Decision Modeling

1. Decision Modeling Integer Programming

2. Introduction • Integer linear programming (ILP) models are mathematical models in which some or all of the variables are required to assume integer values • A few different types exist • All integer linear programs • Mixed integer linear programs (some decision variables are allowed to be continuous) • Binary (0-1) integer linear programs (decision variables typically represent go ― no go decisions)

3. MILP Properties • Optimal solution need not be a corner point of the feasible area defined by the constraints • Optimal solution is always inferior compared to the linear program with the same constraint set and objective function • Rounded LP solution may not be near the optimal MILP solution

4. Max 18E + 6F Example • See GLP demo s.t. E – 3F < 0 (1) 42.8E + 100F < 800 (2) 20E + 6F < 142 (3) 30E + 10F > 135 (4) E, F > 0 and integer

5. Use of Binary Variables • Binary variables make it possible to include yes/no decisions into an optimization model • In a plant location model, let xj=1 if we choose to build a plant in location j and xj=0, otherwise • In a routing model, let xijk=1 if truck k goes from city i to city j and xijk=0, otherwise • The use of binary variables eliminates the need for Excel’s IF() statement, which makes it impossible to solve the model (IF() is a non-linear function!)

6. Example • Capital budgeting at AutoPower: An expansion problem • AutoPower considers the opportunity to expand its operations in several locations and has four investment alternatives to consider • The company would like to maximize expected return subject to capital available for investment over a five-year horizon • The following data are available:

7. CAPTIAL REQUIRED IN YEAR BY ALTERNATIVE ($000s) ALTERNATIVE PRESENT VALUE OF NET RETURN ($000s) 1 2 3 4 5 Expand Belgian Plant 400 100 50 200 100 0 Expand Sm. MachineCapacity in US 700 300 200 100 100 100 Establish New Plant in Chile 800 100 200 270 200 100 Expand Lg. MachineCapacity in US 1000 200 100 400 200 200 Capital Available 500 450 700 400 300 Example: AutoPower Expansion

8. Example: AutoPower Expansion • See board • See Excel solution

9. Use of Binary Variables • Logical conditions:No more than k of n alternatives • Let xi=0 or 1 for i=1,2,…,n • The constraint x1+x2+…+xn ≤ kimplies that at most k alternatives of n possibilities can be selected (not more than k of the xi variables can equal 1) • Adding x1+x3 ≤ 1 in the AutoPower example forces management to choose at most one of the overseas projects

10. Use of Binary Variables • Dependent decisions:i can be chosen only if j is chosen • Let xi=0 or 1 for i=1,2,…,n • The constraint xi ≤ xjimplies that if j is not chosen, i cannot be chosen either (if xj=0, then xi=0)However, if xj =1, then xi can be either 0 or 1 • Adding x4 ≤ x2 in the AutoPower example forces management to invest in the small U.S. machine capacity before considering the large U.S. machine capacity

11. Use of Binary Variables • Lot size requirements:Occasionally, you must meet both a minimum and a maximum requirement if you choose to undertake a particular activity • Let xi (continuous) for i=1,2,…,n represent your quantity decision • If you decide to undertake activity i, thenxi≥a and xi≤b • Introduce yi (binary) representing go/no go:xi≥ayi and xi≤byi are linear constraints

12. Example • A portfolio manager considers the purchase of n different stocks. For each stock, there is a minimum lot size, as well as a maximum investment. If the portfolio manager decides to buy stock i, she must buy at least 200 shares and at most 1,000 shares. The following constraints represent the possibilities: xi≥200yixi≤1000yi xi≥0 yi=0 or 1

13. Use of Binary Variables • k of m constraints:Suppose you have the choice of m different technologies. Each choice results in different technical (LHS) coefficients. Based on your choice(s) of technology, you must select the correct set(s) of constraints • Let xi (continuous) for i=1,2,…,n represent your quantity decision • If you choose technology i, thenai1x1+ai2x2+…+ainxn≤bi • Introduce yi (binary) for technology selection:ai1x1+ai2x2+…+ainxn≤bi+M(1-yi)y1+y2+…+ym ≤ k (or = k or ≥ k)where M is a very large number

14. Example • Assume a company must find production quantities of three products (x1,x2,x3) as part of a large MILP model • Also, the company must choose one or the other (but not both) of two different production technologies for the three products. The constraints are:30x1+20x2+10x3≤100 (technology 1)10x1+30x2+ 5x3≤110 (technology 2)

15. Example • Let y1=0 or 1 and y2=0 or 1 represent the choice of technology 1 and 2, respectively • The resulting constraints are:y1+y2=130x1+20x2+10x3≤100+M(1-y1)10x1+30x2+ 5x3≤110+M(1-y2)

16. Warehouse Selection Example • To conserve capital, STECO, an electronics parts wholesaler, leases regional warehouses for its use • The cost per month to lease warehouse i is Fi. Warehouse i can load a maximum of Ti trucks per month • There are four sales districts, and the typical monthly demand in district j is dj truckloads • The average cost of sending a truck from warehouse i to district j is cij

17. Warehouse Selection Example • STECO wants to know which warehouses to lease and how many trucks to send from each warehouse to each district • STECO pays no lease cost for a given warehouse unless at least one truck is sent from it. If STECO uses the warehouse at all, the company pays the full monthly lease amount • Models incorporating this type of lease charges are commonly called fixed charge models

18. Cost Per TruckSales District ($) Monthly Capacity (No. of Trucks) Monthly Leasing Costs ($) WAREHOUSE 1 2 3 4 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 Selection Example

19. Warehouse Selection Example • See board • See Excel solution

20. Max 10x1 + 100x2 + 1000x3 s.t. 29x1 + 30x2 + 31x3< b1 x1, x2, x3 are binary (0 or 1) Sensitivity Analysis • Solutions to MILPs can be extremely sensitive to changes in coefficient values. Therefore, Solver does not provide such information. Consider the following ILP for different RHS values. What is the meaning of shadow prices?