Criminal Courts A Contemporary Perspective 1st Edition Craig Hemmens, David C. Brody, Cassia C. Spohn - Solutions

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Consider n independent and identically distributed random variables, X1,X2, . . . , Xn. Let Sn 1 nn i1 Xi . Find the variance of the random variable Sn as a function of the variance of Xi .
Consider the following distribution problem. We are given a set of manufacturing facilitiesM, a set of warehouses W and a set of customers S. Each warehouse and each customer has a limit on the amount that can be stored at the facility. Each customer has forecasted demands for a single product for
(Mirchandani and Francis, 1990) It appears that the p -Median problem can be solved by solving the resulting problem UFLP (see Exercise 12.5)for different values of F Fj , ∀j , until a value F∗ is found where the UFLP opens exactly p facilities. Show that this method does not work by giving
(Mirchandani and Francis, 1990) Define the Uncapacitated Facility Location Problem (UFLP) in the following way. Let Fj be the fixed charge of opening a facility at site j , for j 1, 2, . . . , m.Problem UFLP : Minn i1m j1 cijXij +m j1 FjYj s.t.m j1 Xij 1, ∀i ∈ N Xij ≤ Yj , ∀i
Consider Problem P1 of Section 12.4.(a) Show that this formulation can be strengthened by adding the constraints:L(1K k1 skU(j k ≤ qj , ∀j ∈ J.(b) Show that this new formulation can be transformed to a specialized kind of linear program called a transportation problem.(c) Why might we not
Prove Theorem 12.5.1.can be formulated as an instance of the Single-Source CFLP.
In the Distribution System Design Problem, explain how the solution methodology changes when there are a fixed number of possible warehouse capacities. For example, at each site, if we decide to install a warehouse, we can install a small, medium or large one.
Consider the newsboy problem with demand D being a random variable whose density, f (D), is known. Let r be a known selling price per unit and c be a known cost per unit. Assume no initial inventory and no salvage value.The objective is to find an order quantity y that maximizes expected profit.
Prove the following properties.(a) Ifg1(y) and g2(y) areK-convex and L-convex, respectively, then for α, β ≥0, αg1(y) + βg2(y) is (αK + βL)-convex.(b) If g(y) is K-convex, then ED[g(y − D)] is also K-convex.
Consider the following discrete version of the newsboy problem.Demand for product can take the valuesD1,D2, . . . , Dn, n ≥ 3, with probabilities p1, p2, . . . , pn, wheren i1 pi 1. Let r be a known selling price per unit and c be a known cost per unit. Our objective is to find an order
It is now June and your company has to make a decision regarding how many skijackets to produce for the comingWinter season. It costs c dollars to produce one skijacket which can be sold for r dollars. Skijackets not sold during the Winter season are lost. Suppose your marketing department
Consider the singlewarehouse inventory model analyzed in Section 11.5 with leadtime l > 0. Prove that the inventory on hand at the end of period t for some t > l can be written as St−l −t it−l Di , where St−l is the order-up-to-level in period t −l and Di is the demand in period
Prove (11.20).
In (11.1), we assume that F(D) is continuous. Now suppose that F(D) is not necessarily continuous. Does there exist an S such that z(y) is minimized at y S ? If there exists such an S, how can you determine it ?
Determine the complexity of Baker et al.’s algorithm.
The Silver-Meal Heuristic works as follows. Let d1, d2, . . . , dn be the demands in the n period planning horizon. Define C(T ) to be the per period average holding and set-up cost under the condition that the current order covers demand in the next T periods. Then C(1) K, C(2) 1 2 (K + hd2),
Assume order costs are general concave and time-dependent functions of the number of items produced. Also, assume holding costs are general concave and time-dependent functions of the number of items held in inventory.Prove that the Zero-Inventory Ordering Property holds in this general setting as
Prove equation (9.9).
Show that in the proof of the lower bound, B∗, for the single warehouse multi-retailer model we have G0 i≥1 Gi .
