// Numbas version: exam_results_page_options
{"questions": [], "duration": 0, "name": "Linear programming", "showQuestionGroupNames": false, "allQuestions": true, "percentPass": 0, "feedback": {"showanswerstate": true, "advicethreshold": 0, "showactualmark": true, "allowrevealanswer": true, "showtotalmark": true}, "shuffleQuestions": false, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Sensitivity analysis of a linear programming problem", "extensions": ["jsxgraph", "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": [], "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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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>"}, {"name": "Solve a linear programming problem", "extensions": ["jsxgraph", "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": "Chris Graham", "profile_url": 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"variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "mustBeReducedPC": 0}, {"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "yconstraint", "maxValue": "yconstraint", "unitTests": [], "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", "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}, {"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "<p>Now enter&nbsp;the constraints given by the availability of {resource_1}&nbsp;and {resource_2}:</p>\n<p>{capitalise(resource_1)}: [[0]]&nbsp;$x + $&nbsp;[[1]]&nbsp;$y \\le $&nbsp;[[2]]</p>\n<p>{capitalise(resource_2)}: [[3]]&nbsp;$x + $&nbsp;[[4]]&nbsp;$y \\le $&nbsp;[[5]]</p>", "unitTests": [], "sortAnswers": false, "scripts": {}, "gaps": [{"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "resource1x", "maxValue": "resource1x", "unitTests": [], "correctAnswerStyle": "plain", "showFeedbackIcon": true, "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "mustBeReducedPC": 0}, {"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "resource1y", "maxValue": "resource1y", "unitTests": [], "correctAnswerStyle": "plain", "showFeedbackIcon": true, "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "mustBeReducedPC": 0}, {"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "totalresource1", "maxValue": "totalresource1", "unitTests": [], "correctAnswerStyle": "plain", "showFeedbackIcon": true, "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "mustBeReducedPC": 0}, {"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "resource2x", "maxValue": "resource2x", "unitTests": [], "correctAnswerStyle": "plain", "showFeedbackIcon": true, "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "mustBeReducedPC": 0}, {"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "resource2y", "maxValue": "resource2y", "unitTests": [], "correctAnswerStyle": "plain", "showFeedbackIcon": true, "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "mustBeReducedPC": 0}, {"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "totalresource2", "maxValue": "totalresource2", "unitTests": [], "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", "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}, {"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "<p>Given the information on profits for {product_1s} and {product_2s}, write down&nbsp;the objective function which is to be maximised:&nbsp;[[0]]</p>", "unitTests": [], "sortAnswers": false, "scripts": {}, "gaps": [{"answer": "{profitx}*x+{profity}*y", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "checkingType": "absdiff", "vsetRangePoints": 5, "failureRate": 1, "checkVariableNames": false, "unitTests": [], "vsetRange": [0, 1], "type": "jme", "showFeedbackIcon": true, "scripts": {}, "answerSimplification": "all", "expectedVariableNames": [], "showPreview": true, "checkingAccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}, {"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "<p><span class=\"lp-graph\">Input diagram goes here</span></p>\n<p>Drag the objective line on the diagram above to find the position which gives the maximum profit.