// Numbas version: exam_results_page_options
{"questions": [{"variable_groups": [{"variables": ["num_sources", "num_destinations", "total"], "name": "Setup"}, {"variables": ["supply", "demand", "costs"], "name": "Initial data"}, {"variables": ["assignment", "assignment_cost"], "name": "Minimum costs"}], "variables": {"costs": {"group": "Initial data", "templateType": "anything", "definition": "matrix(repeat(\n  repeat(\n    random(1..9),\n    num_destinations\n  ),\n  num_sources\n))", "description": "", "name": "costs"}, "num_sources": {"group": "Setup", "templateType": "anything", "definition": "random(3..5)", "description": "", "name": "num_sources"}, "supply": {"group": "Initial data", "templateType": "anything", "definition": "random_partition(total,num_sources)", "description": "", "name": "supply"}, "num_destinations": {"group": "Setup", "templateType": "anything", "definition": "7-num_sources+random(0,1)", "description": "", "name": "num_destinations"}, "assignment_cost": {"group": "Minimum costs", "templateType": "anything", "definition": "assignment_cost(assignment,costs)", "description": "", "name": "assignment_cost"}, "total": {"group": "Setup", "templateType": "anything", "definition": "random(50..500)", "description": "", "name": "total"}, "assignment": {"group": "Minimum costs", "templateType": "anything", "definition": "minimum_cost(supply,demand,costs)", "description": "", "name": "assignment"}, "demand": {"group": "Initial data", "templateType": "anything", "definition": "random_partition(total,num_destinations)", "description": "", "name": "demand"}}, "ungrouped_variables": [], "name": "Transportation problem - apply minimum cost algorithm - MAS1901", "functions": {"show_variable_list": {"type": "string", "language": "javascript", "definition": "return data.map(function(n,i) {\n  return '$'+name+'_{'+(i+1)+'} = '+Numbas.math.niceNumber(n)+'$';\n}).join(', ');", "parameters": [["data", "list"], ["name", "string"]]}}, "parts": [{"showCorrectAnswer": true, "allowFractions": false, "correctAnswer": "assignment", "variableReplacementStrategy": "originalfirst", "prompt": "<p>Apply the minimum cost algorithm to find an assignment of units of coal to be&nbsp;transported between pits and power stations.</p>\n<p>When there's more than one choice of cell, choose the cell with the maximum available allocation.</p>", "correctAnswerFractions": false, "allowResize": false, "numRows": "num_sources", "scripts": {}, "type": "matrix", "numColumns": "num_destinations", "tolerance": 0, "markPerCell": false, "variableReplacements": [], "marks": 1}, {"correctAnswerFraction": false, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "minValue": "assignment_cost", "prompt": "<p>What is the total cost of this solution?</p>", "showPrecisionHint": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "variableReplacements": [{"part": "p0", "must_go_first": false, "variable": "assignment"}], "marks": 1, "maxValue": "assignment_cost"}], "variablesTest": {"condition": "", "maxRuns": "3000"}, "statement": "<p>Four power stations labelled $j = 1,2,3,4$ require {show_variable_list(demand,'b')} units of coal, respectively. These are to be provided by three pits labelled $i=1,2,3$, which have supplies of {show_variable_list(supply,'a')} units of coal, respectively.</p>\n<p>The cost $c_{ij}$ of transporting one unit of coal from pit $i$ to power station $j$ is in the following table (with supplies and demand shown in the margin).</p>\n<p>{cost_table(supply,demand,costs)}</p>", "tags": [], "rulesets": {}, "preamble": {"css": "", "js": ""}, "metadata": {"notes": "", "licence": "None specified", "description": ""}, "advice": "<h4>a)</h4>\n<p>The following tables show each step of the minimum cost algorithm:</p>\n<p>{minimum_cost_display(supply,demand,costs)}</p>\n<h4>b)</h4>\n<p>The total cost of this solution&nbsp;is $\\var{show_cost_calculation(assignment,costs)}$.