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1 Lecture 5 Linear Programming (6S) and Transportation Problem (8S)

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2 George Dantzig – 1914 -2005 Concerned with optimal allocation of limited resources such as Materials Budgets Labor Machine time among competitive activities under a set of constraints Linear Programming George Dantzig – 1914 -2005

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3 Product Mix Example (from session 1) Type 1Type 2 Profit per unit $60$50 Assembly time per unit 4 hrs10 hrs Inspection time per unit 2 hrs1 hr Storage space per unit 3 cubic ft ResourceAmount available Assembly time100 hours Inspection time22 hours Storage space39 cubic feet

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4 Maximize 60X 1 + 50X 2 Subject to 4X 1 + 10X 2 <= 100 2X 1 + 1X 2 <= 22 3X 1 + 3X 2 <= 39 X 1, X 2 >= 0 Linear Programming Example Variables Objective function Constraints What is a Linear Program? A LP is an optimization model that has continuous variables a single linear objective function, and (almost always) several constraints (linear equalities or inequalities) Non-negativity Constraints

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5 Decision variables unknowns, which is what model seeks to determine for example, amounts of either inputs or outputs Objective Function goal, determines value of best (optimum) solution among all feasible (satisfy constraints) values of the variables either maximization or minimization Constraints restrictions, which limit variables of the model limitations that restrict the available alternatives Parameters: numerical values (for example, RHS of constraints) Feasible solution: is one particular set of values of the decision variables that satisfies the constraints Feasible solution space: the set of all feasible solutions Optimal solution: is a feasible solution that maximizes or minimizes the objective function There could be multiple optimal solutions Linear Programming Model

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6 Another Example of LP: Diet Problem Energy requirement : 2000 kcal Protein requirement : 55 g Calcium requirement : 800 mg FoodEnergy (kcal)Protein(g)Calcium(mg)Price per serving($) Oatmeal110423 Chicken205321224 Eggs160135413 Milk16082859 Pie42042224 Pork260148013

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7 Example of LP : Diet Problem oatmeal: at most 4 servings/day chicken: at most 3 servings/day eggs: at most 2 servings/day milk: at most 8 servings/day pie:at most 2 servings/day pork: at most 2 servings/day Design an optimal diet plan which minimizes the cost per day

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8 Step 1: define decision variables x 1 = # of oatmeal servings x 2 = # of chicken servings x 3 = # of eggs servings x 4 = # of milk servings x 5 = # of pie servings x 6 = # of pork servings Step 2: formulate objective function In this case, minimize total cost minimize z = 3x 1 + 24x 2 + 13x 3 + 9x 4 + 24x 5 + 13x 6

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9 Step 3: Constraints Meet energy requirement 110x 1 + 205x 2 + 160x 3 + 160x 4 + 420x 5 + 260x 6 2000 Meet protein requirement 4x 1 + 32x 2 + 13x 3 + 8x 4 + 4x 5 + 14x 6 55 Meet calcium requirement 2x 1 + 12x 2 + 54x 3 + 285x 4 + 22x 5 + 80x 6 800 Restriction on number of servings 0 x 1 4, 0 x 2 3, 0 x 3 2, 0 x 4 8, 0 x 5 2, 0 x 6 2

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10 So, how does a LP look like? minimize 3x 1 + 24x 2 + 13x 3 + 9x 4 + 24x 5 + 13x 6 subject to 110x 1 + 205x 2 + 160x 3 + 160x 4 + 420x 5 + 260x 6 2000 4x 1 + 32x 2 + 13x 3 + 8x 4 + 4x 5 + 14x 6 55 2x 1 + 12x 2 + 54x 3 + 285x 4 + 22x 5 + 80x 6 800 0 x 1 4 0 x 2 3 0 x 3 2 0 x 4 8 0 x 5 2 0 x 6 2

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11 Optimal Solution – Diet Problem Using LINDO 6.1 Cost of diet = $96.50 per day Food# of servings Oatmeal4 Chicken0 Eggs0 Milk6.5 Pie0 Pork2

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12 Optimal Solution – Diet Problem Using Management Scientist Cost of diet = $96.50 per day Food# of servings Oatmeal4 Chicken0 Eggs0 Milk6.5 Pie0 Pork2

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13 Guidelines for Model Formulation Understand the problem thoroughly. Describe the objective. Describe each constraint. Define the decision variables. Write the objective in terms of the decision variables. Write the constraints in terms of the decision variables Do not forget non-negativity constraints

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14 A Transportation Table Warehouse 47 7 1 100 12 3 8 8 200 8 1016 5 150 450 8090120160 1234 1 2 3 Factory Factory 1 can supply 100 units per period Demand Warehouse B’s demand is 90 units per period Total demand per period Total supply capacity per period

