See the Guidelines provided at the end of this assignment for further details. Submit your assignment electronically as one (1) PDF file via the link (Assignment 3 submission) available on study desk before the deadline. You can resubmit this assignment electronically as long as your assignment is still in draft and the deadline has not passed. Please remember to submit your assignment by the deadline. The last submitted version will be marked so please check your assignment carefully before submitting. Hand-written work is more than welcome, provided you are neat and legible. Do not waste time type-setting and struggling with symbols. Rather show that you can use correct notation by hand and submit a scanned copy of your assignment. You may also type-set your answers if your software offers quality notation if you wish. Any requests for extensions should be made prior to the due date by contacting the examiner. Requests for an extension should be support by documentary evidence. An Assignment submitted after the deadline without an approved extension of time will be penalised. The penalty for late submission is a reduction by 5% of the maximum Assignment Mark, for each University Business Day or part day that the Assignment is late. An Assignment submitted more than ten University Business Days after the deadline will have a Mark of zero recorded for that Assignment. 1Question 1. [40 marks] The MythCo Toy Company manufactures and distributes a wide range of TV merchandise. Currently the company is experiencing some capacity problems in the product and distribution of their very successful Mythbuster figurine lines. Two figurines, Jamie and Adam, are currently produced and sold in a variety of packaged combinations. The table below shows the times required in Construction, Assembly and Packaging stages for the three product combinations, and their respective profits. Product Time Required (in minutes) Profit Construction Assembly Packaging ($ per figurine) Adam Figurine 8 5 2 10 Jamie Figurine 8 5 3 12 Action Pack 12 12 4 16 (Contains Adam and Jamie) Currently 10 hours are available for construction, 8 hours are available for assembly, and 4.5 hours are available for packaging for each day. Also, the production of Jamie Figurines is restricted to 55 per day due to Berets availability. MythCo wishes to maximize its profit, and believes that all of its products can be sold at the current prices. The problem of maximizing MythCo's profit can be expressed as: Maximize z = 10x1 + 12x2 + 16x3 s.t. 8x1 + 8x2 + 12x3 ≤ 600 (Construction) 5x1 + 5x2 + 12x3 ≤ 480 (Assembly) 2x1 + 3x2 + 4x3 ≤ 270 (Packaging) x2 + x3 ≤ 55 (Number of Berets) x1; x2; x3 ≥ 0 where x1, x2, x3 are the number of Adam, Jamie, and Action sets respectively. Solving this L.P. using LINDO produces the following output: LP OPTIMUM FOUND AT STEP 2 OBJECTIVE FUNCTION VALUE 1) 860.0000 VARIABLE VALUE REDUCED COST X1 20.000000 0.000000 X2 55.000000 0.000000 X3 0.000000 1.000000 2ROW SLACK OR SURPLUS DUAL PRICES 2) 0.000000 1.250000 3) 105.000000 0.000000 4) 65.000000 0.000000 5) 0.000000 2.000000 NO. ITERATIONS= 2 RANGES IN WHICH THE BASIS IS UNCHANGED: OBJ COEFFICIENT RANGES VARIABLE CURRENT ALLOWABLE ALLOWABLE COEF INCREASE DECREASE X1 10.000000 2.000000 2.000000 X2 12.000000 INFINITY 1.000000 X3 16.000000 1.000000 INFINITY RIGHTHAND SIDE RANGES ROW CURRENT ALLOWABLE ALLOWABLE RHS INCREASE DECREASE 2 600.000000 168.000000 160.000000 3 480.000000 INFINITY 105.000000 4 270.000000 INFINITY 65.000000 5 55.000000 20.000000 55.000000 THE TABLEAU ROW (BASIS) X1 X2 X3 SLK 2 SLK 3 SLK 4 1 ART 0.000 0.000 1.000 1.250 0.000 0.000 2 X1 1.000 0.000 0.500 0.125 0.000 0.000 3 SLK 3 0.000 0.000 4.500 -0.625 1.000 0.000 4 SLK 4 0.000 0.000 0.000 -0.250 0.000 1.000 5 X2 0.000 1.000 1.000 0.000 0.000 0.000 ROW SLK 5 1 2.000 860.000 2 -1.000 20.000 3 0.000 105.000 4 -1.000 65.000 5 1.000 55.000 3Note the final tableau can written in the following familiar format: cj 10 12 16 0 0 0 0 cB xB x1 x2 x3 s1 s2 s3 s4 b 10 x1 1 0 1 2 1 8 0 0 −1 20 0 s2 0 0 9 2 − 5 8 1 0 0 105 0 s3 0 0 0 − 1 4 0 1 −1 65 12 x2 0 1 1 0 0 0 1 55 zj 10 12 17 5 4 0 0 2 860 zj − cj 0 0 1 5 4 0 0 2 Note: The row index (and hence the slack variable numbering) in the Lindo output is increased by 1. The following parts of this exercise should be answered by using the Lindo output above (including the sensitivity analysis), and not by modifying and running the problem through Lindo (or equivalent software or websolver). Note: Consider any changes in to the original problem formulation independently in parts (b){(f). (a) How many of each product should MythCo produce each day? What is assumed in (6 marks) giving this answer? Remember to state the solution to this problem in plain English so it makes sense when provided to the MythCo's management staff. (b) Is there spare capacity in any of the production stages or Beret availability ? Explain (4 marks) your answer. (c) By how much would the profit on an Action pack have to increase to make its pro- (2 marks) duction worthwhile? (d) Tomorrow the machine used in construction