</p>"}, {"variable_groups": [{"variables": ["num_sources", "num_destinations", "total", "omit_source", "show_sources", "show_destinations"], "name": "Setup"}, {"variables": ["supply", "demand", "costs", "show_supply", "show_demand", "show_costs", "dummy_resource"], "name": "Initial data"}, {"variables": ["assignment", "assignment_cost", "show_assignment"], "name": "Minimum costs"}], "variables": {"show_assignment": {"group": "Minimum costs", "templateType": "anything", "definition": "matrix(map(\n  map(\n    assignment[x][y],\n    y,\n    0..show_destinations-1\n  ),\n  x,\n  0..show_sources-1\n))", "description": "", "name": "show_assignment"}, "costs": {"group": "Initial data", "templateType": "anything", "definition": "matrix(map(\n  map(\n    if(x<show_sources and y<show_destinations,\n        random(1..9),\n        0\n    ),\n    y,\n    0..num_destinations-1\n  ),\n  x,\n  0..num_sources-1\n))", "description": "", "name": "costs"}, "show_sources": {"group": "Setup", "templateType": "anything", "definition": "num_sources-if(omit_source,1,0)", "description": "", "name": "show_sources"}, "omit_source": {"group": "Setup", "templateType": "anything", "definition": "random(true,false)", "description": "", "name": "omit_source"}, "num_destinations": {"group": "Setup", "templateType": "anything", "definition": "4", "description": "", "name": "num_destinations"}, "show_supply": {"group": "Initial data", "templateType": "anything", "definition": "supply[0..show_sources]", "description": "", "name": "show_supply"}, "supply": {"group": "Initial data", "templateType": "anything", "definition": "random_partition(total,num_sources)", "description": "", "name": "supply"}, "show_destinations": {"group": "Setup", "templateType": "anything", "definition": "num_destinations-if(omit_source,0,1)", "description": "", "name": "show_destinations"}, "dummy_resource": {"group": "Initial data", "templateType": "anything", "definition": "if(omit_source,\n  sum(supply)-sum(show_supply),\n  sum(demand)-sum(show_demand)\n)", "description": "", "name": "dummy_resource"}, "num_sources": {"group": "Setup", "templateType": "anything", "definition": "random(3..5)", "description": "", "name": "num_sources"}, "show_demand": {"group": "Initial data", "templateType": "anything", "definition": "demand[0..show_destinations]", "description": "", "name": "show_demand"}, "show_costs": {"group": "Initial data", "templateType": "anything", "definition": "matrix(map(map(costs[x][y],y,0..show_destinations-1),x,0..show_sources-1))", "description": "", "name": "show_costs"}, "assignment_cost": {"group": "Minimum costs", "templateType": "anything", "definition": "assignment_cost(assignment,costs)", "description": "", "name": "assignment_cost"}, "total": {"group": "Setup", "templateType": "anything", "definition": "random(50..500)", "description": "", "name": "total"}, "assignment": {"group": "Minimum costs", "templateType": "anything", "definition": "minimum_cost(supply,demand,costs)", "description": "", "name": "assignment"}, "demand": {"group": "Initial data", "templateType": "anything", "definition": "random_partition(total,num_destinations)", "description": "", "name": "demand"}}, "ungrouped_variables": [], "name": "Transportation problem - apply minimum cost algorithm to unbalanced problem - MAS1901", "functions": {"show_variable_list": {"type": "string", "language": "javascript", "definition": "return data.map(function(n,i) {\n  return '$'+name+'_{'+(i+1)+'} = '+Numbas.math.niceNumber(n)+'$';\n}).join(', ');", "parameters": [["data", "list"], ["name", "string"]]}}, "parts": [{"showCorrectAnswer": true, "scripts": {}, "gaps": [{"displayType": "radiogroup", "choices": ["<p>pit</p>", "<p>power station</p>"], "showCorrectAnswer": true, "matrix": "if(omit_source,[1,0],[0,1])", "distractors": ["", ""], "variableReplacements": [], "shuffleChoices": false, "scripts": {}, "maxMarks": 0, "type": "1_n_2", "minMarks": 0, "variableReplacementStrategy": "originalfirst", "displayColumns": 0, "marks": 0}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "dummy_resource", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "maxValue": "dummy_resource"}, {"showCorrectAnswer": true, "allowFractions": false, "correctAnswer": "costs", "variableReplacementStrategy": "originalfirst", "allowResize": true, "correctAnswerFractions": false, "numRows": 1, "scripts": {}, "type": "matrix", "numColumns": 1, "tolerance": 0, "markPerCell": false, "variableReplacements": [], "marks": 1}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "prompt": "<p>By adding dummy sources or destinations, balance this problem.</p>\n<p>The balanced problem adds&nbsp;another&nbsp;[[0]] with&nbsp;[[1]] units of coal.</p>\n<p>Enter the cost matrix for the balanced problem below.</p>\n<p>[[2]]</p>", "variableReplacements": [], "marks": 0}, {"showCorrectAnswer": true, "allowFractions": false, "correctAnswer": "show_assignment", "variableReplacementStrategy": "originalfirst", "prompt": "<p>Apply the minimum cost algorithm to your balanced problem to find an assignment of units of coal to be&nbsp;transported between pits and power stations.