</p>\n<p>Round your answers down to the nearest integer, end enter them below:</p>\n<p>Produce&nbsp;[[0]] {product_1s} and&nbsp;[[1]] {product_2s}, earning &pound;&nbsp;[[2]]&nbsp;in profit.</p>", "unitTests": [], "sortAnswers": false, "scripts": {}, "gaps": [{"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", 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false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "best_profit", "maxValue": "best_profit", "unitTests": [], "correctAnswerStyle": "plain", "showFeedbackIcon": true, "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [{"part": "p3g0", "must_go_first": false, "variable": "best_x"}, {"part": "p3g1", "must_go_first": false, "variable": "best_y"}], "marks": 1, "mustBeReducedPC": 0}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}], "variables": {"resource_1_units": {"templateType": "string", "group": "Text strings", "definition": "\"kg\"", "name": "resource_1_units", "description": "<p>Dimension of the first resource</p>"}, "resource1y": {"templateType": "anything", "group": "Constraints", "definition": "random(50..70)", "name": "resource1y", "description": "<p>How much of resource 1 the second product uses per unit</p>"}, "product_1": {"templateType": "string", "group": "Text strings", "definition": "\"bulletproof tea cosy\"", "name": "product_1", "description": "<p>Singular name of the first product</p>"}, "resource2x": {"templateType": "anything", "group": "Constraints", "definition": "\nrandom(250..450#10)", "name": "resource2x", "description": "<p>How much of resource 2 the first product uses per unit</p>"}, "business_long": {"templateType": "string", "group": "Text strings", "definition": "\"Amalgamated Mines and Sheep, Inc.\"", "name": "business_long", "description": "<p>The full name of the business</p>"}, "profits": {"templateType": "anything", "group": "Optimum strategy", "definition": "map(let(x,p[0],y,p[1],profitx*x+profity*y),p,intersection_points)", "name": "profits", "description": "<p>Profit earned at each intersection point</p>"}, "best_profit": {"templateType": "anything", "group": "Optimum strategy", "definition": "profit(best_x,best_y)", "name": "best_profit", "description": "<p>The profit at the best intersection point</p>"}, "product_2": {"templateType": "string", "group": "Text strings", "definition": "\"double-edged cardigan\"", "name": "product_2", "description": "<p>Singular name of the second product</p>"}, "constraint2xvalue": {"templateType": "anything", "group": "Bounds", "definition": "round(scalingfactor2*resource2y+xconstraint)", "name": "constraint2xvalue", "description": "<p>Where the constraint line for resource 2 cuts the y-constraint</p>"}, "r1": {"templateType": "anything", "group": "Constraints", "definition": "random(10..20)", "name": "r1", "description": "<p>The second constraint line cuts the x-constraint line at this value more than the y-constraint</p>"}, "constraint1yvalue": {"templateType": "anything", "group": "Bounds", "definition": "r2+constraint2yvalue", "name": "constraint1yvalue", "description": "<p>Where line given by resource 1 constraint cuts the x-constraint line.</p>\n<p>It is always greater than the point at which the second resource line cuts.</p>"}, "totalresource1": {"templateType": "anything", "group": "Constraints", "definition": "siground(\n  scalingfactor1*resource1x*resource1y +\n  resource1y*yconstraint +\n  resource1x*xconstraint\n,3)", "name": "totalresource1", "description": "<p>Total amount of resource 1 available</p>"}, "business_short": {"templateType": "string", "group": "Text strings", "definition": "\"AMS\"", "name": "business_short", "description": "<p>A shorter name for the business, to be repeated throughout the text. If the business name is short enough for repeated use, leave this empty.