stage will be unavailable for part of the (9 marks) day, so that only 8 hours (480 minutes) of time will be available in the construction stage of production. How will the closure change MythCo's profit? How many of each product should MythCo now produce? (e) A new figurine of Buster is being considered, but it would require 10 minutes for (9 marks) Construction, 7 minutes for Assembly and 3 minutes for Packaging. No Beret is required for the proposed figurine. How much profit would be needed on the Buster figurine to make its production worthwhile? (f) If at most 45 Jamie figurines must be produced each day, how should MythCo modify (10 marks) their production plan? 4Question 2. [40 marks] The MythCo Toy Company has signed a contract to supply its hugely successful replica Archimedes Death Rays to two customers A and B for the next 3 months. The requirements for each customer are given in the following Table. Customer September October November A 30 20 20 B 25 20 10 MythCo can produce 40 Death Rays per month at a cost of $200 per unit on regular time and an additional 10 Death Rays per month on overtime at a cost of $240 per unit. They can store units at a cost of $20 per unit per month. The contract allows the company to fall short of its supply commitment to customer A in September and October but this incurs a penalty cost of $10 per unit per month and they are only willing to wait one month at most for the delivery of the replica units. No shortages are allowed for customer B and their demands must be satisfied on the months specified. MythCo wishes to determine the optimal production schedule that will minimize the total cost of production, storage and shortages. (a) Determine the optimal production schedule that will minimize the total cost of pro- (30 marks) duction, storage and shortages. (Suggestion: Start by defining the supply and demand points and forming a cost table. Fill in the cost table completely by assigning a very high cost to any demand that cannot be met from a supply point.) Use Vogel's Method to find your initial feasible solution. (b) MythCo has just found out that there will be no overtime available in September due (10 marks) to labor shortages. How should the company alter their production plan to accommodate this change? 5Question 3. [20 marks] Jamie is trying to decide how best to assign Tory, Grant, and Cary to manage four projects that need doing for the show. The time required to complete each of the projects (in hours) is shown in the table below if Tory, Grant, or Cary manage the project. 17 Project Worker Tory 14 19 16 Grant Cary Project A Project B Project C Project D 21 18 16 12 15 13 15 16 (a) Jamie decides that, to avoid any conflict, only one team member will manage each (10 marks) project. There will be no cost in terms of time incurred if a project is not completed. Determine the assignment which minimizes the total time using the Hungarian method. List this minimum time. Which project is not completed? (b) Suppose that Jamie changes his mind and decides that all projects need to be com- (10 marks) pleted. To achieve this he will allow team members to work on up to two projects. Determine how this affects the assignment of the team members. Which team member should be allocated to manage each project and what is the minimum total time. [Total: 100 marks] 6Assignment Criteria Question 1. 1. Sensitivity analysis • Analysis uses the provided current solution/output and not found from modifying and running the original LP problem through Lindo (or equivalent software or websolver). • Solution for parts (a)-(c) is correctly interpreted from output. • Analysis for parts (d)-(f) is completed by hand with full working shown and is correct. You may use the information provided in the output and in the final tableau for part (d). 2. For each part the answer is given in plain english and makes sense. Question 2. 1. Formulate problem as a Transportation Problem • Problem is correctly balanced, • All appropriate supply and demand points are given and are correct; • The costs are correct in the Tableau; • For (b) the modification of the formulation in (a) is valid. 2. Solution (by hand) • Feasible solution is correctly found using Vogel's approximation method. • All required Tableaux are given and are correct • Optimality is tested correctly using either the Stepping Stone or UV method. All ∆zij are given and are correct. • Transportation plan is correct. 3. Transportation plan (and the associated cost) is given in plain english and makes sense. 7Question 3. 1. Formulate the problem as an assignment problem • Problem is correctly balanced, • The costs are correct in the Tableau, • For (b) the modification of the formulation in (a) is valid. 2. Solution (by hand) • Hungarian method is correctly used, • K¨onigs Theorem is used correctly to test optimality, • New feasible solution obtained if current solution is not optimal, • Assignment of workers is correct, • Associated cost is given and is correct 3. Assignment of workers (and the associated cost) is clearly explained in plain english. 8