# **Study Guide: "What If?" Analysis (Rardin Ch. 4\)** **Context:** In the real world, requirements change. A good modeler doesn't just solve the problem in the book; they adapt the structure when the client adds a new constraint or changes the objective. **How to use this guide:** 1. Read the **Scenario**. 2. Cover the **Solution** and try to write the mathematical modification yourself. 3. Check your answer. ## **1\. Forest Service Allocation (Rardin 4.1)** **Base Model:** Allocate acres (![][image1]) in 7 Analysis Areas to 3 prescriptions (Timber, Grazing, Wilderness) to maximize NPV, subject to Timber Yield, Grazing AUM, and Wilderness Index targets. ### **Scenario A: The "Monoculture" Limit** **The Situation:** Environmental groups are concerned that we might turn the entire forest into a timber farm or a giant pasture. Management introduces a policy: **"No single prescription can occupy more than 50% of the total forest acreage."** **The Modification:** * **Logic:** We need to sum the acres allocated to Prescription ![][image2] across *all* areas (![][image3]) and ensure that sum is less than or equal to ![][image4]. * **New Constraints:** We need one constraint for each prescription ![][image2] (or specifically for Timber/Grazing). ![][image5] *(Where ![][image6] is the fixed size of area ![][image7].)* ### **Scenario B: The "Visual Buffer" (Local Minimum)** **The Situation:** To preserve aesthetics, every specific Analysis Area must maintain at least 10% of its land as **Wilderness**. We cannot just dump all the wilderness in Area 7; Area 1 needs some too. **The Modification:** * **Logic:** This is a **local** constraint. It applies to each ![][image7] individually. * **New Constraints:** ![][image8] *(Note: This tightens the feasible region significantly. It forces ![][image9] to be non-zero everywhere).* ## **2\. Swedish Steel Blending (Rardin 4.2)** **Base Model:** Mix scrap (![][image10]) to make 1000 kg of steel. Meet Carbon/Nickel % limits. Min Cost. ### **Scenario A: Oxidation Loss (Yield Factor)** **The Situation:** The furnace is not 100% efficient. When we melt **Scrap 1**, 10% of its mass is lost to oxidation (burn-off). **Scrap 2** loses 5%. The other scraps lose 0%. We still need **1000 kg of finished steel**. **The Modification:** * **Logic:** The input weight does not equal the output weight. * Input from Scrap 1: ![][image11]. Contribution to final steel: ![][image12]. * Input from Scrap 2: ![][image13]. Contribution to final steel: ![][image14]. * **Updated Weight Constraint:** ![][image15] * **Impact on Ratio Constraints:** * Be careful\! Does the Carbon percentage apply to the *input* mass or the *retained* mass? Usually, chemical elements are retained while "slag" is lost, or they burn off proportionally. * *Assumption:* If Carbon is retained but total mass drops, the % Carbon *increases*. * *New Carbon Constraint:* ![][image16] * (You must calculate Total Carbon Mass based on the physics of the burn-off). ### **Scenario B: The "Recipe" Requirement** **The Situation:** To ensure structural integrity, the metallurgists require that we use **at least twice as much Scrap 2 as Scrap 1**. **The Modification:** * **Logic:** A direct comparison between two variables. * ![][image17] * **New Constraint:** ![][image18] ## **3\. Tubular Products (Rardin 4.3)** **Base Model:** Assign production of Products (![][image19]) to Mills (![][image20]) to minimize cost. Capacity is in Tons. ### **Scenario A: Capacity in Hours (The Unit Trap)** **The Situation:** The previous model assumed Mill Capacity was in **Tons**. Realistically, Mill 1 runs for **200 hours** a month. * Making **Standard** tube takes 0.5 hours/ton. * Making **Pressure** tube takes 0.8 hours/ton. * Making **Special** tube takes 1.2 hours/ton. **The Modification:** * **Logic:** We cannot sum tons (![][image21]) because tons of "Special" eat up more capacity than tons of "Standard". We must convert everything to Hours. * **Parameter:** Let ![][image22] be hours per ton for product ![][image19]. * **New Capacity Constraint (for Mill 1):** ![][image23] * *(Repeat for other mills with their specific parameters).* ### **Scenario B: Risk Diversification** **The Situation:** Management is worried about a strike at Mill 2\. They decree: **"No more than 40% of the total production of 'Pressure' tubing can come from Mill 2."** **The Modification:** * **Logic:** Relate a specific variable (![][image24]) to the total demand for that product (![][image25]). * **New Constraint:** ![][image26] *(Since Demand is a fixed parameter, this is a simple upper bound).* ## **4\. Ohio National Bank (Rardin 4.5)** **Base Model:** Set Covering. Shifts (![][image10]) cover hourly demand (![][image27]). Min Cost. ### **Scenario A: Service Quality Ratio** **The Situation:** Part-timers are cheaper ($40) than Full-timers ($100), so the Solver wants to use only Part-timers. The bank adds a rule: **"At any specific hour, at least 50% of the tellers on duty must be Full-Time employees."** **The Modification:** * **Logic:** This applies to *every hour* ![][image28]. We need to look at the "Staff on Duty" expression. * **Let:** * **![][image29]** (Full timers working at hour ![][image28]). * ![][image30] (Total staff working at hour ![][image28]). * **Constraint:** ![][image31] *Rearranged to Linear Form:* ![][image32] ### **Scenario B: The "Overtime" Option** **The Situation:** Full-time shifts are 9am-5pm. The bank is short-staffed from 5pm-6pm. Management allows Full-Timers starting at 9am to work **one extra hour (OT)** from 5pm-6pm for a cost of $15 (vs regular $12.50/hr). **The Modification:** * **Logic:** We cannot just "extend" the existing variable ![][image33]. That assumes *everyone* works OT. We need a choice. * **New Variables:** Split the original ![][image33] into two: 1. ![][image34]: Works 9-5. Cost $100. 2. ![][image35]: Works 9-6. Cost $115 ($100 \+ $15). * **Matrix Update:** * **![][image34]** has 0 in the 5-6pm slot. * ![][image35] has **1** in the 5-6pm slot. * *Note:* This increases the problem size (more columns) but gives the Solver the granular choice it needs. [image1]: [image2]: [image3]: [image4]: [image5]: [image6]: [image7]: [image8]: [image9]: [image10]: [image11]: [image12]: [image13]: [image14]: [image15]: [image16]: [image17]: [image18]: [image19]: [image20]: [image21]: [image22]: [image23]: [image24]: [image25]: [image26]: [image27]: [image28]: [image29]: [image30]: [image31]: [image32]: [image33]: [image34]: [image35]: