Ingredients R Us, Inc. Order Distribution Plan: Shipping Cos ✓ Solved

Ingredients R Us, Inc. Order Distribution Plan: Shipping cos

Ingredients R Us, Inc. Order Distribution Plan: Shipping costs per pallet from three distribution centers are given for six food retailers with pallet demand as follows — Food Buzz: 60; Healthy Stuff: 78; Smoothie City: 125; Burger Barn: 187; Café Organic: 313; Dessert Center: 567. Total initial inventory: 1615 pallets. The ingredients are ordered by pallet and transported from one of three distribution centers. Depending on order type, the nearest distribution center cannot always be utilized to fill orders. Meet all demand while minimizing total shipping cost. Complete the model by formulating the optimization (decision variables, objective function, constraints), populate any constraint tables needed, and solve the model to determine how many pallets to ship from each distribution center to each retailer and the minimum total cost.

Paper For Above Instructions

Executive summary

This paper formulates and solves a transportation optimization for Ingredients R Us, Inc. using the given per-pallet shipping costs and retailer pallet demands. The model is a classical transportation (linear programming) problem: choose how many pallets to ship from each distribution center (DC) to each retailer in order to meet all retailer demands while minimizing total shipping cost. Because distribution-center capacity allocations were not provided in the assignment, this solution documents explicit assumptions, presents a full mathematical model, implements a cost-minimizing allocation under those assumptions, and reports the optimal shipments and total cost. The approach follows standard operations research practice for transportation problems (Dantzig, 1963; Hillier & Lieberman, 2015).

Data summary

Retailer demands (pallets): Food Buzz 60; Healthy Stuff 78; Smoothie City 125; Burger Barn 187; Café Organic 313; Dessert Center 567. Total demand = 1330 pallets. Total initial inventory (aggregate supply) = 1615 pallets (hence surplus inventory 285 pallets if supply is unrestricted across DCs).

Per-pallet shipping costs by distribution center:

  • Food Buzz: DC1 $20.25, DC2 $78.15, DC3 $27.15
  • Healthy Stuff: DC1 $50.85, DC2 $50.00, DC3 $47.90
  • Smoothie City: DC1 $84.00, DC2 $38.40, DC3 $65.55
  • Burger Barn: DC1 $86.65, DC2 $25.40, DC3 $13.90
  • Café Organic: DC1 $108.95, DC2 $21.50, DC3 $33.95
  • Dessert Center: DC1 $116.45, DC2 $7.95, DC3 $26.20

Model formulation

Decision variables:

x_ij = number of pallets shipped from distribution center i to retailer j, where i ∈ {1,2,3} and j ∈ {FB, HS, SC, BB, CO, DCtr} corresponding to Food Buzz, Healthy Stuff, Smoothie City, Burger Barn, Café Organic, Dessert Center.

Objective function (minimize total cost):

Minimize Z = Σ_i Σ_j c_ij x_ij

where c_ij are the per-pallet shipping costs provided above.

Constraints:

  • Demand satisfaction: For each retailer j, Σ_i x_ij = demand_j (ensures all retailer demand is met) (Hillier & Lieberman, 2015).
  • Supply limits: For each DC i, Σ_j x_ij ≤ supply_i. Because distribution-center supplies were not specified, the aggregate supply is 1615 pallets; to remain conservative and consistent with the data, we assume each DC has sufficient capacity so only the aggregate supply constraint matters: Σ_i Σ_j x_ij ≤ 1615. If individual DC capacities become available, the model is extended by replacing the aggregate constraint with per-DC capacities (Winston, 2004).
  • Nonnegativity and integrality: x_ij ≥ 0 and integer (pallets are discrete).

Notes: If there are business rules that forbid certain DC-to-retailer assignments (the problem statement notes that “the nearest distribution center cannot always be utilized”), they are represented by fixed constraints x_ij = 0 for those prohibited pairs. Because no such prohibitions were explicitly listed, none are applied here; the method is unchanged if prohibitions are later provided (Taha, 2017).

Solution method

The transportation LP is solved by assigning each retailer to the single least-cost source when supply constraints do not bind per-DC or when each DC has adequate inventory (this is a valid optimal policy for an uncapacitated or ample-capacity transportation instance with integer demands) (Dantzig, 1963; Bazaraa, Jarvis & Sherali, 2010). In practice, one would implement the LP in a solver (Excel Solver, Gurobi, CPLEX, or open-source solvers) with decision variables x_ij, the demand equalities, supply inequalities, and integer integrality; run optimization to obtain the minimum-cost allocation (Bertsimas & Tsitsiklis, 1997; Gurobi Optimization, 2020). For instructional clarity and reproducibility here, the direct least-cost assignment is computed because aggregate supply exceeds total demand and no DC capacity bounds are active.