(Porteus, 1985) The Japanese concept of JIT (Just In Time) advocates reducing set up cost as much as possible. To analyze this concept, consider the Economic Lot Size model with constant demand of D items per year, holding cost h $ per item per year and current set up cost K0. Suppose you can lease
Consider the power of two policies described in the single product model of Section 9.1.3. Describe how you could generate a power of three policy(a policy where each Ti 3kTB for some integer k ≥ 0). What is the effectiveness(in terms of worst-case performance) of the best power of three policy?
Prove that the heuristic for the Single Warehouse Multi-Retailer Model described in Section 9.3 provides a solution within 1.06 of the lower bound.
Consider the Economic Lot Size Model over the infinite horizon.Assume that when an order of sizeQis placed the items are delivered by trucks of capacity q and thus the number of trucks used to deliver Q is Q q, where m is the smallest integer greater than or equal to m. The set-up cost is a linear
(Zavi, 1976) Consider the Economic Lot Size Model with infinite horizon and deterministic demand D items per unit of time. When the inventory level is zero, production of Q items starts at a rate of P items per unit of time, P ≥ D. The set-up cost is K$ and holding cost is h$/item/time. Every
Consider the Economic Lot Size Model and let K be the set-up cost, h be the holding cost per item per unit of time and D the demand rate. Shortage is not allowed and the objective is to find an order quantity so as to minimize the long-run average cost. That is, the objective is to minimize C(Q)
Consider an instance of the VRPTW with n customers. Given a subset of the customers S, let b∗(S) be the minimum number of vehicles required to carry the demands of customers in S; that is, b∗(S) is the solution to the Bin-Packing Problem defined on the demands of all customers in S. For i 1,
Develop a dynamic programming procedure for the column generation step in the presence of a distance constraint on the length of any route.What is required of the distance data in order for this to be possible? What is the complexity of your procedure?
Develop a dynamic programming procedure for the column generation step in the presence of time-window constraints. What is required of the time-window data in order for this to be possible? What is the complexity of your procedure?
Develop a dynamic programming procedure for the column generation step similar to fq (i) that avoids two-loops (loops of the type ...i, j, i...). What is the complexity of this procedure?
Recall the dynamic program given by equation (8.6). Let f min i∈N min wi≤q≤Q fq (i).Consider the function defined as follows.gq (i) min wi≤q≤q{fq (i) + fq−q+wi (i)},for each i ∈ N and wi ≤ q ≤ Q. Now define g mini∈N minwi≤q≤Q gq (i). Show that f g.
Consider the Bin-Packing Problem. Let wi be the size of item i, i 1, . . . , n, and assume the bin capacity is 1. An important formulation of the Bin-Packing Problem is as a set-covering problem. Let F {S :i∈S wi ≤ 1}.Define αiS 1, if item i is in S, 0, otherwise, for each i 1, 2, . .
Consider an undirected graphG (V,E) where each edge (i, j ) has a cost cij and each vertex i ∈ V a nonnegative penalty πi . In the Prize-Collecting Traveling Salesman Problem (PCTSP), the objective is to find a tour that visits a subset of the vertices such that the length of the tour plus the
Prove the upper bound on the convergence rate (equation (8.11)).
Consider the Pickup and Delivery Problem. Here customers are pickup customers with probability p and delivery customers with probability 1−p.Assume a vehicle capacity of 1. If customer i is a pickup customer, then a load of size wi ≤ 1 must be picked up at the customer and brought to the depot.
