Write single linear constraint that is equivalent to the statement "x2 = 1 or x3 = 0," but not both. (c) (4 points) Add a binary variable w1, and add two constraints that ensure that w1 = 1 if x5 + x6 70, and w1 = 0 if x5 + x6 69. (d) (4 points) Add 3 binary variables w2, w3, and w4 and at most 4 constraints so as to ensure that at least one of the following constraints is satisfied: (i) x5 92, (ii) x6 40, (iii) x7 + x8 74. (e) (4 points) Add a single binary variable w5 and two constraints to ensure that at least one of the following two constraints are satisfied (i) x9 45, (ii) x10 22. (f) (4 points) Add a single integer variable w6 and a constraint that ensures that x8 is divisible by 2 but not divisible by 4. (The remainder when dividing by 4 must be 2). (g) (4 points) Add three binary variables w7, w8, and w9 and two constraints that ensures that x10 = 13 or 39 or 88.

(h) (4 points) Add variable(s) and constraint(s) that model the cost of x4 as f4(x4), which is defined as follows: If x4 = 0, then f4(x4) = 0. If x4 1, then f4(x4) = 250 + 49x4. (i) (8 points) Add variable(s) and constraint(s) that model the cost of x5 as f5(x5), which is defined as follows: If 0 x5 10, then f5(x5) = 57x5. If 11 x5 20, then f5(x5) = 570. If 21 x5 100, then f5(x5) = 480 + 50x5. Problem 2 As the leader of an oil-exploration drilling venture, you need to determine which 5 sites out of 10 to evaluate for drilling opportunities. The goal is to select 5 sites with the lowest overall cost. Label the sites S1, S2, . . . , S10, and the exploration costs associated with each as c1, c2, . . . , c10. Regional development restrictions are such that: (i) Evaluating sites S2 and S7 will prevent you from evaluating either site S6 or S9. (ii) Evaluating sites S1 and S3 will prevent you from also evaluating both sites S5 and S6. (iii) Evaluating site S3 or S4 prevents you from evaluating site S6. (iv) Of the group S3, S6, S7, S8, at most two sites may be assessed. Formulate an integer program to determine the minimum-cost exploration scheme that satisfies these restrictions. Try to develop a model in which the only variables are x1, . . . , x10, where xj is 1 or 0 according as site j is evaluated or not. (For example, the constraint "Evaluating sites S2 and S7 will prevent you from exploring site S6" can be expressed as x2 + x6 + x7 2 because the only binary solutions prohibited have x2 = x6 = x7 = 1.)

Problem 3 Suppose you want to minimize or maximize a piecewise linear function of one variable, subject to linear constraints. This is a problem that can be solved by resorting to linear constraints only, possibly by adding extra variables. In this example, we consider the function with three pieces shown in Figure 1. (a) Suppose we want to minimize f(x) shown in Figure 1. Assume that x is subject to a set of linear constraints that involve other variables A(x' |x) = b, so that we cannot simply solve the problem by inspection because we do not know what values x will take. How can we formulate this problem in linear form? Do we need integer variables? (In your formulation, you can ignore the additional constraints A(x' |x) = b.) (b) Consider now the problem of maximizing f(x) of Figure 1, subject to a set of linear constraints that involve other variables. We cannot use the same approach of Part 2.A. Explain why and find an alternative way of formulating the problem, adding (binary or integer) variables as needed.

The state vector in the dynamic programming problem is identified as (k,61) but the role of begin ning of period capital is entirely captured through its effect on output. So use this to define the state vector as (v. 51). Then a Recursive Competitive Equilibrium is defined, in general, by three functions: c(vt, di), i (vi, dt), and q (ye, 61). (This is a cursory definition of a RCE. A complete answer would discuss the policy functions for the agent and how the relevant state variables are the individual and aggregate capital stocks, in addition to the & shock. Also a law of motion for the aggregate capital stock would be known by agents and, in equilibrium, there would be consistency between the agents and aggregate law of motion for the capital stock. Something along those lines would be more appropriate.) However, given the resource constraint, once the function for consumption is determined, then the investment function is also determined. And, due to eq.(1), the consumption function and production function also imply the equilibrium function for g (1,5t). So, more precisely, solving for the consumption function, c (t1,61), doen solve for the RCE. This task is made easier by the setup: it is reasonable, as we saw many times in class), to conjecture that this function is homogeneous of degree 1 in y. That is, c(vt, 6) = yew (de). If this conjecture is correct, then the necessary conditions will define w (51). Note that i (p, 61) = (1-w(5)) t and using this in eq (3) yields:

