1. DECISION MODELING WITH MICROSOFT EXCEL Copyright 2001 Prentice Hall Publishers and Ardith E. Baker IntegerOptimization Chapter 6

2. Integer Linear Programming Integer Linear Programming (ILP) models are ___models in which some or all of the variables are required to assume ________ values. modeling ILP has become an important specialized area of _______________modeling. In previous chapters, __________solutions were allowed and often adapted to the integer requirement by rounding (called a ________solution). However, a number of important models require ______solutions. Introduction

3. conditions ___________________is a general term for optimization models with ______________ conditions. The different types of ILP models are described next. Types of Integer Optimization Models Integrality conditions stipulate that some or all of the ____________variables must have integer values.

4. Classifications of Integer Optimization Models All-integer linear program This is a model in which ______of the decision variables are required to be _______________. Mixed integer linear program (MILP) any _______________value This is a model in which __________of the decision variables are required to be integers and others can assume any _______________number (i.e., any _______________value).

5. 01 In this model, __________variables are restricted to the values 0 or 1. The models may be used to represent _________________decisions (yes/no decisions). Binary (or 0-1) integer linear program This LP model results when you start with an _____formulation but then ignore the integer _____________. LP relaxation of the ILP model

6. Graphical Interpretation Of Integer Models In this two variable ILP product mix model, let E and F represent the _____________of two products to make and sell for a______. Optimizing a Two Variable ILP Model Now, assume the represents certain minimum_________________.

7. s.t. E – 3F < 0(1) 42.8E + 100F < 800(2) 20E + 6F < 142(3) 30E + 10F > 135(4) E, F > 0 and integer Max 18E + 6F The symbolic model follows:

8. We will solve this model using the _______ approach following these three steps: 1. Find the ____________set for the LP ____________of the ILP model. 2. Identify the integer points inside the set determined in step 1. 3. Find, among those points determined in step 2, one that ______________the ____________function. We will use the GLP program (covered in Chapter 4) and Excel’s Solver to obtain the solution to the LP relaxation of the ILP.

9. Here are the GLP results: The shaded region is the ________set for the LP relaxation. The blue dots are the _________ points contained within the LP feasible region. (3,6)(4,6)(3,5)(4,5)(5,5)(4,4)(5,4)(4,2)(5,2)(6,2)(4,3)(6,3)(5,3)

10. Here are the Solver results and parameters:

11. To ________the model, determine which of the ILP feasible points yields the _____ value of the objective function. To do this, drag the objective function line in an _____direction until it is not possible to move it farther and still _______ an integer feasible point.

12. Here are the Solver results and parameters:

13. Nearest rounded solution (_____) Optimal solution to LP relaxation (_________) Only feasible rounded integer solution (____) Optimal solution to ILP (___)

14. upper bound 1. In a _______model the OV (optimal value) of the LP ____________always provides an upper bound on the OV of the original ILP. Adding the integer constraints either ________or leaves unchanged the OV for the LP. In a Max model, hurting the OV means making it _______. Comments Graphical Interpretation Of Integer Models

15. bound 2. In a ______model, the OV of the LP relaxation always provides a ________ bound on the OV of the original ILP. Again, adding the integer constraints either hurts or leaves ___________the OV for the LP. In a Min model, hurting the OV means making it___________.

16. Graphical Interpretation Of Integer Models The optimal solution to the LP relaxation is E*=5.39 and F*=5.69. Rounded Solutions Since each of these variables can be _________either up or down, there are four (2 2 ) rounded solutions: (5,5)(5,6) (6,5)(6,6) With n _________variables, there would 2 n rounded points.

17. Of all the rounded points, (5,5) is the only _____________point. All other points are infeasible. Two important facts about rounded solutions: 1. A rounded solution need not be ______________. 2. A rounded solution need not be near the optimal ________solution.

18. Optimal solution to ILP Optimal solution to the LP relaxation. Feasible set for LP relaxation Integer points

19. Recall that Solver’s LP __________method makes use of the fact that the solution to an LP always lies on the ___________of the feasible region. Enumeration Graphical Interpretation Of Integer Models Thus, Solver never has to consider any ___________points of the feasible region. For an ILP optimization, Solver might have to visit many ________points strictly within the interior of the feasible region, and thus, the LP simplex method cannot be used.

20. complete ___________ Once you identify all the integer feasible _________in ILP, you could solve the model by complete ___________(i.e., evaluate the objective function at each integer point and then select the ________one). Unfortunately, complete enumeration of all the feasible integer points is not a ___________procedure for most ILPs. For example, if you had 20 decision variables, each of which could take on an integer value between 1 and 50, then there would be 50 20 (9.5x10 23 ) points to enumerate and test for_______________.

