// Numbas version: exam_results_page_options
{"name": "Sensitivity analysis of a linear programming problem", "extensions": ["jsxgraph", "optimisation"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [{"variables": [], "name": "Testing"}, {"variables": ["product_baseline", "x_bound", "y_bound", "resource_baseline", "profit_baseline"], "name": "Order of magnitude"}, {"variables": ["x_intercept_a", "y_intercept_a"], "name": "Constraint setup"}, {"variables": ["minimum_x", "minimum_y", "x_resource", "y_resource", "total_resource", "profit_x", "profit_y"], "name": "Final constraints"}, {"variables": ["x_intercept", "y_intercept", "inequality_coordinates"], "name": "Diagram data"}, {"variables": ["program", "best", "line_is_binding", "best_coords", "binding_lines"], "name": "Optimal solution"}, {"variables": ["line_names", "change_description", "line_prepositions"], "name": "Text"}, {"variables": ["nonbinding_line", "change", "changing_affects_solution"], "name": "Relax or tighten"}], "variables": {"resource_baseline": {"templateType": "anything", "group": "Order of magnitude", "definition": "repeat(\n  10^random(0..1)*random(1..9)*y_bound,\n  2\n)", "description": "", "name": "resource_baseline"}, "profit_baseline": {"templateType": "anything", "group": "Order of magnitude", "definition": "10^random(1,1,1,2,2,3)", "description": "", "name": "profit_baseline"}, "binding_lines": {"templateType": "anything", "group": "Optimal solution", "definition": "filter(line_is_binding[j],j,0..3)", "description": "", "name": "binding_lines"}, "line_is_binding": {"templateType": "anything", "group": "Optimal solution", "definition": "binding_lines(program)", "description": "", "name": "line_is_binding"}, "changing_affects_solution": {"templateType": "anything", "group": "Relax or tighten", "definition": "[\n  best_coords[0]*x_resource[0]+best_coords[1]*y_resource[0]>total_resource[0]*change-0.0001,\n  best_coords[0]*x_resource[1]+best_coords[1]*y_resource[1]>total_resource[1]*change-0.0001,\n  best_coords[0]<minimum_x*change+0.0001,\n  best_coords[1]<minimum_y*change+0.0001\n][nonbinding_line]\n", "description": "", "name": "changing_affects_solution"}, "best": {"templateType": "anything", "group": "Optimal solution", "definition": "best_point(program)", "description": "", "name": "best"}, "x_intercept_a": {"templateType": "anything", "group": "Constraint setup", "definition": "repeat(\n  random(minimum_x + 0.1*x_bound .. x_bound # 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0.01)*y_bound)", "description": "", "name": "minimum_y"}, "inequality_coordinates": {"templateType": "anything", "group": "Diagram data", "definition": "[\n  [-x_intercept[0],1,x_intercept[0]/y_intercept[0]],\n  [-x_intercept[1],1,x_intercept[1]/y_intercept[1]]\n]", "description": "", "name": "inequality_coordinates"}, "line_names": {"templateType": "list of strings", "group": "Text", "definition": "[ \"availability of resource 1\", \"availability of resource 2\", \"minimum demand for product X\", \"minimum demand for product Y\" ]", "description": "", "name": "line_names"}, "x_intercept": {"templateType": "anything", "group": "Diagram data", "definition": "[\ntotal_resource[0]/x_resource[0],\ntotal_resource[1]/x_resource[1]\n]", "description": "", "name": "x_intercept"}, "best_coords": {"templateType": "anything", "group": "Optimal solution", "definition": "best_coords(program)", "description": "", "name": "best_coords"}, "x_resource": {"templateType": "anything", "group": "Final constraints", "definition": "[\nceil((total_resource[0]-y_intercept_a[0]*y_resource[0])/minimum_x),\nceil((total_resource[1]-y_intercept_a[1]*y_resource[1])/minimum_x)\n]", "description": "", "name": "x_resource"}, "program": {"templateType": "anything", "group": "Optimal solution", "definition": "[minimum_x,minimum_y,x_resource,y_resource,total_resource[0],total_resource[1],profit_x,profit_y]", "description": "", "name": "program"}, "y_resource": {"templateType": "anything", "group": "Final constraints", "definition": "[\nfloor((minimum_x/x_intercept_a[0]-1)*total_resource[0]/(minimum_x*minimum_y/x_intercept_a[0]-y_intercept_a[0])),\nfloor((minimum_x/x_intercept_a[1]-1)*total_resource[1]/(minimum_x*minimum_y/x_intercept_a[1]-y_intercept_a[1]))\n]", "description": "", "name": "y_resource"}, "total_resource": {"templateType": "anything", "group": "Final constraints", "definition": "[\nround(random(resource_baseline[0]*1.1 .. resource_baseline[0]*2 # 1)),\nround(random(resource_baseline[1]*1.1 .. resource_baseline[1]*2 # 1))\n]", "description": "", "name": "total_resource"}, "y_intercept_a": {"templateType": "anything", "group": "Constraint setup", "definition": "repeat(\n  random(minimum_y + 0.1*y_bound .. y_bound # 0.01*y_bound),\n  2\n)", "description": "", "name": "y_intercept_a"}, "product_baseline": {"templateType": "anything", "group": "Order of magnitude", "definition": "10^random(1..3)*random(2..8)", "description": "", "name": "product_baseline"}, "minimum_x": {"templateType": "anything", "group": "Final constraints", "definition": "ceil(random(0.1 .. 