1 Week

• quantitative representation or idealization of a real problem

• key to virtually every management science application.

• Can b ephrased in terms of

  • mathematical expressions

  • series of interrelated cells and spreachsheets

represent the essence of a problem in a concise form, providing several advantages.

• enables managers to undesrstand the problem better

• define scope, possible solutions and data requirements

• allows analysts to apply variety of mathematical solution procedures

• if done correctly, often helps to sell the solution to the end-users

• when equation is used to regulate input -> mathematical/analytical models

• but most situations are too complex, they are intractable -> we rely on simulation model which approximates behavior of the actual system

• management science is a collection of mathematical tools

  • linear programming

  • inventory model

  • queuing models

• course about modelling rather than models

  • modelling is a process where you abstract the essence of a real problem into a model

• problem definition -> specify objectives

• data collection

• model development -> Mathematical or Simulation

• model verification -> reality fit for current situation

• Optimization & Decision making, the ones align with organizations objectives

• model com to management

• model implementation -> must be monitored and updated to enable organization to meet objectives

• 3 Model development

• 4 Optimization & decision making

• decriptive: simply describe a situation

• optimization: suggest a desirable course of action

Inputs:

• The arrival rate of potential customers to the store

• The rate at which customers can be served by a single cashier

• As the arrival rate increases and/or service rate decreases, the waiting line will tend to increase, and customers will wait longer or not enter the line.

Outputs:

• Length of waiting line

• Time in line per customer

• Fraction of customers who do not enter.

• one of the oldest optimization problems

• simple problem: how could a soldier be fed for as little

  money as possible?

• given nine nutrients (calories, protein, Vitamin C, and so on) and 77 candidate foods,

  find the foods that could sustain soldiers at minimum cost

  -> 39.93$ per year (not even the cheapest option)

• used in WW II for logitsic problems -> nine calculator-wielding clerks, and120 person-days to arrive at the optimal solution, today solved in 300 miliseconds

• inputs -> fixed values (at least for model purpose)

• decision variables -> variables to choose/ change/ optimize

• Output/ objective function -> revenue/ profit

• constraints: usually physical, logical, or economic

  restrictions

x1, x2 > 0 {non-negativity}

• model development stage you enter all of the inputs, trial values for the changing cells, and formulas relating these in a spreadsheet

• invoking Solver. At this point, you formally designate the objective cell, the changing cells, the constraints, and selected options, and you tell Solver to find the optimal solution.

• The third stage is sensitivity analysis. Here you see how the optimal solution changes (if at all) as selected inputs are varied. This often provides important insights about the behavior of the model.

two types of analysis:

• on the coefficients of the objectices

• on the right side of the constraints

🧠 Interpretation

• Final Value – the optimal number of units produced.

• Objective Coefficient – current profit contribution per unit ($80 for Basic, $129 for XP).

• Reduced Cost – how much the profit per unit would have to change before this variable enters/leaves the solution.

• Allowable Increase/Decrease – the range in which the current solution remains optimal.

🔍 For Basic:

• Reduced Cost = 0 → Basic is part of the optimal solution.

• Profit per Basic ($80) can increase by $27.5 (to $107.5) or decrease by $80 (to $0) without changing the optimal combination (560 Basics, 1200 XPs).

🔍 For XP:

• Reduced Cost = 33 → XP is at an upper limit (it’s producing at max sales).

  The XP’s profit per unit would have to decrease by $33 before the solution changes.

• The “Allowable Increase = 1E+30” means profit could rise infinitely and XP would still be produced at maximum (as mimited due to constraint)

1️⃣ Binding or not?

• Final value = RHS → Binding constraint.

  (All 10,000 assembly hours are fully used. There’s no slack.)

2️⃣ Shadow Price = 16 → what it means:

• For every extra hour of assembly labor, profit will increase by $16,

  as long as the change stays within the allowable range.

💡 Think of this as:

“Assembly hours are valuable — each one contributes $16 to the bottom line.”

3️⃣ Allowable range

• You can increase assembly hours by up to 200 (to 10,200)

  or decrease them by up to 2,800 (to 7,200)

  before this $16 per hour relationship changes.

Beyond that range, you’d move to a new optimal corner — a new shadow price would apply.

__________________________________

1️⃣ Binding or not?

• Final value (2,960) < RHS (3,000) → Not binding.

• You have 40 hours of unused testing labor (slack = 40).

2️⃣ Shadow Price = 0 → what it means

• An extra hour of testing labor adds no value — total profit won’t change.

  (You’re not limited by testing capacity; you already have enough.)

3️⃣ Allowable range:

• You can add infinitely more testing hours — it won’t affect profit (shadow price = 0 still applies).

• But if you reduce testing hours by more than 40, you’ll hit the limit (the constraint becomes binding).

  After that point, Solver would reoptimize — profit would start to drop.