21. Applications of Binary Variables _______ (0-1) variables make it possible to incorporate yes/no decisions (called __________decisions) into an optimization model. For example: 1. In a plant _______model, let x j = 1 if we choose to have a plant at location j and x j = 0 if we do not. 2. In a __________model, let x ijk = 1 if truck k goes from city i to city j and x ijk = 0 if it does not.

22. The use of 0-1 variables allows many variations of __________conditions in an optimization model without resorting to any of Excel’s “=IF()” statements (which would not allow Solver to __________the model). Applications of Binary Variables

23. The capital budgeting decision is a matter of choosing among n ____________in order to maximize the return, subject to constraints on the amount of ______invested over time. Capital Budgeting: An Expansion Decision Applications of Binary Variables For example, suppose that AutoPower’s board of directors faces the following data:

24. Expand Belgian Plant 400 100 50 200 100 0 Expand Sm. Machine Capacity in US 700 300 200 100 100 100 Establish New Plant in Chile 800 100 200 270 200 100 Expand Lg. Machine Capacity in US 1000 200 100 400 200 200 Capital Available 500 450 700 400 300 ALTERNATIVE PRESENT VALUE OF NET RETURN ($000s) CAPTIAL REQUIRED IN YEAR BY ALTERNATIVE ($000s) 1 2 3 4 5 The Board must select one or more of these alternatives:

25. An ILP Model for Capital Budgeting at AutoPower: 0-1 ILP An ILP Model for Capital Budgeting at AutoPower: This model can be modeled as an ILP (called a ______or 0-1 ILP model) in which all the variables are binary variables. Let x i = 1 if project i is accepted and x i = 0 if project i is not___________. The model becomes:

26. Max 400x 1 + 700x 2 + 800x 3 + 1000x 4 s.t. 100x 1 + 300x 2 + 100x 3 + 200x 4 < 500 50x 1 + 200x 2 + 200x 3 + 100x 4 < 450 200x 1 + 100x 2 + 270x 3 + 400x 4 < 700 100x 1 + 100x 2 + 200x 3 + 200x 4 < 400 100x 2 + 100x 3 + 200x 4 < 300 x i = 0 or 1; i = 1, 2, 3, 4

27. The LP Relaxation: The LP Relaxation: First, approach this model by solving the LP_______________.

28. Note that your solution may differ because the model has _________optimal solutions. In the LP relaxation model, x i was ____________to be less than or equal to 1. This resulted in _________values (0<x i <1) for the decision variables. Since we are looking for 1’s (yes) and 0’s (no), these fractional values are not ___________. In addition, rounding these values would not work very well. Rounding to: x 1 = 1, x 2 = 1, x 3 = 0, x 4 = 1 would result in an ____________solution.

29. The optimal ILP solution can be obtained using Solver’s __________________option.

30. The __________Edition Solver for Education produces a different Solver Results completion ________for ILPs to remind you that the ILP solution may not be optimal.

31. The default ___________field on the Solver Options dialog (relevant only for ILP models) is 5%. This means that the Solver ILP ____________ procedure is continued only until the ILP solution OV is within 5% of the ILP’s optimum_______. A _______Tolerance speeds up Solver at the risk of a reported solution further from the true ILP____________. Setting Tolerance to 0% forces Solver to find the ILP optimum but with much longer ____________times.

32. An important use of _______variables is to impose constraints that arise from _______ conditions. Logical Conditions Applications of Binary Variables No More Than k of n Alternatives: Let x i = 0 or 1, for i = 1, 2, …, n The constraint x 1 + x 2 + … + x n < k implies that, at most, k ___________of n possibilities can be selected (i.e., not more than k x i can equal 1).

33. For example, adding the _____________: x 1 + x 3 < 1 to the previous AutoPower example, implies that the solution can contain _______one of the overseas alternatives. Dependent Decisions: Dependent Decisions: You can use 0-1 variables to force a __________relationship on two or more decisions.

34. For example, suppose that AutoPower’s management does not want to select alternative k unless it first selects alternative m. The following constraint enforces this condition: If alternative m is not selected, then x m = 0 which forces x k to be 0 (i.e., not selected). x k < x m or equivalently x k - x m < 0 If alternative m is selected, then x m = 1 and x k < 1. Solver is then free to select 0 or 1 for x k.