0.2 # 0.01)*x_bound)", "description": "", "name": "minimum_x"}, "x_bound": {"templateType": "anything", "group": "Order of magnitude", "definition": "round(random(0.5..1#0.01)*product_baseline)", "description": "", "name": "x_bound"}, "change": {"templateType": "anything", "group": "Relax or tighten", "definition": "random(2,1/2)", "description": "", "name": "change"}, "line_prepositions": {"templateType": "anything", "group": "Text", "definition": "[\n  ['below','below','to the right of','above'],\n  ['above','above','to the left of','below']\n]", "description": "", "name": "line_prepositions"}}, "ungrouped_variables": [], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Sensitivity analysis of a linear programming problem", "functions": {}, "showQuestionGroupNames": false, "parts": [{"prompt": "<p><span class=\"lp-graph\">Graph goes here</span></p>", "scripts": {}, "type": "information", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"displayType": "radiogroup", "minMarks": 0, "layout": {"type": "all", "expression": ""}, "choices": ["{capitalise(line_names[0])}", "{capitalise(line_names[1])}", "{capitalise(line_names[2])}", "{capitalise(line_names[3])}"], "showCorrectAnswer": true, "matrix": "map(if(binding,[1,0],[0,1]),binding,line_is_binding)", "prompt": "<p>Which of the constraints are binding?</p>", "type": "m_n_x", "maxAnswers": 0, "shuffleChoices": false, "warningType": "none", "scripts": {}, "marks": 0, "minAnswers": 0, "maxMarks": 0, "shuffleAnswers": false, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "answers": ["Binding", "Non-binding"]}, {"displayType": "radiogroup", "choices": ["<p>Yes</p>", "<p>No</p>"], "showCorrectAnswer": true, "matrix": "if(changing_affects_solution,[1,0],[0,1])", "prompt": "<p>Suppose the {line_names[nonbinding_line]} {change_description}. Would that affect the optimal solution?</p>", "distractors": ["", ""], "shuffleChoices": false, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "displayColumns": 0, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}], "statement": "<p>A factory&nbsp;manufractures two products, X and Y. Existing demand&nbsp;means the factory must produce at least <strong>{minimum_x}</strong> units of product X and <strong>{minimum_y}</strong> units of product Y.</p>\n<p>A unit of X requires <strong>{x_resource[0]}</strong> units of resource 1 and <strong>{x_resource[1]}</strong> units of resource 2.</p>\n<p>A unit of Y requires <strong>{y_resource[0]}</strong> units of resource 1 and <strong>{y_resource[1]}</strong> units of resource 2.</p>\n<p>Each unit of product X sold earns <strong>&pound;{profit_x}</strong> profit, and each unit of product Y sold earns <strong>&pound;{profit_y}</strong> profit.</p>", "tags": ["linear programming", "MAS1901", "sensitivity analysis"], "rulesets": {}, "preamble": {"css": "", "js": "var optimisation = Numbas.extensions.optimisation;\n\nquestion.onHTMLAttached(function() {\n  with(question.unwrappedVariables) {\n    var result = optimisation.linear_programming_board({\n      x_bound: x_bound,\n      y_bound: y_bound,\n      objective_line_coordinates: [-(minimum_x*profit_x + minimum_y*profit_y),profit_x,profit_y],\n      minimum_x: minimum_x,\n      minimum_y: minimum_y,\n      inequality_1_coordinates: inequality_coordinates[0],\n      inequality_2_coordinates: inequality_coordinates[1],\n      coord_point: false\n    });\n    result.inequality_line_1.setAttribute({fixed:false, withLabel:true, name: 'Resource 1'});\n    result.inequality_line_2.setAttribute({fixed:false, withLabel:true, name: 'Resource 2'});\n    result.x_constraint_line.setAttribute({fixed:false});\n    result.y_constraint_line.setAttribute({fixed:false});\n    \n    var board = result.board;\n    \n    var shadow_inequality_line_1 = board.create('line',inequality_coordinates[0],{withLabel:false,color:'lightgray', highlight: false, fixed: true});\n    var shadow_inequality_line_2 = board.create('line',inequality_coordinates[1],{withLabel:false,color:'lightgray', highlight: false, fixed: true});\n    var shadow_x_constraint_line = board.create('line',[[minimum_x,0],[minimum_x,1]],{withLabel:false,color:'lightgray', highlight: false, fixed: true});\n    var shadow_y_constraint_line = board.create('line',[[0,minimum_y],[1,minimum_y]],{withLabel:false,color:'lightgray', highlight: false, fixed: true});\n    shadow_inequality_line_1.visProp.layer = 1;\n    shadow_inequality_line_2.visProp.layer = 1;\n    shadow_x_constraint_line.visProp.layer = 1;\n    shadow_y_constraint_line.visProp.layer = 1;\n    \n    board.update();\n    \n    window.result = result;\n\n    $(question.display.html).find('.lp-graph').html('').append(result.div);\n    \n  }\n});"}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Student is shown a linear programming problem, with graphical solution.&nbsp;</p>\n<p>Asked to identify binding constraints, and decide if the optimal solution is changed by a named constraint either doubling or halving.</p>"}, "variablesTest": {"condition": "", "maxRuns": "100"}, "advice": "<h4>a)</h4>\n<p>The optimal solution is found at the intersection of the lines representing {line_names[binding_lines[0]]} and {line_names[binding_lines[1]]}. These constraints are binding; the other constraints are non-binding.</p>\n<h4>b)</h4>\n<p>If&nbsp;the {line_names[nonbinding_line]} {change_description}, the optimal solution found earlier <span data-jme-visible=\"changing_affects_solution\">ends up {line_prepositions[1][nonbinding_line]} the line representing the {line_names[nonbinding_line]}, making it invalid</span><span data-jme-visible=\"not changing_affects_solution\">is still {line_prepositions[0][nonbinding_line]}&nbsp;the line representing the {line_names[nonbinding_line]}, so the optimal solution is not affected</span>.</p>", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}