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15 LP Formulation of Transportation Problem minimize 4x11+7x12+7x13+x14+12x21+3x22+8x23+8x24 +8x31+10x32+16x33+5x34 Subject to x11+x12+x13+x14=100 x21+x22+x23+x24=200 x31+x32+x33+x34=150 x11+x21+x31=80 x12+x22+x32=90 x13+x23+x33=120 x14+x24+x34=160 xij>=0, i=1,2,3; j=1,2,3,4 Supply constraint for factories Demand constraint of warehouses Minimize total cost of transportation

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16 Solution in Management Scientist Total transportation cost = 4(80) + 7(0) + 7(10)+ 1(10) + 12(0) + 3(90) + 8(110) + 8(0) + 8(0) +10(0) + 16(0) +5 (150) = $2300

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17 Solution using LINDO Notice multiple optimal solutions!

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18 Product Mix Problem Floataway Tours has $420,000 that can be used to purchase new rental boats for hire during the summer. The boats can be purchased from two different manufacturers. Floataway Tours would like to purchase at least 50 boats. They would also like to purchase the same number from Sleekboat as from Racer to maintain goodwill. At the same time, Floataway Tours wishes to have a total seating capacity of at least 200. Formulate this problem as a linear program

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19 Maximum Expected Daily Boat Builder Cost Seating Profit Speedhawk Sleekboat $6000 3 $ 70 Silverbird Sleekboat $7000 5 $ 80 Catman Racer $5000 2 $ 50 Classy Racer $9000 6 $110 Product Mix Problem

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20 Define the decision variables x 1 = number of Speedhawks ordered x 2 = number of Silverbirds ordered x 3 = number of Catmans ordered x 4 = number of Classys ordered Define the objective function Maximize total expected daily profit: Max: (Expected daily profit per unit) x (Number of units) Max: 70x 1 + 80x 2 + 50x 3 + 110x 4 Product Mix Problem

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21 Define the constraints (1) Spend no more than $420,000: 6000x 1 + 7000x 2 + 5000x 3 + 9000x 4 < 420,000 (2) Purchase at least 50 boats: x 1 + x 2 + x 3 + x 4 > 50 (3) Number of boats from Sleekboat equals number of boats from Racer: x 1 + x 2 = x 3 + x 4 or x 1 + x 2 - x 3 - x 4 = 0 (4) Capacity at least 200: 3x 1 + 5x 2 + 2x 3 + 6x 4 > 200 Nonnegativity of variables: x j > 0, for j = 1,2,3,4 Product Mix Problem

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22 Max 70x 1 + 80x 2 + 50x 3 + 110x 4 s.t. 6000x 1 + 7000x 2 + 5000x 3 + 9000x 4 < 420,000 x 1 + x 2 + x 3 + x 4 > 50 x 1 + x 2 - x 3 - x 4 = 0 3x 1 + 5x 2 + 2x 3 + 6x 4 > 200 x 1, x 2, x 3, x 4 > 0 Product Mix Problem - Complete Formulation Daily profit = $5040 Boat# purchased Speedhawk28 Silverbird0 Catman0 Classy28

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23 Marketing Application: Media Selection Advertising budget for first month = $30000 At least 10 TV commercials must be used At least 50000 customers must be reached Spend no more than $18000 on TV adverts Determine optimal media selection plan Advertising Media# of potential customers reached Cost ($) per advertisement Max times available per month Exposure Quality Units Day TV100015001565 Evening TV200030001090 Daily newspaper15004002540 Sunday newspaper25001000460 Radio3001003020

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24 Media Selection Formulation Step 1: Define decision variables DTV = # of day time TV adverts ETV = # of evening TV adverts DN = # of daily newspaper adverts SN = # of Sunday newspaper adverts R = # of radio adverts Step 2: Write the objective in terms of the decision variables Maximize 65DTV+90ETV+40DN+60SN+20R Step 3: Write the constraints in terms of the decision variables DTV<=15 ETV<=10 DN<=25 SN<=4 R 30 1500DTV+3000ETV+400DN+1000SN+100R<=30000 DTV+ETV>=10 1500DTV+3000ETV<=18000 1000DTV+2000ETV+1500DN+2500SN+300R>=50000 Budget Customers reached TV Constraints Availability of Media DTV, ETV, DN, SN, R >= 0 Exposure = 2370 units VariableValue DTV10 ETV0 DN25 SN2 R30

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25 Applications of LP Product mix planning Distribution networks Truck routing Staff scheduling Financial portfolios Capacity planning Media selection: marketing