Consider the Vehicle Routing Problem with Distance Constraints.Formally, a set of customers has to be served by vehicles that are all located at a common depot. The customers and the depot are presented as the nodes of an undirected graph G (N,E). Each customer has to be visited by a vehicle.The
A delivery man currently located at the depot must visit each of n customers. Let ZDM be the total waiting time in the optimal delivery man tour through the n points. Let Z∗ be the total time required to travel the optimal traveling salesman tour through the n points.(a) Prove that ZDM ≤n
Consider the Delivery Man Problem described in
You are given a network G (V,A) where |V| n, d(i, j) is the length of edge (i, j ) and a specified vertex a ∈ V . One service unit is located at a and has to visit each vertex in V so that total waiting time of all vertices is as small as possible. Assume the waiting time of a vertex is
Consider an instance of the CVRP with n customers. A customer is red with probability p and blue with probability 1 − p, for some p ∈ [0, 1]. Red customers have loads of size 2 3 , while blue customers have loads of size 13. What is limn→∞Z∗n as a function of p?
Explain why the following constraints can or cannot be integrated into the Savings Algorithm.(a) Distance constraint. Each route must be at most λ miles long.(b) Minimum route size. Each route must pick up at least m points.(c) Mixing constraints. Even indexed points cannot be on the same route as
Prove that the version of the LBH with set-up costs vj 2dj and assignment costs cij di + dij − dj is asymptotically optimal.
Show that Lemma 6.4.5 can be replaced by an equality instead of an inequality.
Construct an example of the single-source CFLP where each facility is a potential site (and vice versa) in which an optimal solution chooses a facility but the demand of that facility is assigned to another chosen site.
Given a seed point i, assume you must estimate the cost of the optimal traveling salesman tour through a set of points S ∪{i} using the following cost approximation. Starting with 2di , when each point j is added to the tour, add the cost cij dj + dij − di . That is, show that for any r ≥ 1
Prove Corollary 6.5.3.
Consider the following heuristic for the CVRP with unequal demands.All customers of demand wi > 1 2 are served individually, one customer per vehicle. To serve the rest, apply the UITP heuristic with vehicle capacity Q.Prove that this solution can be transformed into a feasible solution to the CVRP
Derive equation (5.5).
Consider an n customer instance of the CVRP with equal demands.There are K customer types: a customer is of type k with independent probability pk > 0. Customers of different types cannot be served together in the same vehicle.Devise an asymptotically optimal heuristic for this problem. If K is a
Consider an n customer instance of the CVRP with equal demands.Assume there are m depots and at each depot is an unlimited number of vehicles of limited capacity. Suggest an asymptotically optimal region partitioning scheme for this case.
Derive equation (5.3).
Consider the following version of the Capacitated Vehicle Routing Problem (CVRP). You are given a network G (V,A) with positive arc lengths.Assume that E ⊆ A is a given set of edges that have to be “covered” by vehicles.The vehicles are initially located at a depot p ∈ V . Each vehicle
Prove that if a graph has exactly 2k vertices of odd degree, then the set of edges can be partitioned into k paths such that each edge is used exactly once.
Consider a list L of n items of sizes in (13, 1 2 ]. Let bLP be the optimal fractional solution to the set-partitioning formulation of the Bin-Packing Problem, and let b∗ be the optimal integer solution to the same formulation. Prove that b∗ ≤ bLP + 1.
A function u : [0, 1] → [0, 1] is dual-feasible if for any sets of numbers w1,w2, . . . , wk, we havek i1 wi ≤ 1 ⇒k i1 u(wi ) ≤ 1.(a) Given an instance of the Bin-Packing Problem with item sizesw1,w2, . . . , wn and a dual-feasible function u, prove thatn i1 u(wi ) ≤ b∗.(b) Assume
Consider an instance of the Bin-Packing Problem where there are mj items of size wj ∈ (0, 1] for j 1, 2, . . . , n. Define a bin configuration to be a vector c (c1, c2, . . . , c n) with the property that ci ≥ 0 for i 1, 2, . . . , n and n j1 cjwj ≤ 1. Enumerate all possible bin
Showthat a lower bound on the cost of the optimal traveling salesman tour can be given by:2|N| max i∈Nj∈N dij , where N is the set of cities and dij is the distance from city i to city j .
Prove Lemma 4.3.2.
