ISyE 3133B Sample Final Tests

Transcription

1 ISyE 3133B Sample Final Tests Time: 160 minutes, 100 Points Set A Problem 1 (20 pts). Head & Boulders (H&B) produces two different types of shampoos A and B by mixing 3 raw materials (R1, R2, and R3). The amount of each raw material per pound of each shampoo type is given below. R1 R2 R3 A B H&B is trying decide its shampoo production and raw material procurement plan for the next 4 quarters so as to minimize total production, procurement and inventory holding costs. The shampoos and raw materials can be held in inventory. All initial and final inventories are zero. The costs in each quarter are as follows A B R1 R2 R3 Production/Procument cost (c / lb) Inventory holding cost (c / lb) The demands (in lbs) of shampoos in each quarter are as follows A B H&B has a maximum total production capacity of 500 lbs, and its total inventory capacity (includes shampoos as well as raw material) is 1000 lbs. Formulate a linear program to help H&B decide its shampoo production and raw material procurement plan for the next 4 quarters so as to minimize total production, procurement and inventory holding costs, while satisfying demand, and obeying capacity limits. Problem 2 (20 pts). An auto manufacturer is considering manufacturing three types of cars: compact, midsize and large. The resources required and profits (per car) for each car type is shown below. Compact Midsize Large Steel required (tons) Labor required (hours) Profit ($) Currently, 6000 tons of steel and 60,000 hours of labor are available. The following additional restrictions are specified. (i) If the company decides to produce compact cars then it must produce at least 1000 compact cars. (ii) If the company decides to produce midsize cars then it must produce at least 800 midsize cars. (iii) If the company decides to produce large cars then it can produce at most a total of 1200 compact and midsize cars. Formulate an integer linear program to maximize the companies profits, while satisfies resource limitations, and the above restrictions. The number of cars of each type produced should be integer valued. If you 1

2 use big-m s in your model provide good numerical values for these. Problem 3 (10 pts). The following questions relate to a mixed integer linear program P in minimization form. (a) The optimal objective of P is v 1, the optimal objective value of the LP relaxation of P is v 2, and the optimal objective value corresponding to an integer feasible solution obtained by a heuristic is v 3. Write down the inequality relationship between v 1, v 2, v 3. (b) After prematurely terminating the branch-and-bound algorithm on P, following is the list of terminal nodes along with their status: Node label LP objective value Status n 1 10 Fractional solution. n 2 + Infeasible. n 3 19 Integer feasible solution. n 4 11 Fractional solution. n 5 13 Integer feasible solution. n 6 15 Fractional solution. Which nodes should be further explored and which are fathomed? Why? What is the maximum relative % optimality gap of the best integer solution found so far? Problem 4 (20 pts). A company is considering opening warehouses in four cities: New York,Los Angeles, Chicago, and Atlanta to serve three different demand regions. Each warehouse can ship 100 units per week. The weekly fixed cost of keeping each warehouse open is $400 for New York, $500 for Los Angeles, $300 for Chicago, and $150 for Atlanta. Region 1 of the country requires 80 units per week, region 2 requires 70 units per week, and region 3 requires 70 units per week. The cost (including production and shipping costs) of sending one unit from a plant to a demand region is shown in the following table. From To Region 1 To Region 2 To Region 3 New York $20 $40 $50 Los Angeles $48 $15 $26 Chicago $26 $35 $18 Atlanta $24 $50 $35 The company wants to meet weekly demands at minimum cost, subject to the following additional restrictions: 1. If the New York warehouse is opened, then the Los Angeles warehouse must be opened. 2. At most two warehouse can be opened. 3. Either the Atlanta or the Los Angeles warehouse must be opened. Formulate a mixed-integer linear program to determine which warehouses to open and how to service demand at a minimum total cost. Problem 5 (15 pts). The following table shows the four possible outcomes of an LP as well as the three possible types of feasible region. Put a check in the boxes that can possibly occur. For example, if the feasible region is nonempty and bounded, then it s possible that there is a unique optimal solution, so you should put a check in that box. 2