</p>"}, "resource_2": {"templateType": "string", "group": "Text strings", "definition": "\"iron\"", "name": "resource_2", "description": "<p>The name of the second resource</p>"}, "best_x": {"templateType": "anything", "group": "Optimum strategy", "definition": "floor(best_x_float)", "name": "best_x", "description": "<p>The integer number of units of the first product to produce to earn the highest profit</p>"}, "intersection_points": {"templateType": "anything", "group": "Optimum strategy", "definition": "[\n  [xconstraint, (totalresource1-resource1x*xconstraint)/resource1y],  //resource 1 intersecting with minimum x\n  [xconstraint, (totalresource2-resource2x*xconstraint)/resource2y],  //resource 2 intersecting with minimum x\n  [(totalresource1-resource1y*yconstraint)/resource1x, yconstraint],  //resource 1 intersecting with minimum y\n  [(totalresource2-resource2y*yconstraint)/resource2x, yconstraint],  //resource 2 intersecting with minimum y\n  let(y, (totalresource1-totalresource2*resource1x/resource2x)/(resource1y - resource1x/resource2x * resource2y),\n    [(totalresource1-resource1y*y)/resource1x, y]  // intersection of the two resource constraints\n  )\n]", "name": "intersection_points", "description": "<p>Intersection points of the constraints - the optimum strategy will be one of these</p>"}, "business": {"templateType": "anything", "group": "Text strings", "definition": "if(business_short=\"\",business_long,business_short)", "name": "business", "description": ""}, "boundy": {"templateType": "anything", "group": "Bounds", "definition": "max(constraint1yvalue,constraint2yvalue)", "name": "boundy", "description": "<p>Upper&nbsp;bound of the diagram</p>"}, "product_2s": {"templateType": "string", "group": "Text strings", "definition": "\"double-edged cardigans\"", "name": "product_2s", "description": "<p>Plural name of the second product</p>"}, "constraint1xvalue": {"templateType": "anything", "group": "Bounds", "definition": "round(scalingfactor1*resource1y+xconstraint)", "name": "constraint1xvalue", "description": "<p>Where the constraint line for resource 1 crosses the y-constraint</p>"}, "product_1s": {"templateType": "string", "group": "Text strings", "definition": "\"bulletproof tea cosies\"", "name": "product_1s", "description": "<p>Plural name of the first product</p>"}, "xconstraint": {"templateType": "anything", "group": "Constraints", "definition": "\nrandom(4..10)", "name": "xconstraint", "description": "<p>Minimum production of&nbsp;the first product</p>"}, "profitx": {"templateType": "anything", "group": "Profit", "definition": "//the coeff of x in the objective function\nrandom(2..4)", "name": "profitx", "description": "<p>Profit for each unit of the first product sold</p>"}, "best_y_float": {"templateType": "anything", "group": "Optimum strategy", "definition": "intersection_points[best_point][1]", "name": "best_y_float", "description": "<p>The number of units of the second product to produce to earn the highest profit, if fractions are allowed</p>"}, "possible_points": {"templateType": "anything", "group": "Optimum strategy", "definition": "filter(\n  let(x,floor(intersection_points[j][0]),y,floor(intersection_points[j][1]),\n    resource1x*x+resource1y*y <= totalresource1+1 and \n    resource2x*x+resource2y*y <= totalresource2+1 and\n    x>=xconstraint and\n    y>=yconstraint\n  ),\n  j,\n  0..len(intersection_points)-1\n)", "name": "possible_points", "description": "<p>Indices of intersection points which satisfy all the constraints.</p>"}, "line_names": {"templateType": "list of strings", "group": "Optimum strategy", "definition": "[ \"the amount of resource 1 used\", \"the amount of resource 2 used\", \"the minimum amount of X produced\", \"the maximum amount of Y produced\" ]", "name": "line_names", "description": "<p>Names of the lines, coded as in the definition of <code>best_lines</code></p>"}, "resource1x": {"templateType": "anything", "group": "Constraints", "definition": "random(20..40)", "name": "resource1x", "description": "<p>How much of resource 1 the first product uses per unit</p>"}, "yconstraint": {"templateType": "anything", "group": "Constraints", "definition": "random(4..10)", "name": "yconstraint", "description": "<p>Minimum production of the second