Assume n cities are uniformly distributed in the unit disc. Consider the following heuristic for the n-city TSP. Let di be the distance from city i to the depot. Order the points so that d1 ≤ d2 ≤ · · · ≤ dn. For each i 1, 2, . . . , n, draw a circle of radius di centered at the depot;
(Coffman and Leuker, 1991) Consider flipping a fair coin n times in succession. LetXn represent the random variable denoting the maximum excess of the number of heads over tails at any point in the sequence of n flips. It is known that E(Xn) is 9(√n). From this, argue that E[ZMATCH n ] n 2+
What is the complexity of the dynamic program developed in the previous exercise?
Suggest a method to pack n items drawn randomly from the uniform distribution on [ 1 40 , 59 120 ]. Can you prove that your method is asymptotically optimal? What is the bin-packing constant (γ ) for this distribution?procedure to solve the TSP. Let city 1 be an arbitrary city. Define the
Suggest a method to pack n items drawn randomly from the uniform distribution on [0, 5 12 ]. Can you prove that your method is asymptotically optimal?What is the bin-packing constant (γ ) for this distribution?
Suggest a method to pack n items drawn randomly from the uniform distribution on [13, 1]. Can you prove that your method is asymptotically optimal?What is the bin-packing constant (γ ) for this distribution?
The Harmonic heuristic with parameter M, denoted H(M), is the following. For each k 1, 2, . . . , M − 1, items of size 1 k+1 < wi ≤ 1 k are packed separately, at most k items per bin. That is, items of size greater than 1 2 are packed one per bin, items of size 13< wi ≤ 1 2 are packed two
Prove that the bin-packing constant γ satisfies 1 ≤ γ/E(w) ≤ 2 where E(w) is the expected item size.
Consider the TSP defined on a set of points N indexed 1, 2, . . . , n.Let Z∗ be the length of the optimal tour. Consider now the following strategy:starting with point 1, the salesman moves to the closest point in the set N \{1}, say point 2. The salesman then constructs an optimal traveling
An upper bound on β. (Karp and Steele, 1985) The strips method for constructing a tour through n random points in the unit square dissects the square into 1 7 horizontal strips of width 7, and then follows a zigzag path, visiting the points in the first strip in left-to-right order, then the
A lower bound on β. Let X(n) {x1, x2, . . . , xn} be a set of points uniformly and independently distributed in the unit square. Let (j be the distance from xj ∈ X(n) to the nearest point in X(n) \ xj . Let L(X(n)) be the length of the optimal traveling salesman tour through X(n). Clearly
A complete directed network G (V,A) is a directed graph such that for every pair of vertices u, v ∈ V , there are arcs u → v and v → u in A with nonnegative arc lengths d(u, v) and d(v, u), respectively. The networkG (V,A)satisfies the triangle inequality if for all u, v,w ∈ V , d(u,
Given a network G (V,E), and edge length le for every e ∈ E, assume that edge (u, v) has a variable length x. Find an expression for the length of the shortest path from s to t as a function of x.
(Anily et al., 1994) Consider the Bin-Packing Problem and the Next-Fit Increasing heuristic. In this strategy items are ordered in a nondecreasing order according to their size. Start by packing the first item in bin 1. Then each subsequent item is packed in the last opened bin if possible, or else
Consider the Bin-Packing Problem with items of size wi , i 1, 2, . . . , n, such that 0 < wi ≤ 1. The objective is find the minimum number of unit size bins b∗ needed to pack all the items without violating the capacity constraint.(a) Show thatn i1 wi is a lower bound on b∗.(b) Define a
Suppose there are n jobs that require processing on m machines.Each job must be processed by machine 1, then by machine 2, . . . , and finally by machine m. Each machine can work on at most one job at a time and once it begins work on a job it mustwork on it until completion, without interruption.