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26 Possible Outcomes of a LP A LP is either Infeasible – there exists no solution which satisfies all constraints and optimizes the objective function or, Unbounded – increase/decrease objective function as much as you like without violating any constraint or, Has an Optimal Solution Optimal values of decision variables Optimal objective function value

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27 Infeasible LP – An Example minimize 4x11+7x12+7x13+x14+12x21+3x22+8x23+8x24+8x31+10x32+16 x33+5x34 Subject to x11+x12+x13+x14=100 x21+x22+x23+x24=200 x31+x32+x33+x34=150 x11+x21+x31=80 x12+x22+x32=90 x13+x23+x33=120 x14+x24+x34=170 xij>=0, i=1,2,3; j=1,2,3,4 Total demand exceeds total supply

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28 Unbounded LP – An Example maximize 2x 1 + x 2 subject to - x 1 + x 2 1 x 1 - 2x 2 2 x 1, x 2 0 x 2 can be increased indefinitely without violating any constraint => Objective function value can be increased indefinitely

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29 Multiple Optima – An Example maximize x 1 + 0.5 x 2 subject to 2x 1 + x 2 4 x 1 + 2x 2 3 x 1, x 2 0 x 1 = 2, x 2 = 0, objective function = 2 x 1 = 5/3, x 2 = 2/3, objective function = 2

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30 Operations Scheduling Chapter 16

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31 Establishing the timing of the use of equipment, facilities and human activities in an organization Effective scheduling can yield Cost savings Increases in productivity Scheduling

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32 High-Volume Systems Flow system: High-volume system with Standardized equipment and activities Flow-shop scheduling: Scheduling for high- volume flow system Work Center #1Work Center #2 Output

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33 High-Volume Success Factors Process and product design Preventive maintenance Rapid repair when breakdown occurs Optimal product mixes Minimization of quality problems Reliability and timing of supplies

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34 Scheduling Low-Volume Systems Loading - assignment of jobs to process centers Sequencing - determining the order in which jobs will be processed Job-shop scheduling Scheduling for low-volume systems with many variations in requirements

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35 Gantt Load Chart Gantt chart - used as a visual aid for loading and scheduling Figure 16.2

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36 More Gantt Charts

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37 Assignment Problem Objective: Assign n jobs/workers to n machines such that the total cost of assignment is minimized Special case of transportation problem When # of rows = # of columns in the transportation tableau All supply and demands =1 Plenty of practical applications Job shops Hospitals Airlines, etc.

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38 Cost Table for Assignment Problem 1234 1$1$4$6$3 2$9$7$10$9 3$4$5$11$7 4$8$7$8$5 Pilot (i) Aircraft (j) All assignment costs in thousands of $

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39 Management Scientist Solution PilotAssigned to aircraft # Cost (`000 $) 111 2310 325 445

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40 Formulation of Assignment Problem minimize x 11 +4x 12 +6x 13 +3x 14 + 9x 21 +7x 22 +10x 23 +9x 24 + 4x 31 +5x 32 +11x 33 +7x 34 + 8x 41 +7x 42 +8x 43 +5x 44 subject to x 11 +x 12 +x 13 +x 14 =1 x 21 +x 22 +x 23 +x 24 =1 x 31 +x 32 +x 33 +x 34 =1 x 41 +x 42 +x 43 +x 44 =1 x 11 +x 21 +x 31 +x 41 =1 x 12 +x 22 +x 32 +x 42 =1 x 13 +x 23 +x 33 +x 43 =1 x 14 +x 24 +x 34 +x 44 =1 x ij = 1, if pilot i is assigned to aircraft j, i=1,2,3,4; j=1,2,3,4 0 otherwise PilotAssigned to aircraft # Cost (`000 $) 111 2310 325 445 Optimal Solution: x 11 =1; x 23 =1; x 32 =1; x 44 =1; rest=0 Cost of assignment = 1+10+5+5=$21 (`000)

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41 Sequencing Sequencing: Determine the order in which jobs at a work center will be processed. Workstation: An area where one person works, usually with special equipment, on a specialized job. Priority rules: Simple heuristics used to select the order in which jobs will be processed. FCFS - first come, first served SPT - shortest processing time Minimizes mean flow time EDD - earliest due date In-class example

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42 Performance Measures Job flow time Length of time a job is at a particular workstation Includes actual processing time, waiting time, transportation time etc. Lateness = flow time – due date Tardiness = max {lateness, 0} Makespan Total time needed to complete a group of jobs Length of time between start of first job and completion of last job

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43 Scheduling Difficulties Variability in Setup times Processing times Interruptions Changes in the set of jobs No method for identifying optimal schedule Scheduling is not an exact science Ongoing task for a manager

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44 Minimizing Scheduling Difficulties Set realistic due dates Focus on bottleneck operations Consider lot splitting of large jobs

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