The Midterm Review spring 2013

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Caitlin Small
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1 The Midterm Review spring 213 Featuring a very fine set of example problems Several failed attempts at simplexing

2 Our Feature Presentation 1. Simplexing to glory see previous presentation 2. Transporting in an optimal manner 3. Engineering assignments. Networking on the shortest path. Spanning the tree. Maximum flow 7. Linear Formulating OR students anxious to work the midterm

3 2. The Transportation Algorithm Shipping a single commodity from several sources to several destinations in an optimal manner.

4 (a) The northwest corner factory Store 1 Store 2 Store 3 Store Store Supply ABC DEF XYZ Demand

5 (a) The northwest corner factory Store 1 Store 2 Store 3 Store Store Supply ABC 9 9 DEF XYZ 1 13 Demand

6 (a) The northwest corner factory Store 1 Store 2 Store 3 Store Store Supply ABC 9 DEF 1 + XYZ Demand

7 (a) The northwest corner factory Store 1 Store 2 Store 3 Store Store Supply ABC 9 9 DEF XYZ Demand

8 (b) Our very first (and last) iteration factory Store 1 Store 2 Store 3 Store Store Supply ABC 9 DEF 1 + XYZ Demand (c) Show the reduction in cost: units x -$ = -$3 (d) Is the solution in (b) optimal? No, see cell negative costing

9 PISONIT Ratings Engineer Project A Project B Project C Project D Project E Project F Bill Betty Bob Bertha Bull Determine which engineer is to be assigned to each project in order to maximize the sum of the individual scores. Scores are based upon the job Propensity Index Standard Of the National Institute of Technology (PISONIT). Makes use of the engineer s experience, education, and skill sets and how well they match the engineering requirements of the project.

10 Engineer Project A Project B Project C Project D Project E Project F We start Bill Betty Bob Bertha Bull by subtracting each value in a row or column from the maximum value to convert to a minimization problem and to obtain a zero in each row or column and then add a dummy row:

11 Still starting Need to create more strategic zero cells:

12 We continue

13 We do it again

14 And again

15 Are we done? No!

16 Now are we done? Yes!

17 The Solution Engineer Project A Project B Project C Project D Project E Project F Bill Betty Bob Bertha Bull Dummy Bill project C Betty Project B Bob Project D Bertha Project F Bull Project A Dummy Project E score = = 3

18 . Networking the shortest path Tourist traffic must travel through the county beginning where the interstate system ends (START) and ending at one of Ohio s best state parks (END). Find the shortest route through the network given below from the Start node to the End node using the algorithm discussed in class. Numbers are distances in miles. The following Network worktable for problems, 7, and must 8. be correct to receive full credit. B 9 E 12 8 H 1 START C F I 2 END D 7 G 3 J

19 Networking to a solution Start B C D E F SD-13 BE- CE- DG-7 EC- FJ-2 SB-1 BF-12 CG- DF-11 EB- FH-7 SC-23 CF-12 EI-8 D-11 EH-9 FB-12 FC-12 FI-13 G H I J End GJ-3 HF-7 IE-8 JF-2 GC- HE-9 IG- JG-3 GD-7 HEND-1 IF-13 JEND-18 GI- IEND-1

20 Networking to a solution Start B C D E F SD-13 BE- CE- DG-7 EC- FJ-2 SB-1 BF-12 CG- DF-11 EB- FH-7 SC-23 CF-12 EI-8 D-11 EH G H I J End GJ-3 HF-7 IE-8 JF-2 GC- HE-9 IG- JG-3 GD-7 HEND-1 IF-13 JEND-18 GI- IEND-1 FB-12 FC-12 FI-13 Shortest Route: Start B E I End miles

21 . Spanning the tree Network for problems, 7, and 8. B 9 E 12 8 H 1 START C F I 2 END D 7 G 3 J

22 The tree has been spanned in a minimal way Min total distance = 7 B E 8 H START 7 C F 2 I 1 END 13 D 7 G 3 J

23 . Maximizing Flow Determine the maximal flow and the optimum flow in each arc for the following network:

24 Max flow working 1 8 Path Flow

25 Still working 8 1 Path Flow

26 More still working Path Flow

27 Yes, working Path Flow

28 Almost done Path Flow

29 Done! here are the Arc Flows Path Flow

30 Linear Formulating Formulate but do not solve the following linear program: The CarParts Company manufactures and assembles automobile doors and trunk lids. The plant produces semi-finished products that are then sanded and painted in the company s finishing facility. The finishing facility employs a total of 9 workers in two 8-hour shifts a day, 21 working days a month. The size of the labor force in the finishing facility fluctuates because of the annual leave taken by the employees.