</p>\n<p>When there's more than one choice of cell, choose the cell with the maximum available allocation.</p>", "correctAnswerFractions": false, "allowResize": false, "numRows": "show_sources", "scripts": {}, "type": "matrix", "numColumns": "show_destinations", "tolerance": 0, "markPerCell": false, "variableReplacements": [{"part": "p0g2", "must_go_first": false, "variable": "costs"}], "marks": 1}, {"showCorrectAnswer": true, "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "assignment_cost", "variableReplacementStrategy": "originalfirst", "variableReplacements": [{"part": "p1", "must_go_first": false, "variable": "assignment"}], "marks": 1, "maxValue": "assignment_cost"}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "prompt": "<p>What is the total cost of this solution?</p>\n<p>&pound;&nbsp;[[0]]</p>", "variableReplacements": [], "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>Four power stations labelled $j = 1,2,3,4$ require {show_variable_list(show_demand,'b')} units of coal, respectively. These are to be provided by three pits labelled $i=1,2,3$, which have supplies of {show_variable_list(show_supply,'a')} units of coal, respectively.</p>\n<p>The cost $c_{ij}$, in pounds, of transporting one unit of coal from pit $i$ to power station $j$ is in the following table (with supplies and demand shown in the margin).</p>\n<p>{cost_table(show_supply,show_demand,show_costs)}</p>", "tags": [], "rulesets": {}, "preamble": {"css": "", "js": ""}, "metadata": {"notes": "", "licence": "None specified", "description": ""}, "advice": "<h4>a)</h4>\n<p>The total supply is {sum(show_supply)}, while the total demand is {sum(show_demand)}, so we need to add a dummy {if(omit_source,'pit supplying','power station demanding')} {dummy_resource} units of coal.</p>\n<p>The cost matrix&nbsp;then looks like this:</p>\n<p>{cost_table(supply,demand,costs)}</p>\n<h4>b)</h4>\n<p>The following tables show each step of the minimum cost algorithm:</p>\n<p>{minimum_cost_display(supply,demand,costs)}</p>\n<h4>c)</h4>\n<p>The total cost of this solution&nbsp;is $\\var{show_cost_calculation(assignment,costs)}$.</p>"}, {"variable_groups": [{"variables": ["num_sources", "num_destinations", "total"], "name": "Setup"}, {"variables": ["supply", "demand", "costs"], "name": "Initial data"}, {"variables": ["initial_assignment", "initial_cost"], "name": "Minimum costs"}, {"variables": ["final_assignment", "final_is_optimal", "final_cost"], "name": "Stepping stone"}], "variables": {"final_assignment": {"group": "Stepping stone", "templateType": "anything", "definition": "stepping_stone(initial_assignment,costs)", "description": "", "name": "final_assignment"}, "initial_assignment": {"group": "Minimum costs", "templateType": "anything", "definition": "minimum_cost(supply,demand,costs)", "description": "", "name": "initial_assignment"}, "num_sources": {"group": "Setup", "templateType": "anything", "definition": "3", "description": "", "name": "num_sources"}, "final_cost": {"group": "Stepping stone", "templateType": "anything", "definition": "assignment_cost(final_assignment,costs)", "description": "", "name": "final_cost"}, "final_is_optimal": {"group": "Stepping stone", "templateType": "anything", "definition": "assignment_is_optimal(final_assignment,costs)", "description": "", "name": "final_is_optimal"}, "costs": {"group": "Initial data", "templateType": "anything", "definition": "matrix(repeat(\n  repeat(\n    random(1..9),\n    num_destinations\n  ),\n  num_sources\n))", "description": "", "name": "costs"}, "num_destinations": {"group": "Setup", "templateType": "anything", "definition": "4", "description": "", "name": "num_destinations"}, "initial_cost": {"group": "Minimum costs", "templateType": "anything", "definition": "assignment_cost(initial_assignment,costs)", "description": "", "name": "initial_cost"}, "total": {"group": "Setup", "templateType": "anything", "definition": "random(50..500)", "description": "", "name": "total"}, "supply": {"group": "Initial data", "templateType": "anything", "definition": "random_partition(total,num_sources)", "description": "", "name": "supply"}, "demand": {"group": "Initial data", "templateType": "anything", "definition": "random_partition(total,num_destinations)", "description": "", "name": "demand"}}, "ungrouped_variables": [], "name": "Transportation problem - stepping stones - MAS1901", "functions": {"show_variable_list": {"type": "string", "language": "javascript", "definition": "return data.map(function(n,i) {\n  return '$'+name+'_{'+(i+1)+'} = '+Numbas.math.niceNumber(n)+'$';\n}).join(', ');", "parameters": [["data", "list"], ["name", "string"]]}}, "parts": [{"showCorrectAnswer": true, "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "initial_cost", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "maxValue": "initial_cost"}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "prompt": "<p>What is the cost of this initial assignment?