3 Unique optimal solution Infinitely many optimal solutions Unbounded objective No feasible solution Feasible region Nonempty and bounded Nonempty and unbounded Empty (no feasible points) Problem 6 (15 pts). A camper is considering taking n types of items in his knapsack for a camping trip. An item i weighs a i lbs and earns a benefit to the camper of c i units (i = 1,..., n). His knapsack can hold at most b lbs. (a) Assuming that the camper can take fractional numbers of an item, formulate a linear program to maximize his benefit. (b) How many different type of items will be in an optimal basic feasible solution of the above LP? Why? (c) Suppose that the following holds: c 1 a 1 c 2 a 2... c n a n. Show that it is optimal to take b a n units of item n. 3

4 Set B Problem 1 (30 pts). (a) I have a maximization integer program whose LP relaxation has an optimal value of 100. I obtained a feasible integer solution with an objective value of 80. The relative optimality gap of this solution is at most: (b) Use binary variables x i for i = 1,..., 10 to denote whether project i is selected or not. Write down a linear constraint to enforce that project 1 can be selected only if at least 3 other projects are selected. (c) Use binary variables x i for i = 1,..., 10 to denote whether project i is selected or not. Write down a linear constraint to enforce that if project 1 is selected then at most 3 of the other projects can be selected. (d) If the LP relaxation of an integer program is infeasible then the integer program is infeasible. True or False? (e) If the LP relaxation of an integer program is unbounded then the integer program is unbounded. True or False? (f) Consider the integer program n min{ c j x j : j=1 n a j x j 0, j=1 x j 0 and integer j = 1,..., n}, where c j R and a j R for j = 1,..., n. We can always obtain a feasible integer solution to the above problem by rounding up the optimal solution to the LP relaxation. True or False? (g) Consider the (single variable) nonlinear optimization problem min cx ln x, where c > 0. The optimal objective value of this problem (in terms of c) is: 4

5 (h) Consider the (single variable) nonlinear optimization problem The optimal solution of this problem is x = min x 2 2x + 1. (i) The optimal objective value of an LP is 20. The first constraint of this LP is a constraint with the current right-hand-side value of 3. The optimal dual solution corresponding to this constraint is 2 and the allowable range of the right-hand-side values is [1,5]. Give a range on the optimal objective value if the right-hand-side is changed to 0. (j) If I add a constraint to a maximization problem the objective value can only get worse. True or False? Problem 2 (20 pts). An investor is planning to invest in n assets. The expected $-per-$ return on asset i is r i. There is a transaction fee associated with the investment in an asset according to the following schedule: For each asset, investment up to $ U is charged $ p per dollar and beyond $ U the charge is $ q per dollar, and 0 < p < q. Moreover if the total investment in the set of assets J {1,..., n} is different from $ T (target) there is penalty fee of $ d per dollar of the difference (both excess and shortage relative to the target is penalized). The investor can only invest non-negative amounts. Formulate a linear program to determine the investment in each asset so as to minimize fees while guaranteeing an expected return of $ R. Problem 3 (20 pts). Eddie s Ice Cream Co. has n production facilities and m distribution centers (DCs). The transportation cost from production facility i to DC j is c ij per ton. DC j has an inventory capacity of u j. At the beginning of each month each DC is stocked by supplying from production facilities. For the next T weeks, the DCs supply ice-cream to retailers. The demand at DC j in week t is d jt. Each DC can use its own inventory to satisfy the demand, it can transport ice-cream from other DCs, and it can also transport to other DCs. The weekly inventory holding cost at DC j is h j per ton, and the transportation cost between DC j and DC k is c kj per ton. Formulate a LP to find the minimum cost strategy for satisfying demand at all DCs. Problem 4 (15 pts). A machine shop has a drill press and a milling machine which are used to produce two parts A and B. The required time (in minutes) per unit part on each machine is shown in the table below. Drill press Milling machine A 3 4 B 5 3 The shop must produce at least 50 parts in total (both A and B) and at least 30 parts of type A, and it can make at most 100 units of A and 100 units of B. Formulate a linear program to minimize the absolute difference between the total running time of the drill press and that of the milling machine. You can assume that the shop can make fractional parts. Problem 5 (15 pts). In the following partial branch-and-bound tree, answer the following questions: 5

6 (a) Which nodes do you still need to branch from? Why? (b) Which nodes do you not need to branch from? Why? (c) What is the gap between the best solution and the best bound found so far? (d) In what order were the three integer solutions found in the branch-and-bound process? Figure 1: Branch and bound tree for Problem 5 6