product</p>"}, "best_y": {"templateType": "anything", "group": "Optimum strategy", "definition": "floor(best_y_float)", "name": "best_y", "description": "<p>The integer number of units of the second&nbsp;product to produce to earn the highest profit</p>"}, "resource_1": {"templateType": "string", "group": "Text strings", "definition": "\"wool\"", "name": "resource_1", "description": "<p>The name of the first resource</p>"}, "resource2y": {"templateType": "anything", "group": "Constraints", "definition": "\nrandom(1500..3000#100)", "name": "resource2y", "description": "<p>How much of resource 2 the second product uses per unit</p>"}, "totalresource2": {"templateType": "anything", "group": "Constraints", "definition": "siground(\n  scalingfactor2*resource2x*resource2y +\n  resource2y*yconstraint +\n  resource2x*xconstraint\n,3)", "name": "totalresource2", "description": "<p>Total amount of resource 2 available</p>"}, "scalingfactor1": {"templateType": "anything", "group": "Constraints", "definition": "precround((r1+r2)/resource1x,1)", "name": "scalingfactor1", "description": ""}, "best_x_float": {"templateType": "anything", "group": "Optimum strategy", "definition": "intersection_points[best_point][0]", "name": "best_x_float", "description": "<p>The number of units of X to produce&nbsp;to earn the highest profit, if fractions are allowed</p>"}, "best_profit_float": {"templateType": "anything", "group": "Optimum strategy", "definition": "profit(best_x_float,best_y_float)", "name": "best_profit_float", "description": "<p>The profit at the best intersection point, if fractions of units are allowed</p>"}, "profity": {"templateType": "anything", "group": "Profit", "definition": "//the coeff of y in the objective function\nceil(profitx*(resource1y/resource1x)*random(1.5..3))", "name": "profity", "description": "<p>Profit for each unit of the second product sold</p>"}, "scalingfactor2": {"templateType": "anything", "group": "Constraints", "definition": "precround((r1)/resource2x,2)", "name": "scalingfactor2", "description": ""}, "r2": {"templateType": "anything", "group": "Constraints", "definition": "random(5..15)", "name": "r2", "description": "<p>The first constraint line cuts the x-constraint line at this value greater than where the second constraint cuts.</p>"}, "best_point": {"templateType": "anything", "group": "Optimum strategy", "definition": "best_by(possible_points,profits)", "name": "best_point", "description": "<p>The index of the valid intersection point with the highest profit</p>"}, "boundx": {"templateType": "anything", "group": "Bounds", "definition": "max(constraint1xvalue,constraint2xvalue)", "name": "boundx", "description": "<p>Right bound of the diagram</p>"}, "best_lines": {"templateType": "anything", "group": "Optimum strategy", "definition": "[[0,2],[1,2],[0,3],[1,3],[0,1]][best_point]", "name": "best_lines", "description": "<p>Which lines are involed in the best point? Coded as follows:</p>\n<p>0 - resource 1</p>\n<p>1 - resource 2</p>\n<p>2 - minimum x</p>\n<p>3 - minimum y</p>\n<p>Note that (minimum x,minimum y) is never the best point</p>"}, "constraint2yvalue": {"templateType": "anything", "group": "Bounds", "definition": "r1+yconstraint", "name": "constraint2yvalue", "description": "<p>Where the constraint line for the second resource cuts the x-constraint.</p>\n<p>Second resource line cuts x-constraint at a point greater than the y-constraint</p>"}, "resource_2_units": {"templateType": "string", "group": "Text strings", "definition": "\"tonnes\"", "name": "resource_2_units", "description": "<p>Dimension of the second resource</p>"}}, "ungrouped_variables": [], "variable_groups": [{"variables": ["xconstraint", "yconstraint", "resource1x", "resource1y", "resource2x", "resource2y", "r1", "r2", "scalingfactor1", "scalingfactor2", "totalresource1", "totalresource2"], "name": "Constraints"}, {"variables": ["boundx", "boundy", "constraint1xvalue", "constraint1yvalue", "constraint2xvalue", "constraint2yvalue"], "name": "Bounds"}, {"variables": ["profitx", "profity"], "name": "Profit"}, {"variables": ["intersection_points", "possible_points", "profits", "best_point", "best_x_float", "best_y_float", "best_profit_float", "best_x", "best_y", "best_profit", "best_lines", "line_names"], "name": "Optimum