OVERVIEW OF THE NETWORK OPTIMIZATION PROGRAM (NOP) A. INTRODUCTION

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OVERVIEW OF THE NETWORK OPTIMIZATION PROGRAM (NOP) A. INTRODUCTION NOP is a 32-bit GUI application. This helps the user to use and understand the system quickly.

1 OVERVIEW OF THE NETWORK OPTIMIZATION PROGRAM (NOP) A. INTRODUCTION NOP is a 32-bit GUI application. This helps the user to use and understand the system quickly. Opening and saving input files have become easier. Large amount of data can be transferred between applications using the clipboard. This helps the user to view many input files at the same time. The user can use this approach to compare the input data and output data at the same time, without having to reopen the input file. Most of the forms have scroll bars that allow the user to scroll up and down, in case the file is large. NOP consists of six menu bar options. They are the File menu, Edit menu, Options menu, Procedures menu, Window menu and the Help menu. Figure 1 shows the appearance of NOP on a typical Windows based system. Figure 1. Log and Main Windows for NOP The file menu handles basic file input and output. The File New option allows the user to create a new network model input file. The help system gives the user a detailed description to write procedure 1

2 specific input file. A typical open dialog box opens when the File Open option is chosen. File Close option is used to close input files that are no longer necessary. File name extensions to various network problems can be automatically filtered out. File Save option saves the file into the hard disk. This option prompts the user to specify a file name when it is saved for the first time. The network model can be saved in a different file by using the File Save As option. The program also saves the name of the four most recently used files. The file names can be found in the File menu and keeps on changing as the user opens and closes files. The File Print option that is found on most Windows applications is used to print the contents of windows. The File Exit option enables the user to exit the program by the click of a mouse button. The Edit menu contains buttons that enables data transfer between applications. This menu can also be accessed using the accelerator key ALT + E. Highlighted text from any instance of the SDI form can be placed onto the clipboard using the CUT or COPY keys. Text from the clipboard can be pasted into any of these windows using the PASTE option. The delete button is used to delete highlighted text. The select all button is used to select all the contents of a window. The options menu is used to change the display options in NOP. The toolbar containing commands to open new or and existing file and the edit keys can be turned on or off. The type of font used can also be changed to suit the needs of the user. The procedure option contains all the algorithms available in NOP. This menu is the heart of the package. The procedures contained in NOP are listed in Table 1. Either clicking the procedures item or using keyboard shortcuts can access all the algorithms. For example ALT + P + G can be used to access the Gomory-Hu algorithm and ALT + P + Z to access Generalized networks procedures. The Window option is used to view all the active windows. This includes all active input files, the log window and the output window. These windows can be minimized, maximized and closed using the buttons on the top right-hand corner of the windows. These commands apply to all the four SDI windows and the MDI window. The last option is the Help option. This popup menu bar item contains two buttons. One of them displays an about box with copyright information and the other one brings up a standard windows help system. The help system was developed with a standard Helpfile compiler. NOP contains the algorithms listed in Table 1. 2

3 Table 1. Procedures contained in NOP Name of the algorithm File extension 1. Dijkstra s Algorithm (includes Ford s method).shr 2. Floyd s Algorithm (All-Shortest-Paths Procedure).msc 3. Double-Sweep Algorithm (K-Shortest Paths).ksp. Minimal Spanning Tree.mst 5. Maximum Flow (Labeling Procedure).mxf 6. Multi-Terminal Maximum Capacity.mmc 7. Shortest path with turn penalties.sht 8. Gomory-Hu (Multi-Terminal Maximum Flows).mxf 9. Traveling Salesman Algorithm (Little s Algorithm).tsp 10. Multi-Traveling Salesman algorithm.mts 11. Out of Kilter Algorithm (Ford and Fulkerson).oka 12. Generalized Network (Primal Simplex).gen 13. Time-Cost Trade-Offs Procedure (Fulkerson).cos 1. Critical Path Method (CPM).cpm 15. Project Evaluation and Review Technique (PERT).prt 16. Graphical Evaluation and Review Technique (GERT).grt B. ILLUSTRATION OF EXTERNAL FLOWS (Procedure 12) The problem in this illustration is to determine the optimum shipping schedules from plants to customers. Figure 2 shows a network for a company with plants in cities B, E, and H. The company's products are shipped by air to cities A, C, D, F, and G. Shipping costs per unit are shown adjacent to the arcs connecting the cities. Cities A and D are transshipment points as well as destinations for product shipments. For most problems there are nodes at which flows enter or leave the network. These are called external flows. A node may have three parameters: a fixed external flow, a variable or slack external flow, and a per-unit cost for the variable external flow. Fixed external flows are fixed requirements. Variable external flows are additional maximum requirements (upper bounds) regarding availability and demand 3