(Papadimitriou and Steiglitz, 1982) Which of the following problems remain essentially unchanged (complexity-wise) when they are transformed from minimization to maximization problems? Why?(a) Traveling Salesman Problem.(b) Shortest Path from s to t .(c) Minimum Weight Matching.(d) Minimum Spanning
(Papadimitriou and Steiglitz, 1982) TheWandering Salesman Problem(WSP) is a Traveling Salesman Problem except that the salesman can start wherever he or she wishes and does not have to return to the starting city after visiting all cities.(a) Describe a heuristic for the WSP with worst-case bound 3
Show that a minimum spanning tree T satisfies the following property.When T is compared with any other spanning tree T, the kth shortest edge of T is no longer than the kth shortest edge of T, for k 1, 2, . . . , n − 1.
Consider the TSP with the triangle inequality assumption and two prespecified nodes s and t . Assume that the traveling salesman tour has to include edge (s, t ) (that is, the salesman has to travel from s directly to t ). Modify Christofides’ Heuristic for this model and show that the worst-case
Consider the n-city TSP defined with distances dij . Assume that there exista, b ∈ IRn such that for each i and j , dij ai + bj . What is the length of the optimal traveling salesman tour? Explain your solution.
Let G be a tree with n ≥ 2 nodes. Show that:(a) There exist at least two nodes with degree one.(b) The number of arcs is n − 1.
Prove that for any graph G there exists an even number of nodes with odd degree.
Let ZC be the length of the tour provided by Christofides’ Heuristic, and let Z∗ be the length of the optimal tour. Construct an example with ZC 3 2Z∗.
Prove that in every completely connected directed graph (a graph in which between every pair of vertices there is a directed edge in one of the two possible directions) there is a directed Hamiltonian Path.
(Papadimitriou and Steiglitz, 1982) Consider the n-city TSP in which the triangle inequality assumption holds. Let c∗> 0 be the length of an optimal tour, and let c be the length of the second best tour. Prove: (c − c∗)/c∗ ≤ 2 n .
The 2-TSP is the problem of designing two tours that together visit each of the customers and use the same starting point. Show that any algorithm for the TSP can solve this problem as well.
Prove Lemma 2.3.8.
What do you envision as the future of problem-solving courts? Will additional types of courts develop in the years to come? If so, what types of courts might these be?
What are the pros and cons of prosecuting juveniles in adult criminal courts? Would you favor or oppose a "blanket rule" that all crimes involving youth under a certain age-say 16-must be adjudicated in a juvenile court? Why?
How do juvenile courts differ from adult criminal courts? What accounts for these differences?
If you were a local elected official and could choose whether to fund a community court, a homeless court, a mental health court, or a reentry court, which would you choose? Why?
Why did arrests for domestic violence increase following the implementation of the Lexington County (South Carolina) Domestic Violence Court?
How does the focus of domestic violence courts differ from that of drug treatment courts and other problem-solving courts?
How did the research design used by Gottfredson et al. (2003) improve upon the research design typically used to evaluate drug treatment courts? What did their evaluation of the BCDTC reveal? (See "Current Research.")
Assume that you have been asked to evaluate the effectiveness of a specialized court. How would you measure the court's success?
How does restorative justice differ from the other utilitarian perspectives on punishment? How are the principles of restorative justice incorporated into problem- solving courts?
What are the advantages of adjudicating cases-for example, domestic violence cases or cases involving defendants with mental health problems-in a specialized court? Are there any disadvantages to this approach?
How do specialized courts, such as drug treatment courts, differ from traditional courts? What do you think is the most important difference between these types of courts?
Explain the types of appeals discussed in this chapter that are guaranteed by the Constitution. Do those convicted of crimes have the right to at least one appeal? If so, who guarantees this right?
Why is it important for prosecutors to have the ability to file an interlocutory appeal?
What types of cases are heard by the U.S. Supreme Court? What functions does the U.S. Supreme Court serve?
Describe the processes by which a state case can make its way to the U.S. Supreme Court (see the "Warren McCleskey's Odyssey" box).

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Criminal Courts A Contemporary Perspective 1st Edition Craig Hemmens, David C. Brody, Cassia C. Spohn - Solutions

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