strategy"}, {"variables": ["business_long", "business_short", "business", "resource_1", "resource_1_units", "resource_2", "resource_2_units", "product_1", "product_1s", "product_2", "product_2s"], "name": "Text strings"}], "functions": {"profit": {"type": "number", "language": "jme", "definition": "profitx*x+profity*y", "parameters": [["x", "number"], ["y", "number"]]}, "best_by": {"type": "number", "language": "javascript", "definition": "// l1 is a list of indices on l2\n// return the element of l1 which has the biggest corresponding value in l2 \n// (i.e. i such that l2[l1[i]] is biggest)\n\nvar best_1 = null;\nvar best_2 = null;\nfor(var i=0;i<l1.length;i++) {\n  if(best_1===null || l2[l1[i]]>best_2) {\n    best_2 = l2[l1[i]];\n    best_1 = l1[i];\n  }\n}\nreturn best_1", "parameters": [["l1", "list"], ["l2", "list"]]}}, "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>{business_long}<span data-jme-visible=\"business_short&lt;&gt;&quot;&quot;\"> ({business_short})</span> makes two products: {product_1s} and {product_2s}.</p>\n<p>Existing orders require {business} to make at least <strong>{xconstraint}</strong> {product_1s} and <strong>{yconstraint}</strong> {product_2s} each day.</p>\n<p>Each unit of {product_1s} produced uses&nbsp;<strong>{resource1x}</strong> {resource_1_units} {resource_1}&nbsp;and&nbsp;<strong>{resource2x}</strong> {resource_2_units} {resource_2}.</p>\n<p>Each unit of {product_2s} produced uses&nbsp;<strong>{resource1y}</strong> {resource_1_units}&nbsp;{resource_1}&nbsp;and&nbsp;<strong>{resource2y}</strong> {resource_2_units} {resource_2}.</p>\n<p>However, the resources are limited: {business} can only obtain <strong>{totalresource1}</strong> {resource_1_units} {resource_1} and <strong>{totalresource2}</strong>&nbsp;{resource_2_units} {resource_2}&nbsp;each day.</p>\n<p>Each sale of a unit of {product_1s} gives <strong>&pound;{profitx}</strong> profit. Each sale of a unit of {product_2s} gives <strong>&pound;{profity}</strong> profit.</p>", "tags": ["constraints", "linear programming", "objective function", "template"], "rulesets": {}, "preamble": {"css": "", "js": "function userinput_diagram() {\n\nwith(question.unwrappedVariables) {\n\n  var options = {\n    x_bound: boundx, \n    y_bound: boundy\n  };\n\n  options.objective_line_coordinates = [[0,0],[1,-profitx/profity]];\n  \n  function getStudentAnswer(path) {\n    var part = question.getPart(path);\n    return ko.computed(function() {\n        return parseFloat(part.display.studentAnswer());\n    });\n  }\n  \n  options.minimum_x = getStudentAnswer('p0g0');\n  options.minimum_y = getStudentAnswer('p0g1');\n\n  var x_coefficient_1 = getStudentAnswer('p1g0');\n  var y_coefficient_1 = getStudentAnswer('p1g1');\n  var total_1 = getStudentAnswer('p1g2');\n  options.inequality_1_coordinates = [ko.computed(function() {\n    return [-total_1(),x_coefficient_1(),y_coefficient_1()];\n  })];\n\n  options.inequality_1_ok = ko.computed(function() {\n      return !(isNaN(x_coefficient_1()) || isNaN(y_coefficient_1()) || isNaN(total_1()));\n  });\n\n  var x_coefficient_2 = getStudentAnswer('p1g3');\n  var y_coefficient_2 = getStudentAnswer('p1g4');\n  var total_2 = getStudentAnswer('p1g5');\n  options.inequality_2_coordinates = [ko.computed(function() {\n    return [-total_2(),x_coefficient_2(),y_coefficient_2()];\n  })];\n\n  options.inequality_2_ok = ko.computed(function() {\n      return !