Absolutely! Let's **formulate a general mathematical model** to compare the **Direct Shipping** and **Milk Run** strategies as applied in the numerical example above. We’ll define variables, write down the cost components, and show how to express the total cost for each strategy. --- ## 1. **Define Parameters and Variables** Let: - \( S \): Number of stores - \( P \): Number of supply sources (suppliers) - \( D \): Annual demand per store (units/year) - \( Q \): Batch size per shipment to each store (units) - \( C_t \): Trucking cost per load (per truck, per trip) - \( C_d \): Delivery cost per store (per store, per trip) - \( H \): Holding cost per unit per year - \( K \): Truck capacity (units) - \( n \): Number of stores visited per truck in a milk run Derived: - \( N_{ship} \): Number of shipments per year from each supplier to each store \( = D/Q \) - \( N_{inv} \): Average inventory per store per product \( = Q/2 \) --- ## 2. **Direct Shipment Model** Each truck delivers directly from a supplier to a store. Each shipment fills up to truck capacity. ### a. **Number of Shipments per Year** \[ N_{ship,\,direct} = \frac{D}{Q} \] Where \( Q = K \) (full truckload to each store). ### b. **Annual Trucking Cost** Each shipment incurs one truck cost and one delivery cost per store: \[ \text{Annual Trucking Cost}_{direct} = N_{ship,\,direct} \times (C_t + C_d) \times P \times S \] ### c. **Annual Inventory Cost** \[ \text{Inventory per store per product} = \frac{Q}{2} \] \[ \text{Annual Inventory Cost}_{direct} = \frac{Q}{2} \times H \times P \times S \] ### d. **Total Annual Cost** \[ \text{Total Cost}_{direct} = \text{Annual Trucking Cost}_{direct} + \text{Annual Inventory Cost}_{direct} \] --- ## 3. **Milk Run Model** Each truck visits multiple stores on one trip. Each store receives a smaller shipment. ### a. **Number of Shipments per Year** Batch size per store: \[ Q_{milk} = \frac{K}{n} \] Number of shipments per year from each supplier to each store: \[ N_{ship,\,milk} = \frac{D}{Q_{milk}} = \frac{D \cdot n}{K} \] ### b. **Transportation Cost per Store per Shipment** Each truck serves \( n \) stores, so per store: \[ \text{Transp. Cost per store per shipment} = \frac{C_t}{n} + C_d \] ### c. **Annual Trucking Cost** \[ \text{Annual Trucking Cost}_{milk} = N_{ship,\,milk} \times \left( \frac{C_t}{n} + C_d \right) \times P \times S \] ### d. **Annual Inventory Cost** \[ \text{Average inventory per store per product} = \frac{Q_{milk}}{2} \] \[ \text{Annual Inventory Cost}_{milk} = \frac{Q_{milk}}{2} \times H \times P \times S \] ### e. **Total Annual Cost** \[ \text{Total Cost}_{milk} = \text{Annual Trucking Cost}_{milk} + \text{Annual Inventory Cost}_{milk} \] --- ## 4. **Summary Table** | Strategy | Shipments/yr | Batch size per store | Transp. Cost/shipment | Inventory/store | Annual Trucking Cost | Annual Inventory Cost | Total Cost | |---------------|--------------|---------------------|----------------------|-----------------|---------------------|----------------------|------------| | Direct | \( D/K \) | \( K \) | \( C_t + C_d \) | \( K/2 \) | \( (D/K) \cdot (C_t+C_d) \cdot P \cdot S \) | \( (K/2) \cdot H \cdot P \cdot S \) | sum | | Milk run | \( D/(K/n) \) | \( K/n \) | \( C_t/n + C_d \) | \( (K/n)/2 \) | \( (Dn/K) \cdot (C_t/n + C_d) \cdot P \cdot S \) | \( (K/2n) \cdot H \cdot P \cdot S \) | sum | --- ## 5. **General Mathematical Model** **Direct:** \[ \boxed{ \text{Total Cost}_{direct} = \frac{D}{K} \cdot (C_t + C_d) \cdot P \cdot S + \frac{K}{2} \cdot H \cdot P \cdot S } \] **Milk run:** \[ \boxed{ \text{Total Cost}_{milk} = \frac{D \cdot n}{K} \cdot \left( \frac{C_t}{n} + C_d \right) \cdot P \cdot S + \frac{K}{2n} \cdot H \cdot P \cdot S } \] Where: - \( K \): truck capacity - \( n \): number of stores per milk run (divides \( K \) evenly) --- ## 6. **How to Use the Model** - Plug in the values for \( D, K, n, C_t, C_d, H, P, S \) for your scenario. - Calculate both total costs. - The strategy with the **lower total cost** is preferred. --- ### **Step-by-Step Explanation Summary** 1. **Define variables and parameters.** 2. **For each strategy, write equations for:** - Number of shipments per year - Trucking cost per shipment and per year - Inventory holding cost per year - Total cost per year 3. **Summarize in equations** for comparison. 4. **Plug in numbers** for specific cases as in your example. --- **This model helps you analyze and choose between Direct and Milk Run based on cost components and operational variables.**