The purpose of this recitation is to familiarize students with a variety of integer programming modeling techniques as described in the IP Formulation Guide and in the powerpoint tutorial on IP formulations. We start with an integer program IP1 defined as follows: max 21x1 + 32x2 + 40x3 + 49x4 + 57x5 + +71x6 + 82x7 + 91x8 + 100x9 + 109x10 s.t.: 2x1 + 3x2 + 4x3 + 5x4 + 6x5 + +7x6 + 8x7 + 9x8 + 10x9 + 11x10 900 i = 1, . . . , 3 xi {0, 1} i = 4, . . . , 10 0 xi 100. (IP1) For each of the parts below, you are to add constraint(s) and possibly variables to ensure that the logical condition is satisfied by the integer program. Each part is independent; that is, no part depends on the parts preceding it. You do not need to repeat the integer programming objective or constraints given above. You may use the big M method for formulating constraint when it is appropriate. (a) (4 points) Write a single linear constraint that is equivalent to the statement "If x1 = 1, then x2 = 0."

Problem 1 You create start-up company that caters high-quality organic food directly to a number of customers. You receive a number of tentative orders and you now have to tell your customers which orders you are going to take. Before embarking on this journey, you first want to allocate your production capabilities in order to devise a feasible daily production plan that maximizes your profit. There are only three different kinds of food that you offer at this early stage of the company: Hummus (H) with garlic pitas, an excellent Moussaka (M), and a traditional Tabouleh (T) with parsley and mint. Each meal has to be cooked, packaged and delivered. Each operation is run by yourself. You have to deliver between 12PM and 2PM everyday, and the food is made on the same day, therefore you estimate that the total number of available cooking hours is 4, the total number of packaging hours is 2, and the total number of delivery hours is 2. Cooking sufficient Hummus for 10 portions requires 1 hour of time, packaging is done at the rate of 20 portions per hour, and delivery at the rate of 30 per hour. The cost of the ingredients for 1 portion is $1, and each packaged portion can be sold for $7. Moussaka takes more time to prepare: in one hour, the food cooking team can prepare 5 portions. Packaging is done at the rate of 15 portions per hour. Since the Moussaka has to be delivered while still warm out of the oven, it is delivered in smaller batches, therefore only 15 portions can be delivered in one hour. The cost of the ingredients for 1 portion is $2, and it can be sold for $12. Finally, Tabouleh can be prepared at the rate of 15 portions per hour, it can be packaged at the rate of 25 portions per hour, and delivered at the rate of 30 per hour. Tabouleh is very inexpensive and one portion only costs $0.5 in raw ingredients, and can be sold for $5. Customers expressed interest in having the following products delivered every day: 20 Hummus meals, 10 Moussaka meals, and 30 Tabouleh meals

What appears to have been the study hypothesis in this study? Sometimes the research study question is a generalized statement but it is related to a background study hypothesis even if the hypothesis is not explicitly stated. Try to infer what the study authors hypothesized and note this in relation to the study question. (The hypothesis is considered to be a statement about the association between an exposure and an outcome (disease or illness) that is statistically testable. For example, in a study of alcohol consumption and breast cancer the investigators may have hypothesized that either alcohol consumption in general or high levels of alcohol may increase the risk of breast cancer. Although this is often in the Introduction, it may be explicitly stated in the Results or Discussion section.

7. How were study subjects (participants) recruited and how many subjects were enrolled in the study compared to how many actually participated in the study?

8. How were data collected for the study?

9. What was the main agent, exposure, or risk factor being investigated in relation to the outcome?

10. What types of covariates and confounders were evaluated? These are not the same as what is often referred to as the primary outcome or main effect. Remember a covariate or confounder is a variable that is thought to be relevant and important when evaluating the association of interest between a primary exposure or risk factor and the outcome. For example, alcohol consumption and the risk of breast cancer may be the primary association of interest for a study research question, but it will also be important to

consider other influential factors such as cigarette smoking, obesity, reproductive history, family history of breast cancer, etc. when evaluating this association.