</p>\n<p>&pound;&nbsp;[[0]]</p>", "variableReplacements": [], "marks": 0}, {"showCorrectAnswer": true, "allowFractions": false, "correctAnswer": "final_assignment", "variableReplacementStrategy": "originalfirst", "prompt": "<p>Use the stepping stones method to find an optimal solution, and enter it below.</p>", "correctAnswerFractions": false, "allowResize": false, "numRows": "num_sources", "scripts": {"mark": {"script": "var validation = this.validation;\nvar optimisation = Numbas.extensions.optimisation;\n\nvar studentMatrix = this.studentAnswerAsMatrix();\n\nif(studentMatrix===null) {\n  this.setCredit(0,R('part.matrix.invalid cell'));\n  validation.invalidCell = true;\n  return;\n} else {\n  validation.invalidCell = false;\n}\n\nif(!optimisation.assignment_is_valid(studentMatrix,variables.supply,variables.demand)) {\n  this.setCredit(0,\"Your answer does not satisfy one of the constraints of supply or demand.\");\n  return;\n}\n\nthis.answered = true;\n\nif(optimisation.assignment_is_optimal(studentMatrix,null,variables.costs)) {\n  this.setCredit(1,\"Your answer is optimal.\");\n  return;\n} else {\n  this.setCredt(0,\"Your answer is not optimal.\");\n  return;\n}\n\n", "order": "instead"}}, "type": "matrix", "numColumns": "num_destinations", "tolerance": 0, "markPerCell": false, "variableReplacements": [], "marks": 1}, {"showCorrectAnswer": true, "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "final_cost", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "maxValue": "final_cost"}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "prompt": "<p>What is the cost of the optimal solution?</p>\n<p>&pound;&nbsp;[[0]]</p>", "variableReplacements": [], "marks": 0}], "variablesTest": {"condition": "not assignment_is_optimal(initial_assignment,costs)", "maxRuns": 100}, "statement": "<p>Four power stations labelled $j = 1,2,3,4$ require {show_variable_list(demand,'b')} units of coal, respectively. These are to be provided by three pits labelled $i=1,2,3$, which have supplies of {show_variable_list(supply,'a')} units of coal, respectively.</p>\n<p>The cost $c_{ij}$, in pounds, of transporting one unit of coal from pit $i$ to power station $j$ is in the following table (with supplies and demand shown in the margin).</p>\n<p>{cost_table(supply,demand,costs)}</p>\n<p>The NW corner algorithm has been used to come up with an initial feasible solution, shown below.</p>\n<p>{assignment_table(initial_assignment,supply,demand)}</p>", "tags": [], "rulesets": {}, "preamble": {"css": "", "js": ""}, "metadata": {"notes": "", "licence": "None specified", "description": ""}, "advice": "<h4>a)</h4>\n<p>The cost of the initial solution&nbsp;is $\\var{show_cost_calculation(initial_assignment,costs)}$.</p>\n<h4>b)</h4>\n<p>The steps of the method are shown below.</p>\n<p>{stepping_stone_display(initial_assignment,costs)}</p>\n<p>At this point, no cells have shadow costs greater than their real costs, so the solution is optimal.</p>\n<h4>c)</h4>\n<p>The cost of the optimal solution is $\\var{show_cost_calculation(final_assignment,costs)}$.</p>"}], "duration": 0, "name": "Transportation problems", "showQuestionGroupNames": false, "allQuestions": true, "resources": [], "percentPass": 0, "feedback": {"showanswerstate": true, "advicethreshold": 0, "showactualmark": true, "allowrevealanswer": true, "showtotalmark": true}, "shuffleQuestions": false, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Transportation problem - apply minimum cost algorithm", "extensions": ["optimisation"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [{"variables": ["num_sources", "num_destinations", "total"], "name": "Setup"}, {"variables": ["supply", "demand", "costs"], "name": "Initial data"}, {"variables": ["assignment", "assignment_cost"], "name": "Minimum costs"}], "variables": {"costs": {"templateType": "anything", "group": "Initial data", "definition": "matrix(repeat(\n  