Lecture 2 & 3

1. Lecture 2 & 3 Linear Programming and Transportation Problem

2. Linear Programming • 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

3. Variables Objective function Constraints Non-negativity Constraints Linear Programming Example Maximize 60X1 + 50X2 Subject to 4X1 + 10X2 <= 100 2X1 + 1X2 <= 22 3X1 + 3X2 <= 39 X1, X2 >= 0 • 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)

4. Linear Programming Model • 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

5. Another Example of LP: Diet Problem • Energy requirement : 2000 kcal • Protein requirement : 55 g • Calcium requirement : 800 mg

6. 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

7. Step 1: define decision variables • x1 = # of oatmeal servings • x2 = # of chicken servings • x3 = # of eggs servings • x4 = # of milk servings • x5 = # of pie servings • x6 = # of pork servings Step 2: formulate objective function • In this case, minimize total cost • minimize z = 3x1 + 24x2 + 13x3 + 9x4 + 24x5 + 13x6

8. Step 3: Constraints • Meet energy requirement 110x1 + 205x2 + 160x3 + 160x4 + 420x5 + 260x6 2000 • Meet protein requirement 4x1 + 32x2 + 13x3 + 8x4 + 4x5 + 14x6  55 • Meet calcium requirement 2x1 + 12x2 + 54x3 + 285x4 + 22x5 + 80x6  800 • Restriction on number of servings 0x14, 0x23, 0x32, 0x48, 0x52, 0x62

9. So, how does a LP look like? minimize 3x1 + 24x2 + 13x3 + 9x4 + 24x5 + 13x6 subject to 110x1 + 205x2 + 160x3 + 160x4 + 420x5 + 260x6 2000 4x1 + 32x2 + 13x3 + 8x4 + 4x5 + 14x6  55 2x1 + 12x2 + 54x3 + 285x4 + 22x5 + 80x6  800 0x14 0x23 0x32 0x48 0x52 0x62

10. 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

11. Transportation Problem • Objective: • determination of a transportation plan of a single commodity • from a number of sources • to a number of destinations, • such that total cost of transportation is minimized • Sources may be plants, destinations may be warehouses • Question: • how many units to transport • from source i • to destination j • such that supply and demand constraints are met, and • total transportation cost is minimized

12. Total supply capacity per period Warehouse B’s demand is 90 units per period Total demand per period A Transportation Table Table 8S.1 Warehouse 1 2 3 4 Factory 7 4 7 1 Factory 1 can supply 100 units per period 100 1 3 8 8 12 200 2 10 16 8 5 150 3 450 80 90 120 160 Demand 450

13. Minimize total cost of transportation 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

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

15. Cost Table for Assignment Problem Machine (j) Worker (i)

16. LP Formulation of Assignment Problem • minimize x11+4x12+6x13+3x14 + 9x21+7x22+10x23+9x24 + 4x31+5x32+11x33+7x34 + 8x41+7x42+8x43+5x44 subject to • x11+x12+x13+x14=1 • x21+x22+x23+x24=1 • x31+x32+x33+x34=1 • x41+x42+x43+x44=1 • x11+x21+x31+x41=1 • x12+x22+x32+x42=1 • x13+x23+x33+x43=1 • x14+x24+x34+x44=1 • xij = 1, if worker i is assigned to machine j, i=1,2,3,4; j=1,2,3,4 0 otherwise

17. 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

18. Product Mix Problem 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

19. Product Mix Problem • Define the decision variables x1 = number of Speedhawks ordered x2 = number of Silverbirds ordered x3 = number of Catmans ordered x4 = number of Classys ordered • Define the objective function Maximize total expected daily profit: Max: (Expected daily profit per unit) x (Number of units) Max: 70x1 + 80x2 + 50x3 + 110x4

20. Product Mix Problem • Define the constraints (1) Spend no more than $420,000: 6000x1 + 7000x2 + 5000x3 + 9000x4< 420,000 (2) Purchase at least 50 boats: x1 + x2 + x3 + x4> 50 (3) Number of boats from Sleekboat equals number of boats from Racer: x1 + x2 = x3 + x4 or x1 + x2 - x3 - x4 = 0 (4) Capacity at least 200: 3x1 + 5x2 + 2x3 + 6x4> 200 Nonnegativity of variables: xj> 0, for j = 1,2,3,4

21. Product Mix Problem - Complete Formulation Max 70x1 + 80x2 + 50x3 + 110x4 s.t. 6000x1 + 7000x2 + 5000x3 + 9000x4< 420,000 x1 + x2 + x3 + x4> 50 x1 + x2 - x3 - x4 = 0 3x1 + 5x2 + 2x3 + 6x4> 200 x1, x2, x3, x4> 0

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

23. Graphical Solution of LPs • Consider a Maximization Problem Max 5x1 + 7x2 s.t. x1< 6 2x1 + 3x2< 19 x1 + x2< 8 x1, x2> 0

24. Graphical Solution Example • Constraint #1 Graphed x2 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 x1< 6 (6, 0) x1

25. Graphical Solution Example • Constraint #2 Graphed x2 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 (0, 6 1/3) 2x1 + 3x2< 19 (9 1/2, 0) x1

26. Graphical Solution Example • Constraint #3 Graphed x2 (0, 8) 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 x1 + x2< 8 (8, 0) x1

27. Graphical Solution Example • Combined-Constraint Graph x2 x1 + x2< 8 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 x1< 6 2x1 + 3x2< 19 x1

28. Graphical Solution Example • Feasible Solution Region x2 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 Feasible Region x1

29. Graphical Solution Example • Objective Function Line x2 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 (0, 5) Objective Function 5x1 + 7x2 = 35 (7, 0) x1

30. Graphical Solution Example • Optimal Solution x2 Objective Function 5x1 + 7x2 = 46 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 Optimal Solution (x1 = 5, x2 = 3) x1

31. Graphical Linear Programming • Set up objective function and constraints in mathematical format • Plot the constraints • Identify the feasible solution space • Plot the objective function • Determine the optimum solution

32. 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

33. Infeasible LP – An Example • 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=170 • xij>=0, i=1,2,3; j=1,2,3,4 Total demand exceeds total supply

34. Unbounded LP – An Example maximize 2x1 + x2 subject to -x1 + x2  1 x1 - 2x2  2 x1 , x2  0 x2 can be increased indefinitely without violating any constraint => Objective function value can be increased indefinitely

35. Multiple Optima – An Example maximize x1 + 0.5 x2 subject to 2x1 + x2  4 x1 + 2x2  3 x1 , x2  0 • x1= 2, x2=0, objective function = 2 • x1= 5/3, x2=2/3, objective function = 2

36. Practice Example