35. Lot Size Constraints: Lot Size Constraints: A portfolio manager is considering purchasing security j. Let x j be the number of shares purchased. Consider the following constraints: batch size 1. If he purchases security j, he must purchase at least 200 shares (called a ____________________or batch size constraint) 2. He may not purchase more than 1000 __________of security j. 200 < x j < 1000 These constraints insist that x j always be at least______. We want the conditions to be either x j = 0 or 200 < x j < 1000.

36. To achieve the constraint conditions, use a _________variable, say y j, for security j. If y j = 1, then purchase security j The constraints are: x j < 1000y j x j > 200y j If y =___, then the above constraints imply that 200 < x j < 1000 (purchase j). If y =___, then the above constraints imply that x j = 0 (do not purchase j). If y j = 0, do not purchase security j These 2 constraints together ________ the “minimum lot size” constraint.

37. K of m Constraints: K of m Constraints: In general notation, let the “_________” of m constraints on a model’s (non-binary) decision variables, x i, be g i (x 1, …, x n ) < b i, for i = 1, …, m Now, introduce m ___________0-1 decision variables y i to the model, and let M be chosen as a very large number, so large that, for each i, g i (x i, …, x n ) < M for every x satisfying any set of k _____________taken from the above m.

38. The following m + 1 constraints express the desired condition: m i=1 y i = k i  This constraint forces ______k of the new y i decision variables to have the value____. g i (x 1, …, x n ) < b i y i + M(1- y i ), i=1, …, m Exactly k of the __________constraints are equivalent to: g i (x 1, …, x n ) < b i The remaining m-k inequality constraints are _________to: g i (x 1, …, x n ) < M The very large number choice for M causes each such constraint to be ________and not affect the model’s optimal_____________.

39. K of m Constraints Example: K of m Constraints Example: Assume a company must find production quantities of three products (x 1, x 2, x 3 ) as part of a large LP model. Within the LP formulation, the company must choose one or the other (but not both) of two different production technologies for the 3 products. Here are the two constraints: 30x 1 + 20x 2 + 10x 3 < 100 (Technology 1) 10x 1 + 30x 2 + 5x 3 < 110 (Technology 2)

40. These constraints cannot be added _______ to the LP model. Instead, add two new ______decision variables (y 1 and y 2 ) to the LP model, making it an ILP model. y 1 = 1 “Solver, choose Technology 1” y 1 = 0 “Solver, do not choose Technology 1” y 2 = 1 “Solver, choose Technology 2” y 2 = 0 “Solver, do not choose Technology 2”

41. The original constraints are then modified: 30x 1 + 20x 2 + 10x 3 < 100y 1 + 999999(1-y 1 ) 10x 1 + 30x 2 + 5x 3 < 110y 2 + 999999(1-y 2 ) y 1 + y 2 = 1 Forces Solver to choose exactly one technology option 999999 was chosen to guarantee that one or the other constraint will be redundant for the whole LP model

42. A Fixed Charge Model In order to __________capital, STECO, an electronics parts wholesaler, leases regional warehouses for its use. The cost per month to lease warehouse i is F i. Warehouse i can load a __________of T i trucks per month. There are four sales districts, and the typical monthly __________in district j is d j truckloads. The average cost of sending a truck from _____________i to district j is c ij.

43. STECO wants to know which warehouses to ______and how many trucks to send from each warehouse to each__________. STECO pays no _________cost for a given warehouse unless at least one truck is ___________from it, and then it pays the full monthly lease amount. Lot size models incorporating this cost behavior are common and are called __________________________.

44. Here is the network flow diagram:ABC 1243 Leasing cost/mth for warehouses F A F B F C Capacity (truckloads) T A T B T C Warehouses Districts Demands per month d 1 d 2 d 3 d 4

45. Here are the data for this model: A 170 40 70 160 200 7750 B 150 195 100 10 250 4000 C 100 240 140 60 300 5500 Monthly Demand (truck loads) 100 90 110 60 WAREHOUSE Monthly Capacity (No. of Trucks) Cost Per Truck Sales District ($) 1 2 3 4 Monthly Leasing Costs ($)

46. A Fixed Charge Model Modeling Considerations: Define y i as a binary ___________variable and let: y i = 1 if warehouse i ____leased y i = 0 if warehouse i ______leased Although it may seem ________to treat the number of trucks sent from a warehouse to a district as an ________variable, in reality, there are three arguments as to why we should not:

47. 1. This is a ___________model, not a detailed operating model. 2. Treating the number of trucks as integer variables may make the model more difficult to_____________. 3. It costs _______to lease one of the warehouses than to send a truck from a warehouse to a ________district.