(isNaN(x_coefficient_2()) || isNaN(y_coefficient_2()) || isNaN(total_2()));\n  });\n\n  var result = Numbas.extensions.optimisation.linear_programming_board(options);\n  $(question.display.html).find('.lp-graph').html('').append(result.div);\n}\n}\n\nfunction advice_diagram() {\nwith(question.unwrappedVariables) {\n\n  var result = Numbas.extensions.optimisation.linear_programming_board({\n    x_bound: boundx,\n    y_bound: boundy,\n    objective_line_coordinates: [[0,best_profit_float/profity],[1,(best_profit_float-profitx)/profity]],\n    minimum_x: xconstraint,\n    minimum_y: yconstraint,\n    inequality_1_coordinates: [-totalresource1, resource1x, resource1y],\n    inequality_2_coordinates: [-totalresource2, resource2x, resource2y]\n  });\n  $(question.display.html).find('#advicediagram').html('').append(result.div);\n}\n}\n\nquestion.signals.on('HTMLAttached',function() {\n  userinput_diagram();\n  advice_diagram();\n});\n"}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Student is given a set of constraints for a linear program. Asked to enter the constraints as inequalities, and then to identify the optimal solution.</p>\n<p>Problem with solving the simultaneous equations gven by the constraints - too unwieldy and not given enough marks for doing so. Best if the point of intersection is given graphically by putting the mouse over the intersection.</p>"}, "advice": "<h4>a)</h4>\n<p>The existing orders give two constraints:</p>\n<p>\\begin{align} <br/>x &amp;\\geq \\var{xconstraint} \\\\<br/> y &amp;\\geq \\var{yconstraint}<br/>\\end{align}</p>\n<h4>b)</h4>\n<p>The limits on the two resources, and the amounts of each resource used to produce a unit of each product, give two further constraints:</p>\n<p>\\begin{align} <br/>\\simplify{{resource1x}x + {resource1y}y} &amp;\\leq \\var{totalresource1} \\\\<br/>\\simplify{{resource2x}x + {resource2y}y} &amp;\\leq \\var{totalresource2}<br/>\\end{align}</p>\n<h4>c)</h4>\n<p>The objective function to be maximised is the total profit earned, that is, the sum of the profits earned from each unit made:</p>\n<p>\\[&nbsp;\\simplify{{profitx}x + {profity}y} \\]</p>\n<h4>d)</h4>\n<p><span id=\"advicediagram\">The solution graph goes here</span></p>\n<p>It is clear that the minimum profit is given by producing enough {product_1s} and {product_2s} so that the existing demand is met and no more. You can see this by moving the objective function line down to the intersection of the two minimum constraint lines - there are no other points inside the feasible region under the objective line.</p>\n<p>Next, drag the objective function line upwards until&nbsp;there are no points above the objective line in the feasible region. At this point, the objective line touches the intersection of the lines representing the constraints&nbsp;for {line_names[best_lines[0]]} and {line_names[best_lines[1]]}.</p>\n<p>You can find the coordinates of this point by solving the equations for these two lines:</p>\n<p><span data-jme-visible=\"best_lines[0]=0\">\\[ \\simplify{{resource1x}x + {resource1y}y = {totalresource1}} \\]</span><span data-jme-visible=\"best_lines[0]=1\">\\[ \\simplify{{resource2x}x + {resource2y}y = {totalresource2}} \\]</span></p>\n<p><span data-jme-visible=\"best_lines[1]=1\">\\[ \\simplify{{resource2x}x + {resource2y}y = {totalresource2}} \\]</span><span data-jme-visible=\"best_lines[1]=2\">\\[ x = \\var{xconstraint} \\]</span><span data-jme-visible=\"best_lines[1]=3\">\\[ y = \\var{yconstraint} \\]</span></p>\n<p>The solution to this system of equations is $x = \\var{best_x}$, $y = \\var{best_y}$, giving a profit of</p>\n<p>\\[ \\text{Profit} =&nbsp;\\simplify[]{{profitx}*{best_x} + {profity}*{best_y}} = \\text{&pound;}\\var{best_profit} \\]</p>"}], "name": "", "pickQuestions": 0}], "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Solve a linear programming problem, and perform sensitivity analysis.</p>"}, "type": "exam", "navigation": {"onleave": {"action": "none", "message": ""}, "reverse": true, "browse": true, "showresultspage": "oncompletion", "preventleave": true, "allowregen": true, "showfrontpage": true}, "timing": {"timedwarning": {"action": "none", "message": ""}, "timeout": {"action": "none", "message": ""}, "allowPause": true}, "pickQuestions": 0, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "extensions": ["jsxgraph", "optimisation"], "custom_part_types": [], "resources": []}