repeat(\n    random(1..9),\n    num_destinations\n  ),\n  num_sources\n))", "description": "", "name": "costs"}, "num_sources": {"templateType": "anything", "group": "Setup", "definition": "random(3..5)", "description": "", "name": "num_sources"}, "num_destinations": {"templateType": "anything", "group": "Setup", "definition": "7-num_sources+random(0,1)", "description": "", "name": "num_destinations"}, "assignment": {"templateType": "anything", "group": "Minimum costs", "definition": "minimum_cost(supply,demand,costs)", "description": "", "name": "assignment"}, "assignment_cost": {"templateType": "anything", "group": "Minimum costs", "definition": "assignment_cost(assignment,costs)", "description": "", "name": "assignment_cost"}, "total": {"templateType": "anything", "group": "Setup", "definition": "random(50..500)", "description": "", "name": "total"}, "supply": {"templateType": "anything", "group": "Initial data", "definition": "random_partition(total,num_sources)", "description": "", "name": "supply"}, "demand": {"templateType": "anything", "group": "Initial data", "definition": "random_partition(total,num_destinations)", "description": "", "name": "demand"}}, "ungrouped_variables": [], "rulesets": {}, "functions": {"show_variable_list": {"type": "string", "language": "javascript", "definition": "return data.map(function(n,i) {\n  return '$'+name+'_{'+(i+1)+'} = '+Numbas.math.niceNumber(n)+'$';\n}).join(', ');", "parameters": [["data", "list"], ["name", "string"]]}}, "showQuestionGroupNames": false, "parts": [{"showCorrectAnswer": true, "markPerCell": false, "allowFractions": false, "correctAnswer": "assignment", "allowResize": false, "prompt": "<p>Apply the minimum cost algorithm to find an assignment of units of coal to be&nbsp;transported between pits and power stations.</p>\n<p>When there's more than one choice of cell, choose the cell with the maximum available allocation.</p>", "correctAnswerFractions": false, "numRows": "num_sources", "scripts": {}, "type": "matrix", "numColumns": "num_destinations", "tolerance": 0, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1}, {"showCorrectAnswer": true, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "minValue": "assignment_cost", "prompt": "<p>What is the total cost of this solution?</p>", "showPrecisionHint": false, "scripts": {}, "type": "numberentry", "correctAnswerFraction": false, "variableReplacements": [{"part": "p0", "must_go_first": false, "variable": "assignment"}], "marks": 1, "maxValue": "assignment_cost"}], "variablesTest": {"condition": "", "maxRuns": "3000"}, "statement": "<p>Four power stations labelled $j = 1,2,3,4$ require {show_variable_list(demand,'b')} units of coal, respectively. These are to be provided by three pits labelled $i=1,2,3$, which have supplies of {show_variable_list(supply,'a')} units of coal, respectively.</p>\n<p>The cost $c_{ij}$ of transporting one unit of coal from pit $i$ to power station $j$ is in the following table (with supplies and demand shown in the margin).</p>\n<p>{cost_table(supply,demand,costs)}</p>", "tags": [], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": ""}, "advice": "<h4>a)</h4>\n<p>The following tables show each step of the minimum cost algorithm:</p>\n<p>{minimum_cost_display(supply,demand,costs)}</p>\n<h4>b)</h4>\n<p>The total cost of this solution&nbsp;is $\\var{show_cost_calculation(assignment,costs)}$.</p>"}, {"name": "Transportation problem - apply minimum cost algorithm to unbalanced problem", "extensions": ["optimisation"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [{"variables": ["num_sources", "num_destinations", "total", "omit_source", "show_sources", "show_destinations"], "name": "Setup"}, {"variables": ["supply", "demand", "costs", "show_supply", "show_demand", "show_costs", "dummy_resource"], "name": "Initial data"}, {"variables": ["assignment", "assignment_cost", "show_assignment"], "name": "Minimum costs"}], "variables": {"show_assignment": {"group": "Minimum costs", "templateType": "anything", "definition": "matrix(map(\n  map(\n    assignment[x][y],\n    y,\n    0..show_destinations-1\n  ),\n  x,\n  0..show_sources-1\n))", "description": "", "name": "show_assignment"}, "costs": {"group": "Initial data", "templateType": "anything", "definition": "matrix(map(\n  map(\n    if(x<show_sources and y<show_destinations,\n        random(1..9),\n        0\n    ),\n    y,\n    0..num_destinations-1\n  ),\n  x,\n  