4 figures. Information concerning the demands for the product and production costs at the plant are given below for the network shown in Figure 2. The numbers attached to the arcs are the corresponding shipping costs per unit. F 3 H D E C B 3 5 G 5 A Figure 2. Network with shipping costs per unit. City A: Demand for 100 units which must be met. City B: Maximum production of 200 units with manufacturing cost of $10 per unit. There is no minimum shipment required from Austin. City C: Minimum demand of 200 units. An additional 100 units could be sold if available with a revenue of $20 per unit. City D: Contracted demand for 300 units. No fewer than 175 units must be received. A penalty of $5 must be paid for each unit less than 300 received. City E: Work rules require that all regular-time production of 300 units be shipped. An additional 100 units can be produced using overtime at a cost of $1 per unit. City F: A firm demand of 200 units which must be received. City G: One hundred units left over from previous shipments. These can only be sold in city G. No firm demand but up to 200 units can be sold at $25 each. City H: This plant is to be discontinued. The entire inventory of 500 units must be shipped or sold for scrap. The scrap value is $5 per unit. This demand and supply information can be represented by means of the external flow parameters of the nodes. The signs + and - are used to represent whether flows are in or out, with + describing flow into the network and - indicating flow out of the network. A fixed external flow is forced in or out with no option. Thus the demand at city A is a fixed external flow of -100 at the node representing this city. A variable external flow is an optional quantity in addition to the fixed flow. The absolute magnitude of the parameter indicates the maximum amount that can be inserted or withdrawn at the node in addition to the fixed flow. The sign indicates the direction of flow. For city A the variable external flow is 0. The fixed external flow for city C is -200, and the variable external flow is There is a cost of variable flow that must be considered as part of the optimization of total costs. Since the network model shows costs as positive numbers, revenues appear as negative numbers. The cost of external flow at city C is -$20. Costs

5 are not used for fixed flows since these quantities are not variable and costs or revenues associated with them are irrelevant to the optimization. The parameters defining the external flows for the example problem are listed in Table 2. Note that the fixed and variable external flows may have different signs. Table 2. Parameters for External Flows Node Fixed Variable Cost A B C D E F G H C. OPTIONS ALLOWED IN NOP A simplified flowchart of the Network Optimization Program (NOP) is shown in Figure 3. The flow chart shows the options available to the user. It can be inferred that the user can choose a variety of successive operations according to his preference. Figure 3. Network Optimization Program. 5

6 D. FORMAT FOR DATA FILES DIJKSTRA S SHORTEST ROUTE ALGORITHM (WITH FORD S METHOD). This program always finds the shortest route between the source (NODE 1) and any other node in the network. Negative arc lengths are allowed since the program includes Ford s algorithm. MULTITERMINAL SHORTEST CHAIN PROBLEM. This problem is solved using Floyd s Algorithm. Floyd s algorithm is used to find the shortest distance between any two nodes in a network. K SHORTEST PATH PROBLEM. This problem is solved using the Double-Sweep method. The program computes K shortest path lengths from any node to all the other nodes in the network. You will be asked to enter additional data (values for K and PMAX) using input boxes. PMAX is the maximum number of paths to be traced. MINIMAL SPANNING TREE PROBLEM. This problem is solved using a Greedy algorithm. The program finds the minimum spanning tree of the network. The network must be undirected. Enter the upper triangular matrix only. MAXIMUM FLOW PROBLEM. This problem is solved using a labeling procedure. The program computes the maximum flow along arcs from a source to a sink node, subject to flow conservation constraints. 6