Computed optimal allocation (under stated assumptions)

Choose the minimum cost DC for each retailer:

  • Food Buzz — cheapest: DC1 ($20.25) → ship 60 pallets from DC1
  • Healthy Stuff — cheapest: DC3 ($47.90) → ship 78 pallets from DC3
  • Smoothie City — cheapest: DC2 ($38.40) → ship 125 pallets from DC2
  • Burger Barn — cheapest: DC3 ($13.90) → ship 187 pallets from DC3
  • Café Organic — cheapest: DC2 ($21.50) → ship 313 pallets from DC2
  • Dessert Center — cheapest: DC2 ($7.95) → ship 567 pallets from DC2

Aggregated shipments by distribution center

  • DC1 total shipped = 60 pallets (to Food Buzz)
  • DC2 total shipped = 125 + 313 + 567 = 1005 pallets (to Smoothie City, Café Organic, Dessert Center)
  • DC3 total shipped = 78 + 187 = 265 pallets (to Healthy Stuff, Burger Barn)
  • Total shipped = 60 + 1005 + 265 = 1330 pallets (meets total demand)
  • Remaining inventory = 1615 − 1330 = 285 pallets (surplus)

Minimum total shipping cost

Compute cost = Σ (pallets × per-pallet cost):

  • Food Buzz: 60 × $20.25 = $1,215.00
  • Healthy Stuff: 78 × $47.90 = $3,736.20
  • Smoothie City: 125 × $38.40 = $4,800.00
  • Burger Barn: 187 × $13.90 = $2,599.30
  • Café Organic: 313 × $21.50 = $6,729.50
  • Dessert Center: 567 × $7.95 = $4,507.65

Total minimum shipping cost = $23,587.65.

Discussion and sensitivity

The allocation above is optimal under the assumption that individual DC capacities are not constrained beyond the aggregate supply and that no DC-to-retailer assignments are forbidden. If per-DC supplies are limited, the optimizer (e.g., Excel Solver or any LP solver) will reassign shipments to respect capacity constraints and minimize cost subject to those bounds; this may require splitting retailer demand across multiple DCs (Winston, 2004). If certain DC-to-retailer pairs are prohibited, those variables are set to zero and the solver finds the best feasible allocation (Hillier & Lieberman, 2015). Sensitivity analysis on per-pallet costs or on supply levels can be performed via standard dual-price and reduced-cost outputs available from LP solvers (Bazaraa et al., 2010).

Implementation notes (how to set up Solver)

1) Create a matrix of decision cells x_ij in a spreadsheet; enter cost matrix c_ij and demand vector. 2) Add objective cell = SUMPRODUCT(c_matrix, x_matrix). 3) Add constraints: row sums = demands; column (DC) sums ≤ supply (or leave aggregate supply constraint). 4) Set decision variables as integers and nonnegative. 5) Use Solver (Simplex LP or Integer programming engine) and minimize objective. Solver will output x_ij and objective value (FrontlineSolvers / Microsoft documentation; Gurobi/CPLEX for larger instances) (Microsoft, 2020; Gurobi, 2020).

Conclusions

A clear LP formulation, the least-cost assignment under the stated assumptions, and a verified total cost are provided. The resulting shipment plan ships each retailer from the least-cost DC, satisfying total demand and leaving 285 pallets of inventory. If additional operational constraints (per-DC capacities or forbidden links) are given, the same modeling framework applies and the solver can be rerun to produce the constrained optimal plan. This method follows best practices in transportation optimization and can be operationalized in spreadsheet solvers or commercial optimization software (Dantzig, 1963; Hillier & Lieberman, 2015).

References

  • Dantzig, G. B. (1963). Linear Programming and Extensions. Princeton University Press.
  • Hillier, F. S., & Lieberman, G. J. (2015). Introduction to Operations Research (10th ed.). McGraw-Hill Education.
  • Taha, H. A. (2017). Operations Research: An Introduction (10th ed.). Pearson.
  • Winston, W. L. (2004). Operations Research: Applications and Algorithms (4th ed.). Brooks/Cole.
  • Bazaraa, M. S., Jarvis, J. J., & Sherali, H. D. (2010). Linear Programming and Network Flows (4th ed.). Wiley.
  • Bertsimas, D., & Tsitsiklis, J. N. (1997). Introduction to Linear Optimization. Athena Scientific.
  • Chopra, S., & Meindl, P. (2016). Supply Chain Management: Strategy, Planning, and Operation (6th ed.). Pearson.
  • Gurobi Optimization, LLC. (2020). Gurobi Optimizer Reference Manual. (Use for advanced solver implementation.)
  • Microsoft. (2020). Excel Solver Documentation. Microsoft Support and Documentation.
  • Vanderbei, R. J. (2015). Linear Programming: Foundations and Extensions (4th ed.). Springer.

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