0..num_sources-1\n))", "description": "", "name": "costs"}, "show_sources": {"group": "Setup", "templateType": "anything", "definition": "num_sources-if(omit_source,1,0)", "description": "", "name": "show_sources"}, "omit_source": {"group": "Setup", "templateType": "anything", "definition": "random(true,false)", "description": "", "name": "omit_source"}, "num_destinations": {"group": "Setup", "templateType": "anything", "definition": "4", "description": "", "name": "num_destinations"}, "show_supply": {"group": "Initial data", "templateType": "anything", "definition": "supply[0..show_sources]", "description": "", "name": "show_supply"}, "supply": {"group": "Initial data", "templateType": "anything", "definition": "random_partition(total,num_sources)", "description": "", "name": "supply"}, "show_destinations": {"group": "Setup", "templateType": "anything", "definition": "num_destinations-if(omit_source,0,1)", "description": "", "name": "show_destinations"}, "dummy_resource": {"group": "Initial data", "templateType": "anything", "definition": "if(omit_source,\n  sum(supply)-sum(show_supply),\n  sum(demand)-sum(show_demand)\n)", "description": "", "name": "dummy_resource"}, "num_sources": {"group": "Setup", "templateType": "anything", "definition": "random(3..5)", "description": "", "name": "num_sources"}, "show_demand": {"group": "Initial data", "templateType": "anything", "definition": "demand[0..show_destinations]", "description": "", "name": "show_demand"}, "show_costs": {"group": "Initial data", "templateType": "anything", "definition": "matrix(map(map(costs[x][y],y,0..show_destinations-1),x,0..show_sources-1))", "description": "", "name": "show_costs"}, "assignment_cost": {"group": "Minimum costs", "templateType": "anything", "definition": "assignment_cost(assignment,costs)", "description": "", "name": "assignment_cost"}, "total": {"group": "Setup", "templateType": "anything", "definition": "random(50..500)", "description": "", "name": "total"}, "assignment": {"group": "Minimum costs", "templateType": "anything", "definition": "minimum_cost(supply,demand,costs)", "description": "", "name": "assignment"}, "demand": {"group": "Initial data", "templateType": "anything", "definition": "random_partition(total,num_destinations)", "description": "", "name": "demand"}}, "ungrouped_variables": [], "functions": {"show_variable_list": {"type": "string", "language": "javascript", "definition": "return data.map(function(n,i) {\n  return '$'+name+'_{'+(i+1)+'} = '+Numbas.math.niceNumber(n)+'$';\n}).join(', ');", "parameters": [["data", "list"], ["name", "string"]]}}, "parts": [{"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "<p>By adding dummy sources or destinations, balance this problem.</p>\n<p>The balanced problem adds&nbsp;another&nbsp;[[0]] with&nbsp;[[1]] units of coal.</p>\n<p>Enter the cost matrix for the balanced problem below.</p>\n<p>[[2]]</p>", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"displayType": "radiogroup", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "matrix": "if(omit_source,[1,0],[0,1])", "choices": ["<p>pit</p>", "<p>power station</p>"], "unitTests": [], "variableReplacements": [], "shuffleChoices": false, "showFeedbackIcon": true, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "variableReplacementStrategy": "originalfirst", "displayColumns": 0, "marks": 0}, {"showCorrectAnswer": true, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "dummy_resource", "maxValue": "dummy_resource", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "variableReplacements": [], "marks": 1, "mustBeReducedPC": 0}, {"showCorrectAnswer": true, "markPerCell": false, "allowFractions": false, "correctAnswer": "costs", "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "allowResize": true, "unitTests": [], "correctAnswerFractions": false, "numRows": 1, "scripts": {}, "type": "matrix", "numColumns": 1, "tolerance": 0, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showFeedbackIcon": true}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "sortAnswers": false}, {"scripts": {}, "showCorrectAnswer": true, "markPerCell": false, "allowFractions": false, "correctAnswer": "show_assignment", "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "prompt": "<p>Apply the minimum cost algorithm to your balanced problem to find an assignment of units of coal to be&nbsp;transported between pits and power stations.</p>\n<p>When there's more than one choice of cell, choose the cell with the maximum available allocation.