1 Integer Programming Each year Cross Chek decides

1 Integer Programming • Each year Cross. Chek decides which lines of golf clubs and clothing it will market. Consider that each line of golf clubs is expected to generate $400 K profit for the year, while each new line of golf clothing generates $300 K profit for the year. Each new offering requires time for marketing (clubs 50 hrs, clothing 60 hrs), which is limited to 300 hrs, as well as storage space (clubs 900 sq ft, clothes 400 sq ft), which is limited to 3600 sq ft. Cross. Chek will not consider marketing any more than 4 lines of clothing. How many new lines of golf clubs and clothing should be offered?

2 Integer Programming • Solution: • 2. 82 lines of golf clubs • 2. 65 lines of golf clothing • Profit: • $1, 923, 529

3 Integer Programming • Solution: • 2. 82 lines of golf clubs • 2. 65 lines of golf clothing • Could round down: • 2 lines of golf clubs • 2 lines of golf clothing • Profit = $1. 4 million • Feasible. But optimal?

4 Integer Programming in LINDO Max 400 x 1 + 300 x 2 st 50 x 1 + 60 x 2 < 300 900 x 1 + 400 x 2 < 3600 x 2 < 4 x 1 > 0 x 2 > 0 END GIN x 1 GIN x 2

5 ILP Notes • Removing the integer specifications for an integer linear programming problem gives the LP relaxation • The value of an LP relaxation for a maximization ILP is an upper bound on the value of the ILP • Meaning that the value of the objective function for the optimal integer solution is no higher than that for the LP relaxation • For a minimizing ILP, the LP relaxation gives a lower bound

6 ILP Notes • An integer linear programming problem in which all variables must be integers is referred to as an All Integer Linear Program (AILP) • An integer linear programming problem in which some variables must be integers is referred to as a Mixed Integer Linear Program (MILP)

7 ILP with 0 -1 Variables • Cross. Chek is considering expanding its retail outlets. Possible locations for such outlets are Toronto, Vancouver, Montreal and Fredericton. A maximum of one outlet would exist in each city. Startup costs, advertising and marketing costs, and time required to get each outlet in operation are as follows, as well as expected first-year profits (all dollar values in thousands; all times in number of days). The objective is to maximize first-year profits. Into which cities should Cross. Chek expand? City Start-up costs Time Advertising costs First-year profits Toronto 100 30 20 50 Vancouver 80 40 80 45 Montreal 90 35 25 35 Fredericton 70 25 20 5 Limit 260 110 100

8 Defining 0 -1 Variables in LINDO Max 50 x 1 + 45 x 2 + 35 x 3 + 5 x 4 st 100 x 1 + 80 x 2 + 90 x 3 + 70 x 4 < 260 30 x 1 + 40 x 2 + 35 x 3 + 25 x 4 < 110 20 x 1 + 80 x 2 + 25 x 3 + 20 x 4 < 100 END INT x 1 INT x 2 INT x 3 INT x 4

9 k-out-of-n Alternatives Constraints • Considering the same problem, add each of the following constraints: • Cross. Chek would like to ensure that: a) Exactly one outlet is added b) At least one outlet goes in Montreal or Fredericton c) Exactly three outlets are added d) Two of Montreal, Fredericton and Vancouver are added e) One of Toronto or Vancouver are added but not both

10 Conditional Constraints • Considering the same problem, add each of the following constraints: • Cross. Chek would like to ensure that: a) If an outlet in Toronto is added, then an outlet in Fredericton is added b) If an outlet in Fredericton is not added, then one in Vancouver must be added c) If an outlet in Toronto is added then we cannot add one in Vancouver d) If an outlet in Vancouver is added, then at least one outlet in Montreal or Fredericton must be added

11 Multiple Choice Constraints • Considering the same problem, consider that there are three possible sites for the Vancouver option: downtown, midtown, uptown. We’ve decided to develop in Vancouver for sure. How do we model this choice as a constraint? • Let v 1 be equal to 1 if the downtown location is chosen • Let v 2 be equal to 1 if the midtown location is chosen • Let v 3 be equal to 1 if the uptown location is chosen • What if development in Vancouver is not a certainty?

12 Fixed Cost Problems • Returning to the Cross. Chek golf clubs/clothing strategy example. Suppose there is a fixed cost for licensing fees associated with venturing into the golf clothing market of $200 K. That is, it will cost an extra $200 K if it chooses to sell any golf clothing. What is the optimal course of action? • What if the fixed cost were $100 K?

13 Solution Max 400 x 1 + 300 x 2 - 200 d st 50 x 1 + 60 x 2 < 300 900 x 1 + 400 x 2 < 3600 x 2 - 4 d < 0 x 1 > 0 x 2 > 0 END GIN x 1 GIN x 2 INT d

14 Fixed Cost Problems • Returning to the original Cross. Chek golf clubs/clothing strategy example. Suppose there are two different marketing companies from which to choose. Marketing company 1 costs $70 K, requires 50 hrs for each line of clubs and 60 hrs for each line of clothing, which is limited to 300 hrs, and marketing company 2 costs $80 K, requires 40 hrs for each line of clubs and 45 hrs for each line of clothing, which is limited to 270 hrs. Cross. Check will go with either one marketing company or the other. What is the optimal course of action?

15 Solution Max 400 x 1 + 300 x 2 - 70 d 1 - 80 d 2 st 50 x 1 + 60 x 2 + 1000000 d 1 < 1000300 40 x 1 + 45 x 2 + 1000000 d 2 < 1000270 900 x 1 + 400 x 2 < 3600 d 1 + d 2 = 1 x 2 < 4 x 1 > 0 x 2 > 0 END GIN x 1 GIN x 2 INT d 1 INT d 2