31 Formulating with data May June July Plant Capacity-doors (# units) 2 Plant Capacity-trunk lids (# units) Finishing facility leave requests (worker-months*) 3 Product Unit cost ($) for May and June (including labor) Unit cost ($) for July (including labor)** Doors 23 2 Trunk lids Unit Selling Price($) Product sales orders May June July End-of-April inventory Finishing labor time (minutes) Doors Trunk lids

32 The rest of the story A unit may be produced in one month and held over for sale in a later month. The storage cost is $ per month for each door and $ per month for each trunk lid held over from one month to the next. Work-in-process floor space within the plant is restricted to 7, square feet (i.e. no more than 7, sq. ft. of doors and trunk lids can be produced in a given month). Each door requires 11 square feet and each trunk lid requires 8 square feet. Warehouse floor space for inventory carried over from one month to the next is limited to 3, square feet. There should be no more than doors and trunk lids available in inventory at the end of July. No backorders are permitted The Company desires to maximize its profit over the three-month period. How many doors and trunk lids should be produced each month?

33 Ahhh, the formulation (a) Define all decision variables: X 1j = number of doors produced in month j (j=1,2,3) X 2j = number of trunk lids produced in month j (j=1,2,3) I 1j = number of doors in inventory at the end of of month j (j=1,2,3) I 2j = number of trunk lids in inventory at the end of of month j (j=1,2,3) (b) Formulate the objective function: MAX z = 22X X X 13 +7X 21 +7X 22 +X 23 -I 11 -I 12 -I 13 -I 21 -I 22 -I 23

34 Keep on formulating the constraints! INVENTORY BALANCE CONSTRAINTS!I=7!I2=2 -I11+X11=3-7 -I12+I11+X12=32 -I13+I12+X13=3 I13< -I21+X21=1-2 -I22+I21+X22=1 -I23+I22+X23=12 I23<!LABOR HOUR CONSTRAINTS 1.X11+X21<8 (8 x 21 x ) 1.X12+X22<8 (8 x 21 x ) 1.X13+X23<72(8 x 21 x )!STORAGE SPACE CONSTRAINTS 11X11+8X21<7 11X12+8X22<7 11X13+8X23<7 11I11+8I21<3 11I12+8I22<3 11I13+8I23<3!PLANT CAPACITY CONSTRAINTS X11<2 X12< X13< X21<3 X22<2 X23<2

35 Another Formulation The Make-it-Rite Company produces 3 products in 2 factories from iron ore mined in the hills of Pennsylvania. Given the following data, determine how many tons of each product should be produced monthly in each factory to maximize total profit (selling price minus production and raw material (iron ore) costs. Iron ore costs $ per ton and a maximum of, tons a month is available for distribution to the two factories. Because of shipping limitations, tons of product A cannot be more than half the total tons of products B and C combined.

The Data Product Selling price ($) per ton Maximum monthly sales tons of iron ore per ton of finished product produced A 3 2 tons 2. B 1 tons 3.1 C 2 18 tons 1.

36 The Data Product Selling price ($) per ton Maximum monthly sales tons of iron ore per ton of finished product produced A 3 2 tons 2. B 1 tons 3.1 C 2 18 tons 1. Number Factory Maximum monthly production Production budget ($) 1 Allentown 2 tons 1, 2 Pittsburgh 3 tons 2, Factory A B C 1 Allentown 7 2 Pittsburgh 7 Unit production costs ($/ton)

37 The Objective Function Define x i,j = the number of tons of product i produced monthly in factory j where i = A, B, C and j = 1,2 Max z = (3--2) xa1 + (-7-3)xb1 + (2- -1)xc1 + (3--2) xa2 + (-7-3)xb2 + (2--1)xc2 = 3xa1 + 2xb1 + 2xc1 + xa2 + 2xb2 + 2xc2 Iron ore costs $ per ton Product Selling price ($) per ton tons of iron ore per ton of finished product produced A 3 2. B 3.1 C 2 1. Unit production costs ($/ton) Factory A B C 1 Allentown 7 2 Pittsburgh 7

38 Prod uct The Constraints Max production xa1 + xb1 + xc1 2 xa2 + xb2 + xc2 3 Max sales xa1 + xa2 2 xb1 + xb2 1 xc1 + xc2 18 Maximum monthly sales A 2 tons 2. B 1 tons 3.1 C 18 tons 1. tons of iron ore per ton of finished product produced Numbe r Factory Maximum production budget ($) 1 Allentown 2 tons 1, 2 Pittsburgh 3 tons 2, Production budget xa1 +7 xb1 + xc1 1, xa2 + 7xb2 + xc2 2, Available iron ore 2.xa xb1 + 1.xc1 + 2.xa xb2 + 1.xc2, Limit on product A xa1 + xa2.(xb1 + xb2 + xc1 + xc2)

39 This has been a Midterm Review Brought to you by your friendly engineering management department Preparing for the midterm

ANNA UNIVERSITY - M.C.A IV SEMESTER Problem Set MC1752/MC9242/ Resource Management Techniques

ANNA UNIVERSITY - M.C.A IV SEMESTER Problem Set MC1752/MC9242/ Resource Management Techniques Sou rce S r i V e n k a t e s w a r a C o l l e g e o f E n g i n e e r i n g, S r i p e r u m b u d u r D e p a r t m e n t o f C o m p u t e r A p p l i c a t i o n s ---------------------------------------------------------------------------------