</p>", "unitTests": [], "correctAnswerFractions": false, "numRows": "show_sources", "allowResize": false, "type": "matrix", "numColumns": "show_destinations", "tolerance": 0, "variableReplacementStrategy": "originalfirst", "variableReplacements": [{"part": "p0g2", "must_go_first": false, "variable": "costs"}], "marks": 1, "showFeedbackIcon": true}, {"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "<p>What is the total cost of this solution?</p>\n<p>&pound;&nbsp;[[0]]</p>", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "assignment_cost", "maxValue": "assignment_cost", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "variableReplacements": [{"part": "p1", "must_go_first": false, "variable": "assignment"}], "marks": 1, "mustBeReducedPC": 0}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "sortAnswers": false}], "statement": "<p>Four power stations labelled $j = 1,2,3,4$ require {show_variable_list(show_demand,'b')} units of coal, respectively. These are to be provided by three pits labelled $i=1,2,3$, which have supplies of {show_variable_list(show_supply,'a')} units of coal, respectively.</p>\n<p>The cost $c_{ij}$, in pounds, of transporting one unit of coal from pit $i$ to power station $j$ is in the following table (with supplies and demand shown in the margin).</p>\n<p>{cost_table(show_supply,show_demand,show_costs)}</p>", "tags": [], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Although the statement has 4 power stations and 3 pits, when the question is run sometimes 3 power stations are given and sometimes 4 pits.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<h4>a)</h4>\n<p>The total supply is {sum(show_supply)}, while the total demand is {sum(show_demand)}, so we need to add a dummy {if(omit_source,'pit supplying','power station demanding')} {dummy_resource} units of coal.</p>\n<p>The cost matrix&nbsp;then looks like this:</p>\n<p>{cost_table(supply,demand,costs)}</p>\n<h4>b)</h4>\n<p>The following tables show each step of the minimum cost algorithm:</p>\n<p>{minimum_cost_display(supply,demand,costs)}</p>\n<h4>c)</h4>\n<p>The total cost of this solution&nbsp;is $\\var{show_cost_calculation(assignment,costs)}$.</p>"}, {"name": "Transportation problem - stepping stones", "extensions": ["optimisation"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [{"variables": ["num_sources", "num_destinations", "total"], "name": "Setup"}, {"variables": ["supply", "demand", "costs"], "name": "Initial data"}, {"variables": ["initial_assignment", "initial_cost"], "name": "Minimum costs"}, {"variables": ["final_assignment", "final_is_optimal", "final_cost"], "name": "Stepping stone"}], "variables": {"final_cost": {"group": "Stepping stone", "templateType": "anything", "definition": "assignment_cost(final_assignment,costs)", "description": "", "name": "final_cost"}, "costs": {"group": "Initial data", "templateType": "anything", "definition": "matrix(repeat(\n  repeat(\n    random(1..9),\n    num_destinations\n  ),\n  num_sources\n))", "description": "", "name": "costs"}, "num_sources": {"group": "Setup", "templateType": "anything", "definition": "3", "description": "", "name": "num_sources"}, "final_is_optimal": {"group": "Stepping stone", "templateType": "anything", "definition": "assignment_is_optimal(final_assignment,costs)", "description": "", "name": "final_is_optimal"}, "initial_assignment": {"group": "Minimum costs", "templateType": "anything", "definition": "minimum_cost(supply,demand,costs)", "description": "", "name": "initial_assignment"}, "total": {"group": "Setup", "templateType": "anything", "definition": "random(50..500)", "description": "", "name": "total"}, "num_destinations": {"group": "Setup", "templateType": "anything", "definition": "4", "description": "", "name": "num_destinations"}, "initial_cost": {"group": "Minimum costs", "templateType": "anything", "definition": "assignment_cost(initial_assignment,costs)", "description": "", "name": "initial_cost"}, "final_assignment": {"group": "Stepping stone", "templateType": "anything", "definition": "stepping_stone(initial_assignment,costs)", "description": "", "name": "final_assignment"}, "supply": {"group": "Initial data", "templateType": "anything", "definition": "random_partition(total,num_sources)", "description": "", "name": "supply"}, "demand": {"group": "Initial data", "templateType": "anything", "definition": "random_partition(total,num_destinations)", "description": "", "name": "demand"}}, "ungrouped_variables": [], "functions": {"show_variable_list": {"type": "string", "language": "javascript", "definition": "return data.map(function(n,i) {\n  return '$'+name+'_{'+(i+1)+'} = '+Numbas.math.niceNumber(n)+'$';\n}).join(', ');", "parameters": [["data", "list"], ["name", "string"]]}}, "parts": [{"prompt": "<p>What is the cost of this initial assignment?