48. A Fixed Charge Model The MILP Model To model STECO’s model as an MILP, let y i = 1 if warehouse i is leased, i=A, B, C y i = 0 if warehouse i is not leased x ij = number of trucks sent from warehouse i to district j i=A, B, C; j=1, 2, 3, 4

49. Min 7750y A + 4000y B + 5500y C + 170x A1 + … + 60x C4 x A1 + x B1 + x C1 > 100demand at district 1 Demand Constraints: x A2 + x B2 + x C2 > 90demand at district 2 x A3 + x B3 + x C3 > 110demand at district 3 x A4 + x B4 + x C4 > 60demand at district 4 These four constraints guarantee that demand will be _________at the respective sales district. Objective Function:

50. x A1 + x A2 + x A3 + x A4 < 200y A or Capacity Constraints (for each warehouse): These ____________serve 2 purposes: x A1 + x A2 + x A3 + x A4 - 200y A < 0 1. It guarantees that capacity at warehouse i is not____________. 2. It forces STECO to _____warehouse i if anything is sent out of it. x B1 + x B2 + x B3 + x B4 < 250y B x C1 + x C2 + x C3 + x C4 < 300y C

51. Here is the Solved spreadsheet model:

52. Here are the Solver parameters:

53. In conclusion, the __________solution to this model with integer supplies and demands will always include an _________ allocation of trucks. A Fixed Charge Model The argument involves two steps: 1. The _________solution must lease some set of warehouses, and 2. Every possible set of leased warehouses yields an integer ______________of trucks.

54. Integer Optimization Methods Solver’s solution procedure for an ILP model (called________________) is as follows for a maximization model: 1. Solve the _______ILP formulation as a relaxed LP. The OV for the relaxation is the ILP ______________. If the optimal solution is_________, it is optimal for the ILP, and so, quit.

55. 2. If the LP relaxation has some _______ variable at a fractional value, form two sub models from this parent ______so as to create two new unsolved LP sub- models (the_____________) with the property that the optimal solution to one of the successor ILPs will be the optimal solution to the _________ILP. Let x i * be any ________variable of the optimal solution to the parent’s relaxation. Let [x i * ] be the __________of x i * to its integer part.

56. Then, [x i * ] + 1 is the next integer _________than x i *. One successor ___________will be the parent’s LP model augmented by the ____________, x i < [x i * ]. The other successor sub-model is formed by _____________the parent’s LP model with x i > [x i * ] + 1.

57. 3. Commence with any ___________sub- model in step 2. And optimize it as a _________LP sub-model. If the optimal solution is all-integer, ___________its objective function. Compare the _____of the best ILP model’s solution found so far with this relaxed sub-model’s OV. If the relaxed sub-model’s OV is _____ than the best ILP solution found so far, don’t_________. Instead, continue with another____________.

58. If the relaxed sub-model’s OV is not worse than the best ILP _______found so far, then proceed to eliminate any other ________variables it may have using the constraint augmentation procedure by going back to step_____. If all remaining relaxed sub-models have integer solutions, go to step 4; otherwise, go back to step 2. 4. The optimal solution to the _________ ILP is the all-integer solution of some sub-model that ____________the best value of the OV found so far.

59. In the branch-and-bound method, the original ILP is ____________into a growing sequence of LP sub-models. Solver uses ___________optimizations of increasingly augmented LP sub-model _______________to solve a given ILP. Therefore, it is more time-consuming to optimize ILPs than_______. The operation of the branch-and-bound procedure in Solver displays the following message in Excel’s lower left corner: Branch: 1422 Trial Solution: 11 Set Cell: $39,366

60. Integer Optimization Methods Sensitivity Analysis for ILPs Sensitivity The Solver solution to an ILP does not contain any ____________information (as evidenced by the lack of a Sensitivity option in the ___________________dialog). An ILP solution does not include _________ that is equivalent to the_______________, reduced cost and objective coefficient sensitivity information in an LP.

61. Solutions to ILPs can be extremely _______ to changes in ___________values. To illustrate, consider the following capital budgeting model: Although this model is easily solved by ____________, consider the following optimal solution: Max 10x 1 + 100x 2 + 1000x 3 29x 1 + 30x 2 + 31x 3 < b 1 x 1, x 2, x 3 are binary (0 or 1) s.t.

62. x 1 x 2 x 3 29 1 0 0 10 30 0 1 0 100 31 0 0 1 1000 b1b1 Optimal Solution OV Here are the sensitivity data for the model: A change in the ______________side of the constraint increases the OV by a factor of 1000%.

63. Integer Optimization Methods Heuristic Methods: ___________methods are designed to efficiently produce “_______,” although not necessarily optimal, solutions. These methods will be discussed in a the next chapter.