7 START_NODE END_NODE ARC_CAPACITY; (for each arc) MULTI TERMINAL MAXIMUM CAPACITY PROBLEM. This problem is solved using a modified version of Floyd s method. The program computes the route with maximum possible capacity between pairs of nodes in a directed network. START_NODE END_NODE ARC_CAPACITY; (for each arc) SHORTEST PATH PROBLEM WITH TURN PENALTIES. This problem is solved using a modified version of Dijkstra s or Ford s method. The program computes the shortest paths between the source and sink nodes while accounting for turn penalties. You will then be asked to enter turn penalties via an input form. Remember that three nodes characterize a turn. Enter the corresponding penalty and the three nodes after running the algorithm. Penalties and lengths must be specified in the same unit. MULTITERMINAL MAXIMAL FLOW PROBLEM. This problem is solved using Gomory- Hu s method. The program computes the maximal flow between nodes of the network. It constructs a maximal spanning tree using the minimum cut/maximum flow rule. Enter the upper triangular matrix only. START_NODE END_NODE ARC_CAPACITY; (for each arc) TRAVELING SALESMAN PROBLEM. This problem is solved using Little s Branch-and-Bound algorithm. Make sure that all nodes have a predecessor and a successor node. MULTITRAVELING SALESPERSON PROBLEM. This problem is solved using Garcia- Diaz s method. The program computes the route after transforming the given network and applying the Outof-Kilter algorithm to the modified network. Negative flow limits are not allowed. 7

8 When you run the algorithm, it will ask you to enter the number of salesmen, and their corresponding fixed costs. OUT OF KILTER ALGORITHM. The algorithm is used to solve a variety of MIN-COST problems in pure networks. Negative flows are not allowed. If upper bounds are INFINITE type The network must be first transformed into an equivalent circulation network. START_NODE END_NODE UPPER_LIMIT LOWER_LIMIT COST_PER_UNIT; (for each arc) GENERALIZED NETWORKS. This program uses Network Primal Simplex procedures to solve a min-cost problem in a generalized (or pure) network. The data for fixed and variable external flows starts with STN; and it is finished with ENDN; It is not necessary to enter zero external node flows. The arc data starts with STA; and it is finished with ENDA; Type 9999 for infinite values. STN; NODE_NUMBER FIXED_FLOW VARIABLE_FLOW UNIT_COST_FOR_VARIABLE_FLOW; (for all nodes with at least one type of non-zero external flows) ENDN; STA; ST_NODE END_NODE UPPER_BOUND LOWER_BOUND UNIT_COST ARC_MULTIPLIER; (for aach arc) ENDA; PERT PROBLEM. This problem is used to find the CRITICAL (LONGEST) path in a CPM activity network. The START NODE label must be less than the label of the END NODE for each arc (no circuits allowed). If an arc duration is INFINITE type START_NODE END_NODE MOST_LIKELY PESSIMISTIC OPTIMISTIC; (for each arc) CPM PROBLEM. This problem is used to find the CRITICAL (LONGEST) path of a CPM network with a specified duration for each arc. START_NODE value must be less than END_NODE value for each arc (no circuits). 8

9 START_NODE END_NODE ACTIVITY_DURATION; (for each arc) TIME-COST TRADE-OFFS PROCEDURE. This program uses Fulkerson s algorithm to shorten the duration of a project assuming linear costs for the activity durations. START_NODE END_NODE UPPER_LIMIT LOWER_LIMIT ABS_VAL_SLOPE; (for each arc) GRAPHICAL EVALUATION AND REVIEW TECHNIQUE (GERT). This program uses Prietsker s GERT algorithm to find the estimated time and the probability of reaching desired node(s) in a stochastic network. Negative arc lengths are not allowed. START_NODE END_NODE PROBABILITY_DISTRIBUTION PARAMETER 1 PARAMETER2; (for each arc) Codes and parameters for distributions supported by GERT (µ = mean; σ = standard deviation) PARAMETER 1 PARAMETER 2 BIN Binomial n p CON Constant a 0 EXP Exponent (µ=a) a 0 GAM Gamma (µ=b/a) 1/a b GEO Geometric p 0 NOR Normal µ σ POI Poisson (µ=a) a 0 UNI Uniform a b NB Negative r p NOTE: If a loop ends at the first node, a dummy node should be created and used as the source of the network. This dummy node is then connected to the original first node with an arc having probability equal to one and duration equal to zero. 9

56:272 Integer Programming & Network Flows Final Examination -- December 14, 1998

56:272 Integer Programming & Network Flows Final Examination -- December 14, 1998 Part A: Answer any four of the five problems. (15 points each) 1. Transportation problem 2. Integer LP Model Formulation