</p>\n<p>&pound;&nbsp;[[0]]</p>", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "useCustomName": false, "customName": "", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"correctAnswerFraction": false, "customName": "", "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "initial_cost", "useCustomName": false, "maxValue": "initial_cost", "unitTests": [], "showFractionHint": true, "correctAnswerStyle": "plain", "showFeedbackIcon": true, "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "mustBeReducedPC": 0}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}, {"prompt": "<p>Use the stepping stones method to find an optimal solution, and enter it below.</p>", "markPerCell": false, "useCustomName": false, "customName": "", "allowFractions": false, "correctAnswer": "final_assignment", "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "constraints_unsatisfied:\n  assert(assignment_is_valid(studentmatrix,supply,demand),\n    incorrect(\"Your answer does not satisfy one of the constraints of supply or demand.\");\n    end();\n    true\n  )\n\nis_optimal:\n  if(assignment_is_optimal(studentmatrix, costs),\n    correct(\"Your answer is optimal.\")\n  ,\n    incorrect(\"Your answer is not optimal.\")\n  )\n\nmark:\n  apply(any_empty);\n  apply(any_invalid);\n  apply(constraints_unsatisfied);\n  apply(is_optimal)", "allowResize": false, "unitTests": [], "correctAnswerFractions": false, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "type": "matrix", "numColumns": "num_destinations", "tolerance": 0, "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "numRows": "num_sources"}, {"prompt": "<p>What is the cost of the optimal solution?</p>\n<p>&pound;&nbsp;[[0]]</p>", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "useCustomName": false, "customName": "", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"correctAnswerFraction": false, "customName": "", "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "final_cost", "useCustomName": false, "maxValue": "final_cost", "unitTests": [], "showFractionHint": true, "correctAnswerStyle": "plain", "showFeedbackIcon": true, "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "mustBeReducedPC": 0}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}], "variablesTest": {"condition": "not assignment_is_optimal(initial_assignment,costs)", "maxRuns": 100}, "statement": "<p>Four power stations labelled $j = 1,2,3,4$ require {show_variable_list(demand,'b')} units of coal, respectively. These are to be provided by three pits labelled $i=1,2,3$, which have supplies of {show_variable_list(supply,'a')} units of coal, respectively.</p>\n<p>The cost $c_{ij}$, in pounds, of transporting one unit of coal from pit $i$ to power station $j$ is in the following table (with supplies and demand shown in the margin).</p>\n<p>{cost_table(supply,demand,costs)}</p>\n<p>The NW corner algorithm has been used to come up with an initial feasible solution, shown below.</p>\n<p>{assignment_table(initial_assignment,supply,demand)}</p>", "tags": [], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": ""}, "advice": "<h4>a)</h4>\n<p>The cost of the initial solution&nbsp;is $\\var{show_cost_calculation(initial_assignment,costs)}$.</p>\n<h4>b)</h4>\n<p>The steps of the method are shown below.</p>\n<p>{stepping_stone_display(initial_assignment,costs)}</p>\n<p>At this point, no cells have shadow costs greater than their real costs, so the solution is optimal.</p>\n<h4>c)</h4>\n<p>The cost of the optimal solution is $\\var{show_cost_calculation(final_assignment,costs)}$.</p>"}], "name": "", "pickQuestions": 0}], "navigation": {"onleave": {"message": "", "action": "none"}, "browse": true, "reverse": true, "preventleave": true, "showresultspage": "oncompletion", "allowregen": true, "showfrontpage": true}, "type": "exam", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": ""}, "extensions": ["optimisation"], "timing": {"timedwarning": {"message": "", "action": "none"}, "timeout": {"message": "", "action": "none"}, "allowPause": true}, "pickQuestions": 0, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "custom_part_types": []}